Overlap Functions for Measures in Conformal Iterated Function Systems

Abstract

We employ thermodynamic formalism for the study of conformal iterated function systems (IFS) $\mathcal S = \{\phi _i\}_{i \in I}$ with arbitrary overlaps, and of measures $\mu$ on limit sets $\Lambda$, which are projections of equilibrium measures $\hat{\mu }$ with respect to a certain lift map $\Phi$ on $\Sigma _I^+ \times \Lambda$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat{\mu }$ with respect to $\mathcal {S}$; and, in particular a notion of (topological) overlap number $o(\mathcal {S})$. These notions take in consideration the n-chains between points in the limit set. We prove that $o(\mathcal {S}, \hat{\mu })$ is related to a conditional entropy of $\hat{\mu }$ with respect to the lift $\Phi$. Various types of projections to $\Lambda$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(\mu )$ of $\mu$ on $\Lambda$, by using pressure functions and $o(\mathcal {S}, \hat{\mu })$. In particular, this applies to projections of Bernoulli measures on $\Sigma _I^+$. Next, we apply the results to Bernoulli convolutions $\nu _\lambda$ for $\lambda \in (\frac{1}{2}, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal {S}_\lambda$. We prove that for all $\lambda \in (\frac{1}{2}, 1)$, there exists a relation between $HD(\nu _\lambda )$ and the overlap number $o(\mathcal {S}_\lambda )$. We also estimate $o(\mathcal {S}_\lambda )$ for certain values of $\lambda$.

1 Introduction and Outline

Iterated function systems (IFS) have been studied by many authors, and a lot about their theory is known. In many instances, systems which satisfy the Open Set Condition were studied. When arbitrary overlaps of the images of the contractions are allowed, the theory is different and the results from the case of Open Set Condition do not work anymore.

Let us consider a finite set I and an iterated function system $\mathcal {S} = \{\phi _i, i \in I\}$ consisting of injective conformal contractions $\phi _i$ defined on the closure of an open set $V \subset \mathbb {R}^q, q \ge 1$. Denote by $\Sigma _I^+$ the one-sided space $\{\omega = (\omega _1, \omega _2, \ldots ), \omega _j \in I, j \ge 1\}$, with its shift endomorphism $\sigma : \Sigma _I^+ \rightarrow \Sigma _I^+, \sigma (\omega ) = (\omega _2, \omega _3, \ldots )$. For an arbitrary sequence $\omega$ and for an integer $n \ge 1$, let the n-truncation $\omega |_n$ be the finite sequence $(\omega _1, \ldots , \omega _n)$. Also by $[i_1\ldots i_n]$ we denote the n-cylinder $\{\omega \in \Sigma _I^+, \ \omega _1=i_1, \ldots , \omega _n = i_n\}, \ n \ge 1, i_1, \ldots , i_n \in I$.

Let denote now by $\Lambda$ the fractal limit set of the iterated function system $\mathcal {S}$, where:

$$\begin{aligned} \Lambda := \mathop {\cup }\limits _{\omega \in \Sigma _I^+} \mathop {\cap }\limits _{n \ge 1} \phi _{\omega |_n}(V) \end{aligned}$$

Since all the maps $\phi _i$ are contractions, we can define the canonical coding map $\pi : \Sigma _I^+ \rightarrow \Lambda , \ \pi (\omega ) = \mathop {\lim }\limits _{n \rightarrow \infty } \phi _{\omega _1}\circ \phi _{\omega _2}\circ \cdots \circ \phi _{\omega _n}(V)$, for all $\omega = (\omega _1, \omega _2, \ldots ) \in \Sigma _I^+$. The singleton $\pi (\omega )$ will also be denoted by $\phi _{\omega _1}\circ \phi _{\omega _2}\circ \cdots$, as this infinite composition is in fact a point. We will denote the composition $\phi _{i_1}\circ \cdots \circ \phi _{i_m}$ also by $\phi _{i_1\ldots i_m}$, for $m \ge 1, i_j \in I, 1\le j \le m$. The map $\pi$ is called the canonical projection onto the limit set $\Lambda$ of the system $\mathcal {S}$. Various properties of conformal IFS’s with overlaps were studied by several authors, for e.g. in [4, 12, 15, 16, 21], etc. Let us fix now some more terminology and notation.

Definition 1

By overlaps we mean intersections of type $\phi _i( \Lambda ) \cap \phi _j(\Lambda ) \ne \emptyset , \ i \ne j$. If for a point $x \in \Lambda$ and an integer $m \ge 1$, there exists a point $\zeta \in \Lambda$ and a finite sequence $i_1, \ldots i_m \in I$ such that $\phi _{i_1}\circ \cdots \circ \phi _{i_m}(\zeta ) = x$, then $\zeta$ is called an m-root of x, and $(i_1, \ldots , i_m)$ is called an m-chain from $\zeta$ to x.

In general, the number of roots/overlaps depends on the point $x \in \Lambda$, so it is not constant. Notice also that the m-chain from a certain root $\zeta$ to x is not uniquely defined, i.e. there may exist two different m-chains $(i_1, \ldots , i_m)$ and $(j_1, \ldots , j_m)$ so that $\phi _{i_1\ldots i_m}(\zeta ) = \phi _{j_1\ldots j_m}(\zeta ) = x$. Considering the above, how can we define a good notion of average number of overlaps of the IFS $\mathcal {S}$, and how is such a notion dependent on a probability measure $\mu$ on $\Lambda$; also, how does such a number of overlaps affect the Hausdorff dimension of $\mu$? It is clear that we have to look at n-roots of points, since the limit set $\Lambda$ is invariant under the system $\mathcal {S}$, i.e. $\Lambda = \mathop {\cup }\limits _{i \in I} \phi _i(\Lambda )$, thus for k-iterations of $\mathcal {S}$ we have $\Lambda = \mathop {\cup }\limits _{i_1, \ldots , i_k \in I} \phi _{i_1\ldots i_k}(\Lambda )$, for any $k \ge 2$. In [12] we studied the effect, of the bounds for the number of overlaps, on the Hausdorff dimension of the limit set $\Lambda$. This hints to the fact that the overlap number should be given by an average rate of growth of the number of n-chains between points in $\Lambda$. Another question is, what probabilities $\mu$ on $\Lambda$ should be considered, and what roots in $\Lambda$ should we use. Some n-roots and n-chains which are non-generic with respect to $\mu$ and to a lift map $\Phi : \Sigma _I^+ \times \Lambda \rightarrow \Sigma _I^+ \times \Lambda$ will thus be ignored when defining the overlap number relative to $\mu$.

Besides the canonical coding projection $\pi : \Sigma _I^+ \rightarrow \Lambda$, one can consider also the projection $\pi _2: \Sigma _I^+ \times \Lambda \rightarrow \Lambda , \ \pi _2(\omega , x) = x$, and the projection $\tilde{\pi }: \Sigma _I^+ \times \Sigma _I^+ \rightarrow \Sigma _I^+ \times \Lambda , \ \tilde{\pi }(\omega , \eta ) = (\omega , \pi \eta )$; so we obtain projections of $\sigma$-invariant measures on $\Sigma _I^+$, $\Phi$-invariant measures on $\Sigma _I^+ \times \Lambda$ or $\tilde{\Phi }$-invariant measures on $\Sigma _I^+ \times \Sigma _I^+$ (where $\tilde{\Phi }$ is a lift of $\Phi$ to $\Sigma _I^+ \times \Sigma _I^+$). In Theorem 1 we will prove that, for Bernoulli measures, the corresponding projection measures on $\Lambda$ are in fact the same.

We introduce a notion of overlap number $o(\mathcal {S}, \hat{\mu }_\psi )$ associated to a $\Phi$-invariant Gibbs state $\hat{\mu }_\psi$ on $\Sigma _I^+ \times \Lambda$ (and to its $\pi _2$-projection $\mu _\psi$ on $\Lambda$), and we use thermodynamic formalism to relate it to the dimension of $\mu _\psi$. In Theorem 2 and Corollary 1 we show that the overlap number $o(\mathcal {S}, \hat{\mu }_\psi )$ is related to the folding entropy of $\hat{\mu }_\psi$ with respect to the lift map $\Phi$. In particular, this applies to Bernoulli measures on $\Sigma _I^+$ and their lifts on $\Sigma _I^+ \times \Lambda$. When $\mu =\mu _0$ is the projection of the measure of maximal entropy $\hat{\mu }_0$ from $\Sigma _I^+\times \Lambda$, one obtains a topological overlap number $o(\mathcal {S})$ of $\mathcal {S}$, which quantifies the average level of overlapping in $\mathcal {S}$, and indicates how far is $\mathcal {S}$ from satisfying the Open Set Condition. By using Theorem 1, we compute in Corollary 2 the overlap number $o(\mathcal {S})$ as a limit of integrals over $\Sigma _I^+$ w.r.t the uniform Bernoulli measure $\nu _{(\frac{1}{|I|}, \ldots , \frac{1}{|I|})}$. And in general for Bernoulli measures $\nu _\mathbf{p}$, Corollary 2 gives a simpler formula for $o(\mathcal {S}, \hat{\mu }_\mathbf{p})$.

Next, in Theorem 3 we use the overlap number of $\hat{\mu }_\psi$ to obtain estimates for the Hausdorff dimension of a set of full $\mu _\psi$-measure in $\Lambda$, which set is constructed explicitly. This gives upper bounds for $HD(\mu _\psi )$, by using zeros of pressure functions associated to $o(\mathcal {S}, \hat{\mu }_\psi )$, which are computable in certain cases of interest.

In Sect. 3 we apply the results to the case of Bernoulli convolutions $\nu _\lambda$ for $\lambda \in (\frac{1}{2}, 1)$, where $\nu _\lambda$ gives the distribution of the random series $\mathop {\sum }\nolimits _{n\ge 0} \pm \lambda ^n$ with the $+, -$ signs taken independently and with equal probabilities. In this case, one has an iterated function system with overlaps $\mathcal {S}_\lambda$, whose limit set is an interval $I_\lambda$, and $\nu _\lambda$ appears as the projection of the measure of maximal entropy $\nu _{(\frac{1}{2}, \frac{1}{2})}$ from $\Sigma _2^+$ to $I_\lambda$. Bernoulli convolutions have attracted a lot of attention (see [15]), starting with Erdös [3] who showed that $\nu _\lambda$ is singular for $\lambda ^{-1}$ Pisot; then, continuing with the result of Solomyak [21] about the absolute continuity of $\nu _\lambda$ for Lebesgue-a.e $\lambda \in (\frac{1}{2}, 1)$, and the result of Przytycki and Urbański [17] that $HD(\nu _\lambda ) < 1$ for $\lambda ^{-1}$ Pisot, and other more recent results. In Theorem 4 we find a relation between $HD(\nu _\lambda )$ and the overlap number $o(\mathcal {S}_\lambda )$, for all $\lambda \in (\frac{1}{2}, 1)$. We show how to approximate $o(\mathcal {S}_\lambda )$ with integrals on $\Sigma _2^+$ with respect to the uniform Bernoulli measure $\nu _{(\frac{1}{2}, \frac{1}{2})}$. By using known results on $HD(\nu _\lambda )$, one obtains then upper estimates for $o(\mathcal {S}_\lambda )$; in particular, one can estimate $o(\mathcal {S}_\lambda )$ more precisely for specific values of $\lambda$, like $\lambda = 2^{-\frac{1}{m}}, m \ge 2$ (i.e. $\frac{1}{\lambda }$ non-Pisot), or $\lambda = \frac{\sqrt{5} -1}{2}$ (i.e. $\frac{1}{\lambda }$ Pisot). In Corollary 3 we prove that $o(\mathcal {S}_\lambda )$ is strictly less than 2, for all $\lambda \in (\frac{1}{2}, 1)$. In the end, we obtain dimension estimates for biased Bernoulli convolutions $\nu _{\lambda , p}$, for $\lambda \in (\frac{1}{2}, 1)$ and $p \in (0, 1)$. The results about overlap numbers can be applied also to other conformal iterated function systems with overlaps.

2 Overlap Numbers of Measures and Dimension Estimates

First, let us define an overlap lift function which allows to associate the dynamics of a map to our IFS $\mathcal {S}$. With regard to this function, the contractions $\phi _i$ appear as restrictions to cylinders $[i], i \in I$.

Definition 2

In the above setting, for the finite IFS $\mathcal {S} = \{\phi _i\}_{i \in I}$, define the overlap lift map

$$\begin{aligned} \Phi : \Sigma _I^+ \times \Lambda \rightarrow \Sigma ^+_I \times \Lambda , \ \Phi (\omega , x) = (\sigma \omega , \phi _{\omega _1}(x)), \ (\omega , x) \in \Sigma _I^+ \times \Lambda \end{aligned}$$

Let us now consider a Hölder continuous function $\psi : \Sigma _I^+\times \Lambda \rightarrow \mathbb {R}$. Since the lift map $\Phi$ is distance-expanding in the first coordinate and contracting in the second coordinate, it follows that it is expansive and we can apply the theory of equilibrium states (for e.g. [7, 22]). As $\psi$ is Hölder, there exists a unique equilibrium measure for $\psi$ with respect to $\Phi$ on $\Sigma _I^+ \times \Lambda$, denoted by $\hat{\mu }_\psi$.

In particular, if we take a Hölder continuous function $g: \Lambda \rightarrow \mathbb R$ and the associated function $\psi _g: \Sigma _I^+\times \Lambda \rightarrow \mathbb R, \ \psi _g(\omega , x) = g(x)$, then we have the equilibrium measure $\hat{\mu }_{\psi _g}$ on $\Sigma _I^+\times \Lambda$ (relative to $\Phi$) and its projection $(\pi _2)_*(\hat{\mu }_{\psi _g})$ on $\Lambda$, where $\pi _2$ is the projection on the second coordinate. In general this measure is different from the projection $\pi _*(\bar{\mu }_{g\circ \pi })$, where $\pi : \Sigma _I^+ \rightarrow \Lambda , \pi (\omega ) = \phi _{\omega _1}\circ \cdots ,$ and where in general $\bar{\mu }_\chi$ denotes the equilibrium measure of a Hölder continuous $\chi$ on $\Sigma _I^+$ (relative to the shift $\sigma$).

For any $n \ge 1$ and any $(\omega , x) \in \Sigma _I^+ \times \Lambda$, we have $\Phi ^n(\omega , x) = (\sigma ^n\omega , \phi _{\omega _n}\circ \phi _{\omega _{n-1}} \circ \cdots \circ \phi _{\omega _1}(x))$. Notice that, if $\eta _1, \cdots , \eta _n$ are given and if $\phi _{\omega _n}\circ \cdots \circ \phi _{\omega _1}(x) = \phi _{\eta _n}\circ \cdots \circ \phi _{\eta _1}(y)$, then from the injectivity of the contractions $\phi _i, i \in I$, there exists exactly one point y with this property. By Definition 1, this means that, given the n-chain $(\eta _n, \ldots , \eta _1)$ as above, the corresponding n-root y is uniquely defined such that $(\eta _n, \ldots , \eta _1)$ is an n-chain from y to $\phi _{\omega _n\ldots \omega _1}(x)$.

Given now a measure $\hat{\mu }_\psi$ as above, an arbitrary point $(\omega , x) \in \Sigma _I^+ \times \Lambda$, and $\tau >0$, define the set of n-chains from points in $\Lambda$ to $\phi _{\omega _n\ldots \omega _1}(x)$, which are $\tau$-generic relative to $\hat{\mu }_\psi$:

$$\begin{aligned} \Delta _n\big ((\omega , x), \tau , \hat{\mu }_\psi \big )= & {} \{(\eta _1, \ldots , \eta _n) \in I^n, \ \exists y \in \Lambda , \ \phi _{\eta _n\ldots \eta _1}(y) \\ \nonumber= & {} \phi _{\omega _n\ldots \omega _1}(x) \ \text {and} \ |\frac{S_n\psi (\eta , y)}{n} - \int _{\Sigma _I^+\times \Lambda } \psi d\hat{\mu }_\psi | < \tau \}, \end{aligned}$$

(1)

where $\eta = (\eta _1, \ldots , \eta _n, \omega _{n+1}, \omega _{n+2}, \ldots ) \in \Sigma _I^+$, and where $S_n\psi (\eta , y) = \psi (\eta , y) + \psi (\Phi (\eta , y)) + \cdots +\psi (\Phi ^n(\eta , y))$. We denote the cardinality of the set $\Delta _n$ by $b_n$, so

$$\begin{aligned} b_n((\omega , x), \tau , \hat{\mu }_\psi ) := \text {Card} \ \Delta _n\big ((\omega , x), \tau , \hat{\mu }_\psi \big ), \ \forall (\omega , x) \in \Sigma _I^+ \times \Lambda \end{aligned}$$

Remark that, if $(i_1, \ldots , i_n) \in \Delta _n\big ((\omega , x), \tau , \hat{\mu }_\psi \big )$ with corresponding n-root y of $\phi _{\omega _n\ldots \omega _1}(x)$, then $\Delta _n\Big (\big ((i_1, \ldots , i_n, \omega _{n+1}, \omega _{n+2}, \ldots ), y\big ), \tau , \hat{\mu }_\psi \Big ) = \Delta _n\Big ((\omega , x), \tau , \hat{\mu }_\psi \Big )$.

Definition 3

Given a Hölder continuous potential $\psi$ on $\Sigma _I^+\times \Lambda$ and $\tau >0$, we call $b_n(\cdot , \tau , \hat{\mu }_\psi ): \Sigma _I^+\times \Lambda \rightarrow \mathbb N$ the n-overlap function associated to the measure $\hat{\mu }_\psi$ and $\tau$.

The function $b_n(\cdot , \tau , \hat{\mu }_\psi )$ is measurable and bounded, but in general discontinuous on $\Sigma _I^+\times \Lambda$. In the sequel, we will use the folding entropy of a $\Phi$-invariant measure $\hat{\mu }$ on $\Sigma _I^+ \times \Lambda$; the notion of folding entropy of a measure was introduced by Ruelle [19]. The folding entropy of a $\Phi$-invariant probability $\mu$ with respect to $\Phi : \Sigma _I^+ \times \Lambda \rightarrow \Sigma _I^+ \times \Lambda$, is defined as the conditional entropy $F_{\Phi }(\mu ) := H_{\mu }(\epsilon |\Phi ^{-1}\epsilon )$, where $\epsilon$ is the point partition of the Lebesgue space $\Sigma _I^+ \times \Lambda$. For entropy production see also [13, 19, 20]. And for measures invariant under endomorphisms, for e.g. [9–11, 14, 20]. In [14] Parry introduced a notion of Jacobian of an invariant measure for an endomorphism, and studied its properties; in particular, the Jacobian satisfies the Chain Rule. Given a map $f:X \rightarrow X$ on a Lebesgue space X and an f-invariant probability $\mu$, such that f is essentially countable-to-one, we denote the Jacobian of $\mu$ by $J_f(\mu )$. From [14, 19] it follows that, in general, the folding entropy of a measure $\mu$ is equal to the integral of the logarithm of the Jacobian of $\mu$. So in our case, the folding entropy of $\hat{\mu }_\psi$ with respect to $\Phi$ is given by:

$$\begin{aligned} F_\Phi (\hat{\mu }_\psi ) = \int _{\Sigma _I^+\times \Lambda }\log J_{\Phi }(\hat{\mu }_\psi ) \ d\hat{\mu }_\psi \end{aligned}$$

We investigate now the structure of the $\Phi$-invariant probabilities on the product space $\Sigma _I^+\times \Lambda$. Let define also the lift homeomorphism $\tilde{\Phi }$ on $\Sigma _I^+ \times \Sigma _I^+$, namely:

$$\begin{aligned} \tilde{\Phi }:\Sigma _I^+\times \Sigma _I^+\rightarrow \Sigma _I^+\times \Sigma _I^+, \quad \tilde{\Phi }(\omega , \eta ) = (\sigma \omega , \omega _1\eta ) \end{aligned}$$

If $\tilde{\pi }(\omega , \eta ):=(\omega , \pi (\eta ))$, for $(\omega , \eta ) \in \Sigma _I^+\times \Sigma _I^+$, then we obtain the following diagram of maps on $\Sigma _I^+\times \Sigma _I^+$, respectively $\Sigma _I^+\times \Lambda$, where both vertical maps below are equal to $\tilde{\pi }: \Sigma _I^+\times \Sigma _I^+ \rightarrow \Sigma _I^+\times \Lambda$:

$$\begin{aligned} \begin{array}{clclcr} \Sigma _I^+\times \Sigma _I^+ &{} \ &{}\mathop {\longrightarrow }\limits ^{\tilde{\Phi }} &{} \ &{}\Sigma _I^+\times \Sigma _I^+ \\ \downarrow &{} &{} \ \ \ &{} &{} \downarrow \\ \Sigma _I^+\times \Lambda &{} \ &{} \mathop {\longrightarrow }\limits ^{\Phi } &{} \ &{} \Sigma _I^+\times \Lambda \end{array} \end{aligned}$$

(2)

This diagram is commutative. Indeed, $\tilde{\pi }\circ \tilde{\Phi }(\omega , \eta ) = (\sigma \omega , \pi (\omega _1\eta )=(\sigma \omega , \phi _{\omega _1}\circ \phi _{\eta _1}\circ \phi _{\eta _2}\circ \cdots )$; on the other hand, $\Phi \circ \tilde{\pi }(\omega , \eta ) = \Phi (\omega , \phi _{\eta _1}\circ \phi _{\eta _2}\circ \cdots ) = (\sigma \omega , \phi _{\omega _1}\circ \phi _{\eta _1}\circ \cdots )$. Hence $\tilde{\pi }\circ \tilde{\Phi }= \Phi \circ \tilde{\pi }$.

Also $\tilde{\Phi }$ is a homeomorphism. Then as in [18], by using Hahn–Banach theorem and Markov–Kakutani theorem and by approximating integrals of functions from $\mathcal {C}(\Sigma _I^+\times \Sigma _I^+, \mathbb R)$ with integrals of functions $g\circ \tilde{\pi }\circ \tilde{\Phi }^n, n\in \mathbb Z$, for $g \in \mathcal {C}(\Sigma _I^+\times \Lambda , \mathbb R)$, it follows that for any $\Phi$-invariant probability $\nu$ on $\Sigma _I^+\times \Lambda$, there exists a unique $\tilde{\Phi }$-invariant probability $\tilde{\nu }$ on $\Sigma _I^+\times \Sigma _I^+$ such that $\tilde{\pi }_*(\tilde{\nu }) = \nu$. In particular, the equilibrium measure $\hat{\mu }_\psi$ of the Hölder continuous $\psi$ on $\Sigma _I^+\times \Lambda$, is the $\tilde{\pi }$-projection of the equilibrium measure $\tilde{\mu }_{\tilde{\psi }}$ of $\tilde{\psi }:=\psi \circ \tilde{\pi }$ on $\Sigma _I^+\times \Sigma _I^+$. Hence, the measure of maximal entropy $\hat{\mu }_0$ on $\Sigma _I^+\times \Lambda$ is the $\tilde{\pi }$-projection of the measure of maximal entropy $\tilde{\mu }_0$ for $\tilde{\Phi }$ on $\Sigma _I^+\times \Sigma _I^+$, i.e.

$$\begin{aligned} \hat{\mu }_0 = \tilde{\pi }_*(\tilde{\mu }_0) \end{aligned}$$

Moreover, the topological entropy of the map $\Phi$ is equal to the topological entropy of the shift $\sigma : \Sigma _I^+ \rightarrow \Sigma _I^+$, i.e. $\log |I|$, because in the second coordinate we have contractions, so the separated sets are determined only by the expansion $\sigma$ in the first coordinate. With the canonical distance on $\Sigma _I^+$, $d(\omega , \eta ) = \mathop {\sum }\limits _{i\ge 1}\frac{|\omega _i- \eta _i|}{2^i}$, the ball of center $\omega$ and radius $\frac{1}{2^n}$ is the cylinder $[\omega _1, \ldots , \omega _n]$, so $B((\omega , x), \frac{1}{2^n}) = [\omega _1, \ldots , \omega _n] \times B(x, \frac{1}{2^n})$. If we consider n-roots of x and the measure of maximal entropy $\hat{\mu }_0$ w.r.t $\Phi$, then all these n-roots are generic. Since in this case the overlap function $b_n$ does not depend on $\tau$, we denote it simply by $b_n(\omega , x)$, for $(\omega , x) \in \Sigma _I^+\times \Lambda$.

In general, there are several ways to define projections of invariant measures on the fractal limit set $\Lambda$, depending whether we project $\sigma$-invariant measures on $\Sigma _I^+$, or $\Phi$-invariant measures on $\Sigma _I^+ \times \Lambda$, or $\tilde{\Phi }$-invariant measures on $\Sigma _I^+ \times \Sigma _I^+$. In many cases, for example for Bernoulli measures, these projections will be shown to coincide. Let us first consider a Hölder continuous potential $\psi$ on $\Sigma _I^+\times \Lambda$, and as above let $\hat{\mu }_\psi$ its (unique) equilibrium state on $\Sigma _I^+\times \Lambda$; if $\pi _2: \Sigma _I^+\times \Lambda \rightarrow \Lambda$ is the projection on the second coordinate $\pi _2(\omega , x) = x$, denote the projection measure on $\Lambda$ by:

$$\begin{aligned} \mu _\psi :=(\pi _2)_*(\hat{\mu }_\psi ) \end{aligned}$$

(3)

Consider next g a Hölder continuous potential on $\Sigma _I^+$, and let $\bar{\mu }_g$ be its unique equilibrium measure on $\Sigma _I^+$. Then we can define two kinds of projection measures on $\Lambda$. The first type is $\mu _\psi$ defined above in (3), where $\psi = g \circ \pi _1$; so $\mu _\psi = (\pi _2)_*(\hat{\mu }_\psi )$. The second type is the self-conformal measure:

$$\begin{aligned} \pi _*(\bar{\mu }_g), \end{aligned}$$

(4)

where $\pi : \Sigma _I^+ \rightarrow \Lambda , \ \pi (\omega _1\omega _2\ldots ) = \phi _{\omega _1}\circ \phi _{\omega _2} \circ \cdots$ is the canonical coding map for $\Lambda$.

We now prove that, for Bernoulli measures on $\Sigma _I^+$, the two types of projection measures defined above, are in fact equal. This will make our results about overlap numbers apply to $\pi$-projections of Bernoulli measures onto $\Lambda$. Consider then a Bernoulli measure $\nu _{\mathbf {p}}$ on $\Sigma _I^+$ determined by an arbitrary probabilistic vector $\mathbf p = (p_1, \ldots , p_{|I|})$. Thus the $\nu _{\mathbf p}$-measure of the cylinder $[\omega _1, \ldots , \omega _n]= \{\eta \in \Sigma _I^+, \eta _1=\omega _1, \ldots , \eta _n = \omega _n\}$, is equal to $p_{\omega _1} \ldots p_{\omega _n}$ for any $n \ge 1$ and $\omega _i \in I, 1 \le i \le n.$ Consider the potential $\phi : \Sigma _I^+ \rightarrow \mathbb R, \ \phi (\omega _1\omega _2 \ldots ) = \log p_{\omega _1}$, for $\omega = (\omega _1, \omega _2, \ldots ) \in \Sigma _I^+$. Then $S_n\phi (\omega ) = \phi (\omega ) + \phi (\sigma (\omega )) + \cdots + \phi (\sigma ^{n-1}(\omega )) = \log p_{\omega _1} \ldots p_{\omega _n}$. By taking Bowen balls for the shift $\sigma$ (which are cylinders in our case), we see immediately that

$$\begin{aligned} P_\sigma (\phi ) = 0 \end{aligned}$$

Clearly, $\phi$ is Hölder continuous on $\Sigma _I^+$ and its unique equilibrium measure $\bar{\mu }_\phi$ is equal to the Bernoulli measure $\nu _{\mathbf p}$; this is due to the expression of $\bar{\mu }_\phi$ on cylinders $[\omega _1 \ldots \omega _n]$ (see [2, 7]), i.e.

$$\begin{aligned} \frac{1}{C} e^{S_n\phi (\omega ) - nP_\sigma (\phi )} \le \bar{\mu }_\phi (B_n(\omega , \varepsilon )) \le C e^{S_n\phi (\omega ) - nP_\sigma (\phi )}, \end{aligned}$$

so we conclude that

$$\begin{aligned} \bar{\mu }_\phi = \nu _{\mathbf p} \end{aligned}$$

In case of Bernoulli measures, we can now prove that the various projection measures are equal on $\Lambda$:

Theorem 1

In the above setting, let $\mathbf p = (p_1, \ldots , p_{|I|})$ an arbitrary probabilistic vector, and $\psi : \Sigma _I^+\times \Lambda \rightarrow \mathbb R, \ \psi ((\omega _1\ldots ), x) := \log p_{\omega _1}$, with $\hat{\mu }_{\psi }$ denoting the unique equilibrium measure of $\psi$ with respect to $\Phi : \Sigma _I^+ \times \Lambda \rightarrow \Sigma _I^+ \times \Lambda$. Then the following measures are equal on $\Lambda$:

$$\begin{aligned} \pi _* \nu _{\mathbf p} = \pi _{2*} \hat{\mu }_{\psi } = (\pi _2 \circ \tilde{\pi })_*(\nu _\mathbf{p } \times \nu _\mathbf{p }), \end{aligned}$$

where $\pi _2: \Sigma _I^+ \times \Lambda \rightarrow \Lambda , \ \pi _2(\omega , x) = x$, and $\pi : \Sigma _I^+ \rightarrow \Lambda$ is the canonical coding map, and where $\tilde{\pi }: \Sigma _I^+ \times \Sigma _I^+ \rightarrow \Sigma _I^+ \times \Lambda , \ \tilde{\pi }(\omega , \eta ) = (\omega , \pi (\eta ))$.

Proof

In order to prove the first equality, let us define $\tilde{\psi }= \psi \circ \tilde{\pi }$, where $\tilde{\pi }(\omega , \eta ) = (\omega , \pi \eta )$. So $\tilde{\psi }$ is a Hölder continuous potential on $\Sigma _I^+ \times \Sigma _I^+$. Then recalling that $\tilde{\Phi }(\omega , \eta ) = (\sigma \omega , \omega _1 \eta )$ is an expansive homeomorphism with specification property, it follows [7] that there exists a unique equilibrium measure $\tilde{\mu }_{\tilde{\psi }}$ on $\Sigma _I^+ \times \Sigma _I^+$. Also we have the projection $\tilde{\pi }(\omega , \eta ) = (\omega , \pi \eta )$ from $\Sigma _I^+ \times \Sigma _I^+$ to $\Sigma _I^+ \times \Lambda$. Moreover, from definitions it can be seen that

$$\begin{aligned} \tilde{\pi }\tilde{\Phi }(\omega , \eta ) = (\sigma \omega , \phi _{\omega _1}(\pi \eta )) = \Phi \circ \tilde{\pi }(\omega , \eta ), \end{aligned}$$

so $\tilde{\pi }\circ \tilde{\Phi }= \Phi \circ \tilde{\pi }$. This implies that $\tilde{\pi }_*(\tilde{\mu }_{\tilde{\psi }}) = \hat{\mu }_\psi$, i.e. the projection to $\Sigma _I^+ \times \Lambda$ of the equilibrium measure of $\tilde{\psi }$ on $\Sigma _I^+ \times \Sigma _I^+$, is equal to the equilibrium measure of $\psi$. Hence from above,

$$\begin{aligned} \pi _{2*}(\hat{\mu }_\psi )(A) = \hat{\mu }_\psi (\pi _2^{-1}(A)) = \tilde{\mu }_{\tilde{\psi }}(\Sigma _I^+\times \pi ^{-1}(A)) \end{aligned}$$

(5)

On the other hand, notice that the Bowen ball for $\tilde{\Phi }$ is given by $B_n((\omega , \eta ), \varepsilon ) = [\omega _1 \ldots \omega _n] \times \Sigma _I^+$, and for any $1 \le i \le n$, we have $\tilde{\Phi }^i(B_n((\omega , \eta ), \varepsilon )) = [\omega _{i+1} \ldots \omega _n] \times [\omega _i \ldots \omega _1]$. From the $\tilde{\Phi }$-invariance of the equilibrium measure $\tilde{\mu }_{\tilde{\psi }}$, it follows that for any $1 \le i \le n$,

$$\begin{aligned} \tilde{\mu }_{\tilde{\psi }}(\tilde{\Phi }^i(B_n((\omega , \eta ), \varepsilon ))) = \tilde{\mu }_{\tilde{\psi }}([\omega _1 \ldots \omega _n] \times \Sigma _I^+) = \tilde{\mu }_{\tilde{\psi }}([\omega _{i+1} \ldots \omega _n] \times [\omega _i \ldots \omega _1]) \end{aligned}$$

(6)

However recall that $\pi _{1*}\hat{\mu }_\psi = \bar{\mu }_\phi = \nu _{\mathbf p}$, and thus $(\pi _1 \circ \tilde{\pi })_* \tilde{\mu }_{\tilde{\psi }} = \nu _{\mathbf p}$. Therefore using also (6) we obtain that, for any $j \ge 1$ and any $\omega , \eta \in \Sigma _I^+$,

$$\begin{aligned} \tilde{\mu }_{\tilde{\psi }}([\omega _1] \times [\eta _1 \ldots \eta _j]) = \nu _{\mathbf p}([\eta _j \ldots \eta _1\omega _1]) = p_{\eta _j} \cdot \cdots \cdot p_{\eta _1} p_{\omega _1} \end{aligned}$$

(7)

By adding over $\omega _1 \in \Sigma _I^+$ we obtain that, for any $j \ge 1$ and for any $\eta = (\eta _1 \eta _2 \ldots ) \in \Sigma _I^+$,

$$\begin{aligned} \tilde{\mu }_{\tilde{\psi }}(\Sigma _I^+ \times [\eta _1 \ldots \eta _j]) = p_{\eta _1} \ldots p_{\eta _j} = \nu _{\mathbf p}([\eta _1 \ldots \eta _j] \end{aligned}$$

But this works for any cylinder in $\Sigma _I^+$. Also, for any Borel set $A \subset \Lambda$, we have $\pi _{*}\nu _{\mathbf p}(A) = \nu _{\mathbf p}(\pi ^{-1}(A))$. Hence from the above, and by using also (5), we can infer that $\pi _{2*}\hat{\mu }_\psi$ is in fact a self-conformal measure on $\Lambda$, namely,

$$\begin{aligned} \pi _{2*}\hat{\mu }_\psi = \pi _* \nu _{\mathbf p} \end{aligned}$$

We now prove the second equality. From before, $\tilde{\Phi }: \Sigma _I^+ \times \Sigma _I^+ \rightarrow \Sigma _I^+ \times \Sigma _I^+$ is a homeomorphism which preserves $\tilde{\mu }_{\tilde{\psi }}$. Also notice that for any $\omega _1, \omega _2, \eta _1, \ldots , \eta _m \in I$, one has $\tilde{\Phi }([\omega _1\omega _2] \times [\eta _1\ldots \eta _m]) = [\omega _2] \times [\omega _1\eta _1\eta _2\ldots \eta _m]$. But, from (7), $\tilde{\mu }_{\tilde{\psi }}([\omega _2]\times [\omega _1\eta _1\ldots \eta _m]) = p_{\omega _2}p_{\omega _1}p_{\eta _1}\ldots p_{\eta _m}$, and from the $\tilde{\Phi }$-invariance of $\tilde{\mu }_{\tilde{\psi }}$, it follows that $\tilde{\mu }_{\tilde{\psi }}([\omega _1\omega _2] \times [\eta _1\ldots \eta _m]) = \tilde{\mu }_{\tilde{\psi }}(\tilde{\Phi }([\omega _1\omega _2] \times [\eta _1\ldots \eta _m])) = p_{\omega _1}p_{\omega _2}p_{\eta _1}\ldots p_{\eta _m}$. Hence by induction it follows similarly that, for any $k, m \ge 1$,

$$\begin{aligned} \tilde{\mu }_{\tilde{\psi }}([\omega _1\ldots \omega _k]\times [\eta _1\ldots \eta _m]) = p_{\omega _1}\ldots p_{\omega _k} \cdot p_{\eta _1}\ldots p_{\eta _m} \end{aligned}$$

This means that $\tilde{\mu }_{\tilde{\psi }} = \nu _\mathbf{p } \times \nu _\mathbf{p }$, and that $\pi _* \nu _\mathbf{p } = (\pi _2 \circ \tilde{\pi })_*(\nu _\mathbf{p } \times \nu _\mathbf{p })$. $\square$

The equality of the projection measures for Bernoulli probabilities has useful consequences when computing the associated overlap numbers, see Corollary 2.

For any conformal iterated function system $\mathcal {S}$, we want to prove now that the exponential rate of growth in n, of the number of generic n-chains/roots from $\Delta _n$, is approaching the folding entropy of the measure $\hat{\mu }_\psi$. In particular it follows that, on average, the number of n-chains associated to the n-overlaps of $\Lambda$ grows exponentially like $e^{nF_\Phi (\hat{\mu }_0)}$.

Theorem 2

Let a finite conformal IFS $\mathcal {S} = \{\phi _i, i \in I\}$ with limit set $\Lambda$, and a Hölder continuous potential $\psi$ on the lift space $\Sigma _I^+ \times \Lambda$; denote the equilibrium measure of $\psi$ on $\Sigma _I^+\times \Lambda$ by $\hat{\mu }_\psi$. Then,

$$\begin{aligned} \mathop {\lim }\limits _{\tau \rightarrow 0}\mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n} \int _{\Sigma _I^+\times \Lambda } \log b_n((\omega , x), \tau , \hat{\mu }_\psi ) \ d\hat{\mu }_\psi (\omega , x) = F_\Phi (\hat{\mu }_\psi ) \end{aligned}$$

Proof

In our case the map $\Phi : \Sigma _I^+ \times \Lambda \rightarrow \Sigma _I^+\times \Lambda$ is distance-expanding in the first coordinate, and distance contracting in the second coordinate. Let $B_m(z, \varepsilon )$ denote the $(m, \varepsilon )$-Bowen ball around z in the canonical product metric on the compact metric space $\Sigma _I^+\times \Lambda$ with respect to the endomorphism $\Phi$; hence in particular it is expansive. Since $\hat{\mu }_\psi$ is the equilibrium measure of a Hölder continuous potential on $\Sigma _I^+\times \Lambda$, we can apply the properties of equilibrium measures with respect to expansive maps on compact metric spaces (see [7]). We will use first the ideas of Theorem 1 from [9], giving the comparison between the (equilibrium) measure of various parts of the preimage set. So, from [9] there exists a constant $C>0$ such that, for any positive integer m and for any sets $A_1, A_2$ satisfying $A_1 \subset B_m(z_1, \varepsilon ), A_2 \subset B_m(z_2, \varepsilon )$ and $\Phi ^m(A_1) = \Phi ^m(A_2)$, we have:

$$\begin{aligned} \frac{1}{C}\frac{\hat{\mu }_\psi (A_2)}{e^{S_m\psi (z_2)}} \le \frac{\hat{\mu }_\psi (A_1)}{e^{S_m\psi (z_1)}} \le C \frac{\hat{\mu }_\psi (A_2)}{e^{S_m\psi (z_2)}} \end{aligned}$$

(8)

Now the Jacobian of the measure $\hat{\mu }_\psi$ with respect to $\Phi ^n$ gives the change in the measure of a set by applying the map $\Phi ^n$ (see [14]); hence for any integer $n \ge 1$, $\hat{\mu }_\psi (\Phi ^n(\mathcal {A})) = \int _{\mathcal {A}}J_{\Phi ^n}(\hat{\mu }_\psi ) d\hat{\mu }_\psi$, for any measurable set $\mathcal {A} \subset \Sigma _I^+\times \Lambda$, on which $\Phi ^n$ is injective. But in fact, $J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x) = \mathop {\lim }\limits _{r \rightarrow 0} \frac{\hat{\mu }_\psi (\Phi ^n(B((\omega , x), r)}{\hat{\mu }_\psi (B((\omega , x), r)}$, for $\hat{\mu }_\psi$-a.e $(\omega , x) \in \Sigma _I^+\times \Lambda$. However from the $\Phi$-invariance of the measure $\hat{\mu }_\psi$ it follows that $\hat{\mu }_\psi (\Phi ^{n}(\mathcal {A})) = \hat{\mu }_\psi (\Phi ^{-n}(\Phi ^n(\mathcal {A})))$, for any Borel set $\mathcal {A}$. Hence we can apply the above comparison between the various parts of the preimage set $\Phi ^{-n}(\Phi ^n(\mathcal {A}))$ for n arbitrary (i.e. in fact the comparison between various sets taken by different compositions $\phi _{j_1}\circ \cdots \circ \phi _{j_n}$ to the same image), in order to obtain that there exists a constant $C>0$ independent of n such that:

$$\begin{aligned} \ \frac{\mathop {\sum }\nolimits _{(\eta , y), \Phi ^n(\eta , y) = \Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{C\cdot \exp (S_n\psi (\omega , x))}\le & {} J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x)\nonumber \\\le & {} C \cdot \frac{\mathop {\sum }\nolimits _{(\eta , y), \Phi ^n(\eta , y) = \Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))},\nonumber \\ \end{aligned}$$

(9)

for $\hat{\mu }_\psi$-a.e pair $(\omega , x) \in \Sigma _I^+\times \Lambda$. Now, as the probability $\hat{\mu }_\psi$ is $\Phi$-invariant on the product space $\Sigma _I^+ \times \Lambda$, it follows from (9) and from the properties of the folding entropy that

$$\begin{aligned} F_\Phi (\hat{\mu }_\psi )= & {} \frac{1}{n} \int _{\Sigma _I^+\times \Lambda }\log J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x) d\hat{\mu }_\psi (\omega , x) \nonumber \\= & {} \mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n}\int _{\Sigma _I^+\times \Lambda } \log \frac{\mathop {\sum }\limits _{\Phi ^n(\eta , y) = \Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))} d\hat{\mu }_\psi (\omega , x) \end{aligned}$$

(10)

From Birkhoff Ergodic Theorem we know that, $\hat{\mu }_\psi \big ((\omega , x) \in \Sigma _I^+\times \Lambda , |\frac{S_n\psi (\omega , x)}{n}-\int _{\Sigma _I^+\times \Lambda } \psi d\hat{\mu }_\psi | > \tau /2\big ) \mathop {\rightarrow }\limits _{n\rightarrow \infty } 0$. Then, for any positive small number $\xi$, there exists an integer $n = n(\xi ) \ge 1$ so that for all integers $n \ge n(\xi )$, we have

$$\begin{aligned} \mu _\psi \left( (\omega , x) \in \Sigma _I^+\times \Lambda , \left| \frac{S_n\psi (\omega , x)}{n}-\int _{\Sigma _I^+\times \Lambda } \psi d\hat{\mu }_\psi \right| > \tau /2\right) < \xi \end{aligned}$$

(11)

Recall that, if $(\eta _1, \ldots , \eta _n) \in \Delta _n((\omega , x), \tau , \hat{\mu }_\psi )$, then the n-chain $(\eta _n, \ldots , \eta _1)$ uniquely determines an n-root y of $\phi _{\omega _n\ldots \omega _1}(x)$. Hence with $\eta _{n+i} = \omega _{n+i}, i \ge 1$, we can consider also the finite set

$$\begin{aligned} \Delta _n'((\omega , x), \tau , \hat{\mu }_\psi )= & {} \left\{ (\eta , y) \in \Sigma _I^+ \times \Lambda , \ \Phi ^n(\eta , y)\right. \\&\left. = \Phi ^n(\omega , x), \ \left| \frac{S_n\psi (\eta , y)}{n} - \int \psi \ d\hat{\mu }_\psi \right| < \tau \right\} , \end{aligned}$$

and there exists a bijection between $\Delta _n((\omega , x), \tau , \hat{\mu }_\psi )$ and $\Delta _n'((\omega , x), \tau , \hat{\mu }_\psi )$, taking $(\eta _1, \ldots , \eta _n)$ to $((\eta _1, \ldots , \eta _n, \omega _{n+1}, \omega _{n+2}, \ldots ), y)$. Thus $b_n((\omega , x), \tau , \hat{\mu }_\psi ) = \text {Card}\Delta _n'((\omega , x), \tau , \hat{\mu }_\psi )$. We now define the following set of n-roots,

$$\begin{aligned} \Gamma _n((\omega , x), \tau , \hat{\mu }_\psi ):= & {} \{(\eta , y) \in \Sigma _I^+\times \Lambda , \Phi ^n(\eta , y)\\= & {} \Phi ^n(\omega , x), (\eta _1, \ldots , \eta _n) \notin \Delta _n((\omega , x), \tau , \hat{\mu }_\psi )\} \end{aligned}$$

Denote the sum corresponding to the roots from $\Gamma _n((\omega , x), \tau , \hat{\mu }_\psi )$ by

$$\begin{aligned} \vartheta _n((\omega , x), \tau , \hat{\mu }_\psi ):= \mathop {\sum }\limits _{(\eta , y) \in \Gamma _n((\omega , x), \tau , \hat{\mu }_\psi )} \exp (S_n\psi (\eta , y)) \end{aligned}$$

Let us now see what a typical Bowen ball for the map $\Phi : \Sigma _I^+\times \Lambda \rightarrow \Sigma _I^+\times \Lambda$ looks like. If $d(\cdot , \cdot )$ denotes the product metric, and if $d(\Phi ^i(\omega , x), \Phi ^i(\eta , y)) < \varepsilon , 0 \le i \le n-1$, then there exists an integer $N(\varepsilon )$ so that $\omega _i=\eta _i, i= 1, \ldots , n+N(\varepsilon )$, and $d(x, y)<\varepsilon$, since the maps $\phi _j$ are all contractions. For an arbitrary $n \ge 2$, we now consider a measurable partition of $\Sigma _I^+\times \Lambda$ modulo $\hat{\mu }_\psi$, into sets $L_i^n, 1 \le i \le p_n$, such that for any $1 \le i \le p_n$ there exists a point $\zeta _i \in L_i^n$ so that for any point $\zeta _{ij} \in \Phi ^{-n}(\zeta _i), 1 \le j \le p_{i, n}$, we have $L_i^n \subset \Phi ^n(B_n(\zeta _{ij}, \varepsilon ))$. The integer $p_{i, n} \ge 1$ depends on i for $1 \le i \le p_n$, and it is given by the number of n-roots of $\zeta _i$ in $\Lambda$, with respect to $\mathcal {S}$. This is possible to do if we take the sets $L_i^n$ small enough. Then, let us denote by $L_{ij}^n:= \Phi ^{-n}(L_i^n)\cap B_n(\zeta _{ij}, \varepsilon )$, for $1 \le i \le p_n, 1\le j \le p_{i, n}$. Notice that if $\Phi (\eta , y) = \Phi (\eta ', y') = (\omega , x) \in \Sigma _I^+\times \Lambda$, then $\sigma \eta =\sigma \eta '=\omega$, i.e. $\eta _2 =\omega _2, \ldots$, and $\phi _{\eta _1}(y) = \phi _{\eta _1'}(y')=x$. If $\eta _1 \ne \eta _1'$, then $d((\eta , y), (\eta ', y')) \ge d(\eta _1, \eta _1') >\varepsilon _0>\varepsilon$, for some $\varepsilon _0>0$. If $\eta _1 = \eta _1'$, then $\phi _{\eta _1}(y) = \phi _{\eta _1'}(y')$; but $\phi _\eta , \eta \in I$ are injective and thus $y = y'$. This implies that the sets $L_{ij}^n$ are mutually disjoint in i, j. We now decompose the integral of the logarithm of the Jacobian of $\hat{\mu }_\psi$ with respect to $\Phi ^n$, along this partition with the sets $L_{ij}^n, 1 \le i \le p_n, 1 \le j \le p_{i, n}$. Therefore, for an arbitrary $n \ge 2$, we have:

$$\begin{aligned}&\int _{\Sigma _I^+\times \Lambda }\log \frac{\mathop {\sum }\limits _{\Phi ^n(\eta , y) = \Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))} d\hat{\mu }_\psi (\omega , x)\\ \nonumber&\quad =\,\mathop {\mathop {\sum }\limits _{1\le i\le p_n}}\limits _{ 1\le j\le p_{i, n}} \int _{L_{ij}^n} \log \frac{\mathop {\sum }\limits _{\Phi ^n(\eta , y) = \Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))} d\hat{\mu }_\psi (\omega , x) \end{aligned}$$

(12)

Now, in regards to formula (9), we can write in general

$$\begin{aligned} \mathop {\sum }\limits _{(\eta , y) \in \Phi ^{-n}\Phi ^n(\omega , x)} e^{S_n\psi (\eta , y)} = \mathop {\sum }\limits _{(\eta _1, \ldots , \eta _n) \in \Delta _n((\omega , x), \tau , \hat{\mu }_\psi )} e^{S_n\psi (\eta , y)} + \vartheta _n((\omega , x), \tau , \hat{\mu }_\psi ) \end{aligned}$$

Denote also $\rho _n(i, \tau , \hat{\mu }_\psi ):= \mathop {\sum }\nolimits _{j, \zeta _{ij} \notin \Delta _n'(\zeta _{i1}, \tau , \hat{\mu }_\psi )} \hat{\mu }_\psi (L_{ij}^n)$. Thus by using (8), the definition of $\Delta _n'((\omega , x), \tau , \hat{\mu }_\psi )$ and the fact that $b_n((\omega , x), \tau , \hat{\mu }_\psi ) = \text {Card}(\Delta _n'((\omega , x), \tau , \hat{\mu }_\psi ))$, we obtain that the above sum in (12) is comparable to the sum:

$$\begin{aligned} \mathop {\sum }\limits _{i, j} \hat{\mu }_\psi (L_{ij}^n) \log \frac{b_n(\zeta _{ij}, \tau , \hat{\mu }_\psi ) \hat{\mu }_\psi (L_{ij}^n) +\rho _n(i, \tau , \hat{\mu }_\psi )}{\hat{\mu }_\psi (L_{ij}^n)}, \end{aligned}$$

where we recall that the comparability constant $C>0$ does not depend on n, nor on $L_{i j}^n$. Now in general, if $(\eta , y)\in \Delta _n'((\omega , x), \tau , \hat{\mu }_\psi )$, and if $0 < \varepsilon < \tau$ and $(\eta , y) \in B_n(\zeta _{ij}, \varepsilon )$, then since the potential $\psi$ is Hölder continuous, it follows that

$$\begin{aligned} \Big |\frac{S_n\psi (\eta , y)}{n} - \frac{S_n\psi (\zeta _{ij})}{n}\Big | \le v(\tau ), \end{aligned}$$

for some small $v(\tau )>0$ where $\mathop {\lim }\limits _{\tau \rightarrow 0}v(\tau ) = 0$. Also, if $K:= \sup _{\Sigma _I^+\times \Lambda }|\psi |$, then $e^{S_n\psi (\eta , y)} \le e^{n K}$. Notice in addition, that the set $\Phi ^{-n}\Phi ^n(\omega , x)$ has at most $|I|^n$ elements in $\Sigma _I^+ \times \Lambda$. Denote the set of indices j corresponding to nongeneric roots by $Q(n, i, \tau , \hat{\mu }_\psi ) := \{j, 1 \le j \le p_{i, n}, \ \zeta _{ij} \in \Gamma _n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\}$. Then if $j \in Q(n, i, \tau , \hat{\mu }_\psi )$, then $\frac{1}{n} |S_n\psi (\zeta _{ij})- \int _{\Sigma _I^+\times \Lambda } \psi d\hat{\mu }_\psi | > \tau$. Hence we can use the measure estimate in (11) to obtain that:

$$\begin{aligned}&\mathop {\sum }\limits _{1 \le i \le p_n, \ j \in Q(n, i, \tau , \hat{\mu }_\psi )} \frac{1}{n} \int _{L_{ij}^n} \log \frac{\mathop {\sum }\limits _{(\eta , y) \in \Phi ^{-n}\Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))} d\hat{\mu }_\psi (\omega , x)\\&\quad \le \frac{1}{n} \xi \log (2K|I|^n) \end{aligned}$$

Therefore, from the comparison in (8) and from the above discussion, it follows that there exists a positive constant C, independent of n, of the partition $\{L_i^n\}_{1\le i\le p_n}$ and of the points $\zeta _i\in L_i^n$, such that:

$$\begin{aligned}&\frac{1}{n}\mathop {\mathop {\sum }\limits _{1\le i\le p_n}}\limits _{ j \notin Q(n, i, \tau , \hat{\mu }_\psi )} \hat{\mu }_\psi (L_{ij}^n) \log b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\nonumber \\&\qquad +\,\frac{1}{n} \mathop {\sum }\limits _{i, j\notin Q(n, i, \tau , \hat{\mu }_\psi )} \hat{\mu }_\psi (L_{ij}^n)\log \left( 1+\frac{\rho _n(i, \tau , \hat{\mu }_\psi )}{b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\hat{\mu }_\psi (L_{ij}^n)}\right) - v(\tau )-C\xi \nonumber \\&\quad \le \,\int _{\Sigma _I^+\times \Lambda } \frac{1}{n} \log \frac{\mathop {\sum }\limits _{(\eta , y) \in \Phi ^{-n}\Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x))} \ d\hat{\mu }_\psi (\omega , x) \nonumber \\&\quad \le \, \frac{1}{n}\mathop {\mathop {\sum }\limits _{1\le i\le p_n}}\limits _{ j \notin Q(n, i, \tau , \hat{\mu }_\psi )} \hat{\mu }_\psi (L_{ij}^n) \log b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi ) \nonumber \\&\qquad +\,\frac{1}{n} \mathop {\sum }\limits _{i, j\notin Q(n, i, \tau , \hat{\mu }_\psi )} \hat{\mu }_\psi (L_{ij}^n)\log \left( 1+\frac{\rho _n(i, \tau , \hat{\mu }_\psi )}{b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\hat{\mu }_\psi (L_{ij}^n)}\right) +v(\tau ) +C\xi ,\qquad \end{aligned}$$

(13)

where we recall that $\xi$ is the bound on the measure of non-generic points in (11). But in general, $\log (1+x) \le x$ for any $x >0$, hence $\log (1+\frac{\rho _n(i, \tau , \hat{\mu }_\psi )}{b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\hat{\mu }_\psi (L_{ij}^n)}) \le \frac{\rho _n(i, \tau , \hat{\mu }_\psi )}{b_n(\zeta _{i1}, \tau , \hat{\mu }_\psi )\hat{\mu }_\psi (L_{ij}^n)}$. Therefore from (11), the second sum in the right-hand term of (13) is less than $\xi$, which implies that:

$$\begin{aligned}&\left| \frac{1}{n} \int _{\Sigma _I^+\times \Lambda } \frac{1}{n} \log \frac{\mathop {\sum }\limits _{(\eta , y) \in \Phi ^{-n}\Phi ^n(\omega , x)} \exp (S_n\psi (\eta , y))}{\exp (S_n\psi (\omega , x)} d\hat{\mu }_\psi (\omega , x) \right. \\&\left. \quad -\,\frac{1}{n} \int _{\Sigma _I^+\times \Lambda } \log b_n((\omega , x), \tau , \hat{\mu }_\psi ) d\hat{\mu }_\psi (\omega , x)\right| \le v(\tau ) +C\xi \end{aligned}$$

Therefore, using the expression for the folding entropy $F_\Phi (\hat{\mu }_\psi )$ from (10), and the fact that $\xi$ converges to 0 when $\tau$ converge to 0 (and also that $v(\tau )$ converges to 0 at the same time), we obtain the conclusion of the Theorem. $\square$

We now want to define a notion of overlap number of $\mathcal {S}$ associated to an equilibrium state $\hat{\mu }_\psi$. This notion will take into consideration the $\hat{\mu }_\psi$-generic n-roots in $\Lambda$ and all the corresponding n-chains starting from them, for n large. In particular, we obtain a (topological) overlap number of the system $\mathcal {S}$, which gives the average rate of growth of the number of n-chains from n-roots to points in $\Lambda$.

Corollary 1

If $\mathcal {S} = \{\phi _i, i \in I\}$ is an arbitrary finite conformal iterated function system with overlaps and $\Lambda$ is its limit set, and if $\psi$ is a Hölder continuous potential on $\Sigma _I^+ \times \Lambda$ with equilibrium measure $\hat{\mu }_\psi$, we call the overlap number of $\mathcal {S}$ with respect to $\hat{\mu }_\psi$,

$$\begin{aligned} o(\mathcal {S}, \hat{\mu }_\psi ):= \exp \left( \mathop {\lim }\limits _{\tau \rightarrow 0}\mathop {\lim }\limits _{n\rightarrow \infty } \frac{1}{n} \int _{\Sigma _I^+\times \Lambda } \log b_n((\omega , x), \tau , \hat{\mu }_\psi ) \ d\hat{\mu }_\psi (\omega , x)\right) \end{aligned}$$

(14)

If $\hat{\mu }_0$ is the measure of maximal entropy for $\Phi$ on $\Sigma _I^+\times \Lambda$, then the (topological) overlap number of $\mathcal {S}$ is given by:

$$\begin{aligned} \begin{aligned} o(\mathcal {S}) := o(\mathcal {S}, \hat{\mu }_0)&= \exp \left( \mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n} \int _{\Sigma _I^+\times \Lambda } \log b_n(\omega , x) \ d\hat{\mu }_0(\omega , x)\right) = \exp \big (F_\Phi (\hat{\mu }_0)\big ) \\&=\exp \left( \int _{\Sigma _I^+\times \Lambda } \log \mathop {\lim }\limits _{n\rightarrow \infty } \frac{\hat{\mu }_0([\omega _2, \ldots , \omega _n] \times \phi _{\omega _1}\left( B\left( x, \frac{1}{2^n}\right) \right) }{\hat{\mu }_0\left( [\omega _1, \ldots , \omega _n]\times B\left( x, \frac{1}{2^n}\right) \right) } \ d\hat{\mu }_0(\omega , x)\right) \end{aligned} \end{aligned}$$

The $\Phi$-invariant measure $\hat{\mu }_0$ on $\Sigma _I^+ \times \Lambda$ is the projection of the measure of maximal entropy $\tilde{\mu }_0$ of $\tilde{\Phi }$ on $\Sigma _I^+\times \Sigma _I^+$, i.e. $\tilde{\pi }_*( {\tilde{\mu }}_0 ) = \hat{\mu }_0$. But in general, for $n \ge 2$, $\tilde{\Phi }^n(\omega , \eta ) = (\sigma ^n(\omega ), \omega _n\ldots \omega _1\eta )$. This means that for the product metric on $\Sigma _I^+\times \Sigma _I^+$ the Bowen ball $B_n((\omega , \eta ), \varepsilon )$ is equal to $[\omega _1 \ldots \omega _n] \times \Sigma _I^+$ (where $[\omega _1 \ldots \omega _n]$ is the cylinder of sequences in $\Sigma _I^+$ with starting coordinates $\omega _1, \ldots , \omega _n$), i.e. the second coordinate does not matter in Bowen balls for $\tilde{\Phi }$. Moreover, we have for any $1 \le i \le n$, $\tilde{\Phi }^i(B_n((\omega , \eta ), \varepsilon )) = [\omega _{i+1} \ldots \omega _n] \times [\omega _i \ldots \omega _1]$, and we know that the measure $\tilde{\mu }_0$ is $\tilde{\Phi }$-invariant, hence

$$\begin{aligned} \tilde{\mu }_0\big ([\omega _1 \ldots \omega _n] \times \Sigma _I^+\big ) = \tilde{\mu }_0\big ([\omega _{i+1}\ldots \omega _n] \times [\omega _i \ldots \omega _1]\big ) \end{aligned}$$

On the other hand, we know that $\pi _1\circ \tilde{\Phi }(\omega , \eta ) = \sigma \omega = \sigma \circ \pi _1(\omega , \eta )$, so $\pi _{1*}(\tilde{\mu }_0) = \bar{\mu }_0$, where $\bar{\mu }_0$ is the measure of maximal entropy for the shift on $\Sigma _I^+$. Therefore we have $\tilde{\mu }_0([\omega _1\ldots \omega _n] \times \Sigma _I^+) = \bar{\mu }_0([\omega _1 \ldots \omega _n])$. This implies, from above, that we can say exactly what is $\tilde{\mu }_0$ on a neighbourhood basis in the product space $\Sigma _I^+\times \Sigma _I^+$; namely for any $i, j \ge 1$, the measure $\tilde{\mu }_0$ on the product of any two cylinders $[\omega _1\ldots \omega _i], [\eta _1\ldots \eta _j]$ in $\Sigma _I^+\times \Sigma _I^+$ is given by:

$$\begin{aligned}\tilde{\mu }_0([\omega _1 \ldots \omega _i] \times [\eta _1 \ldots \eta _j]) = \bar{\mu }_0([\omega _1\ldots \omega _i]) \cdot \bar{\mu }_0([\eta _1 \ldots \eta _j]) = \frac{1}{|I|^{i+j}}. \end{aligned}$$

In the case of projections of Bernoulli measures, we can use now Theorem 1 to compute more easily the overlap numbers. Let us take an arbitrary probability vector $\mathbf p = (p_1, \ldots , p_{|I|})$, which gives a Bernoulli measure $\nu _\mathbf{p}$ on $\Sigma _I^+$. According to the discussion before Theorem 1, there exists an equilibrium measure denoted $\hat{\mu }_\mathbf{p}$ of the potential $\psi ((\omega _1, \ldots ), x)= \log p_{\omega _1}, \ (\omega , x) \in \Sigma _I^+\times \Lambda$, with respect to $\Phi$ on $\Sigma _I^+\times \Lambda$, so that $\pi _*\nu _\mathbf{p} = \pi _{2*}\hat{\mu }_\mathbf{p}$. The measure $\hat{\mu }_\mathbf{p}$ is called the equilibrium measure (with respect to $\Phi$) associated to $\mathbf p$. Denote also by $h(\mathbf p ):= \mathop {\sum }\nolimits _{1 \le j \le |I|} p_j \log p_j$, and notice that $h(\mathbf{p}) = \int \psi \ d{\hat{\mu }}_\mathbf{p}$. Let us denote now by

$$\begin{aligned} \beta _n(x):= \text {Card}\{(\eta _1, \ldots , \eta _n) \in I^n, \ x \in \phi _{\eta _1}\circ \cdots \circ \phi _{\eta _n}(\Lambda )\}, \ \forall x \in \Lambda \end{aligned}$$

More generally, we define for $\tau >0$,

$$\begin{aligned}&\beta _n(x, \tau , \mathbf p )\nonumber \\&\quad := \text {Card}\left\{ (\eta _1, \ldots , \eta _n) \in I^n, \ x \in \phi _{\eta _1}\circ \cdots \circ \phi _{\eta _n}(\Lambda ), \ \left| \frac{\log (p_{\eta _1}\ldots p_{\eta _n})}{n} - h(\mathbf p)\right| < \tau \right\} \nonumber \\ \end{aligned}$$

(15)

As before if $x \in \phi _{\eta _1}\circ \cdots \circ \phi _{\eta _n}(\Lambda )$, then there exists a unique point $y \in \Lambda$ with $x = \phi _{\eta _1}\circ \cdots \circ \phi _{\eta _n}(y)$. When the system $\mathcal {S}$ satisfies Open Set Condition, then the overlap number $o(\mathcal {S}, \hat{\mu }_\mathbf{p})$ is equal to 1.

We prove now the following simpler expression for the overlap number in the case of Bernoulli projections for conformal IFS’s with overlaps $\mathcal {S}$, by employing the function $\beta _n(\cdot )$, that counts the number of n-chains from n-roots in the limit set $\Lambda$:

Corollary 2

Let a conformal iterated function system with overlaps $\mathcal {S} = \{\phi _i, i \in I\}$ with limit set $\Lambda$, and consider $\mathbf p$ an arbitrary probabilistic vector, with $\hat{\mu }_\mathbf{p}$ being the equilibrium measure on $\Sigma _I^+ \times \Lambda$ associated to $\mathbf p$. Then, the overlap number $o(\mathcal {S}, \hat{\mu }_\mathbf{p})$ can be computed as:

$$\begin{aligned} o(\mathcal {S}, \hat{\mu }_\mathbf{p}) = \exp \left( \mathop {\lim }\limits _{\tau \rightarrow 0}\mathop {\lim }\limits _n \frac{1}{n} \int _{\Sigma _I^+} \log \beta _n(\pi \omega , \tau , \mathbf p ) \ d\nu _\mathbf{p}(\omega )\right) \end{aligned}$$

In particular, we obtain the (topological) overlap number of $\mathcal {S}$, by integrating with respect to the uniform Bernoulli measure $\nu _{(\frac{1}{|I|}, \ldots , \frac{1}{|I|})}$,

$$\begin{aligned} o(\mathcal {S}) = \exp \left( \mathop {\lim }\limits _n \frac{1}{n} \int _{\Sigma _I^+} \log \beta _n(\pi \omega ) \ d\nu _{(\frac{1}{|I|}, \ldots , \frac{1}{|I|})}(\omega )\right) \end{aligned}$$

Proof

We prove here the second part of the statement, about the topological overlap number; the first part follows similarly. Let us denote by $\mathbf p = (\frac{1}{|I|}, \ldots , \frac{1}{|I|})$, and consider $\mu _\mathbf{p} = \pi _*\nu _\mathbf{p}$. As in Theorem 1 there exists a corresponding $\Phi$-invariant measure $\hat{\mu }_\mathbf{p}$ on $\Sigma _I^+ \times \Lambda$. We have from Theorem 1 that $\pi _*\nu _\mathbf{p} = \pi _{2*}\hat{\mu }_\mathbf{p}$, hence

$$\begin{aligned} \int _{\Lambda } \log \beta _n(x) \ d\mu _\mathbf{p}(x)= & {} \int _{\Sigma _I^+\times \Lambda }\log \beta _n \circ \pi _2(\omega , x) \ d\hat{\mu }_\mathbf{p}(\omega , x)\\= & {} \int _{\Sigma _I^+\times \Lambda }\log \beta _n \circ \pi _2\circ \Phi ^n(\omega , x) \ d\hat{\mu }_\mathbf{p}(\omega , x) \end{aligned}$$

But notice that $\beta _n\circ \pi _2\circ \Phi ^n(\omega , x) = \beta _n(\phi _{\omega _n}\circ \cdots \circ \phi _{\omega _1}(x)) = \text {Card}\{(\eta _1, \ldots , \eta _n) \in I^n, \ \phi _{\omega _n}\circ \cdots \circ \phi _{\omega _1}(x) \in \phi _{\eta _1} \circ \cdots \circ \phi _{\eta _n}(\Lambda ) \} = b_n(\omega , x)$, for any $(\omega , x)$. Therefore, from the last displayed equality, it follows that:

$$\begin{aligned} \int _{\Sigma _I^+} \log \beta _n(\pi \omega ) \ d\nu _{(\frac{1}{|I|}, \ldots , \frac{1}{|I|})} (\omega ) = \int _\Lambda \log \beta _n(x)\ d\mu _\mathbf{p}(x) = \int _{\Sigma _I^+\times \Lambda } \log b_n(\omega , x) \ d\hat{\mu }_\mathbf{p}(\omega , x) \end{aligned}$$

$\square$

We now show that overlap numbers of conformal IFS and of equilibrium measures on $\Sigma _I^+ \times \Lambda$, can be used to estimate the dimensions of the associated projection measures on $\Lambda$. Denote the Hausdorff dimension (for sets or measures) by HD. Recall that, in general for a measure $\mu$ on a metric space X, its Hausdorff dimension is defined by:

$$\begin{aligned} HD(\mu ):= \inf \{HD(Z), Z \subset X \ \text {with} \ \mu (X{\setminus } Z)=0\} \end{aligned}$$

In the following Theorem, we give an upper estimate for $HD(\mu _\psi )$, by estimating $HD(\Lambda {\setminus } Z(\psi ))$ for some set $Z(\psi )\subset \Lambda$ of $\mu _\psi$-measure zero with the help of the overlap number $o(\mathcal {S}, \hat{\mu }_\psi )$. Moreover, we will construct explicitly this set of $\mu _\psi$-measure zero $Z(\psi )$ below.

Theorem 3

Consider a finite conformal iterated function system $\mathcal {S} = \{\phi _i\}_{i \in I}$ with limit set $\Lambda$, $\pi : \Sigma _I^+\rightarrow \Lambda$ be the canonical projection, and let a Hölder continuous potential $\psi : \Sigma _I^+\times \Lambda \rightarrow \mathbb R$, with its (unique) equilibrium measure $\hat{\mu }_\psi$; let $\mu _\psi := \pi _{2*}\hat{\mu }_\psi$ be the projection as in (3). Then,

$$\begin{aligned} HD(\mu _\psi ) \le t(\mathcal {S}, \psi ), \end{aligned}$$

where $t(\mathcal {S}, \psi )$ is the unique zero of the pressure function with respect to the shift $\sigma : \Sigma _I^+ \rightarrow \Sigma _I^+$,

$$\begin{aligned} t \rightarrow P_\sigma (t\log |\phi _{\omega _1}'(\pi (\sigma \omega ))| - \log o(\mathcal {S}, \hat{\mu }_\psi )) \end{aligned}$$

Proof

Let denote by $R_n(\hat{\mu }_\psi , \delta )$ the set of points $(\omega , x) \in \Sigma _I^+\times \Lambda$ for which the number of generic roots satisfies $b_n((\omega , x), \tau , \hat{\mu }_\psi ) < \frac{1}{2}\cdot e^{n(F_\Phi (\hat{\mu }_\psi )-\delta )}$. We want to show that the $\hat{\mu }_\psi$-measure of these sets converges to 0, when $n\rightarrow \infty$. If this does not happen, then there exist an infinite sequence $\{k_n\}_n$ and a number $\beta >0$, such that $\hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta )) > \beta >0, \forall n \ge 1$. Then, for all pairs $(\omega , x) \in R_{k_n}(\hat{\mu }_\psi , \delta )$,

$$\begin{aligned} \frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n} < \frac{-\log 2}{k_n} + F_\Phi (\hat{\mu }_\psi ) - \delta \end{aligned}$$

Therefore, after integrating with respect to $\hat{\mu }_\psi$,

$$\begin{aligned} \int _{R_{k_n}(\hat{\mu }_\psi , \delta )} \frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n} d\hat{\mu }_\psi (\omega , x) < \hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta ))\cdot \left( F_\Phi (\hat{\mu }_\psi )-\delta -\frac{\log 2}{k_n}\right) \end{aligned}$$

We now use the last displayed inequality, and the properties of $J_{\Phi ^n}(\hat{\mu }_\psi )$ from the proof of Theorem 2 (namely relation (9)); thus by adding the integral of $\frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n}$ over $R_{k_n}$ and the integral of $\frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n}$ over the complement of $R_{k_n}$, we obtain that:

$$\begin{aligned} \begin{aligned} \int _{\Sigma _I^+\times \Lambda } \frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n} d\hat{\mu }_\psi (\omega , x)&< \hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta ))\cdot \left( F_\Phi (\hat{\mu }_\psi )-\delta -\frac{\log 2}{k_n}\right) \ \\&\quad + \int _{\Sigma _I^+\times \Lambda {\setminus } R_{k_n}(\hat{\mu }_\psi , \delta )} \frac{\log J_{\Phi ^{k_n}}(\hat{\mu }_\psi )}{k_n} d\hat{\mu }_\psi (\omega , x) \end{aligned} \end{aligned}$$

(16)

On the other hand, from the Chain rule we know that $\log J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x) = \log J_\Phi (\omega , x) + \cdots + \log J_\Phi (\hat{\mu }_\psi )(\Phi ^{n-1}(\omega , x))$, for all $n \ge 1$. Therefore from the Birkhoff Ergodic Theorem,

$$\begin{aligned} \frac{\log J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x)}{n} \mathop {\rightarrow }\limits _{n \rightarrow \infty } F_\Phi (\hat{\mu }_\psi ), \end{aligned}$$

for $\hat{\mu }_\psi$-almost all $(\omega , x) \in \Sigma _I^+\times \Lambda$. Moreover, from (9) we have that

$$\begin{aligned} J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x) \le C \cdot \frac{\mathop {\sum }\nolimits _{\Phi ^n(\eta , y) = \Phi ^n(\omega , x)} e^{S_n\psi (\eta , y)}}{e^{S_n\psi (\omega , x)}} \le C |I|^n \cdot e^{n(C_1 - C_2)}, \end{aligned}$$

(17)

for all $n \ge 1$, where $C_2 \le \psi \le C_1$ on $\Sigma _I^+\times \Lambda$ (as the potential $\psi$ is continuous). This implies that the sequence $\{\frac{1}{n} \log J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x)\}_n$ is bounded by $\log C+\log |I| + C_1 - C_1$, independently of $(\omega , x)$. Since $\log J_\Phi (\hat{\mu }_\psi )$ is integrable, we obtain then from the Birkhoff Ergodic Theorem, that $\int _{\Sigma _I^+\times \Lambda } \frac{\log J_{\Phi ^n}(\hat{\mu }_\psi )(\omega , x)}{n} \ d\hat{\mu }_\psi (\omega , x) \mathop {\rightarrow }\limits _{n \rightarrow \infty } F_\Phi (\hat{\mu }_\psi )$, and similarly,

$$\begin{aligned} \begin{aligned} \gamma _n(\hat{\mu }_\psi , \delta ):=&\int _{\Sigma _I^+\times \Lambda {\setminus } R_n(\hat{\mu }_\psi , \delta )} \left( \frac{\log J_{\Phi ^n}(\hat{\mu }_\psi )}{n} - F_\Phi (\hat{\mu }_\psi )\right) \ d\hat{\mu }_\psi (\omega , x) = \\&=\int _{\Sigma _I^+\times \Lambda } \left( \frac{\log J_{\Phi ^n}(\hat{\mu }_\psi )}{n} - F_\Phi (\hat{\mu }_\psi )\right) \cdot \chi _{\Sigma _I^+\times \Lambda {\setminus } R_n(\hat{\mu }_\psi , \delta )} d\hat{\mu }_\psi (\omega , x) \mathop {\rightarrow }\limits _{n \rightarrow \infty } 0 \end{aligned} \end{aligned}$$

Hence for any integer $n \ge 1$,

$$\begin{aligned} \int _{\Sigma _I^+ \times \Lambda \setminus R_n(\hat{\mu }_\psi , \delta )} \frac{\log J_{\Phi ^n}(\hat{\mu }_\psi )}{n} d\hat{\mu }_\psi = \gamma _n(\hat{\mu }_\psi , \delta )+ F_\Phi (\hat{\mu }_\psi )\cdot \hat{\mu }_\psi (\Sigma _I^+\times \Lambda {\setminus } R_n(\hat{\mu }_\psi , \delta )) \end{aligned}$$

Therefore, we obtain from (16) that:

$$\begin{aligned} \begin{aligned}&\int _{\Sigma _I^+\times \Lambda } \frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n} d\hat{\mu }_\psi (\omega , x) \\&\quad <\,\hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta )) \left( F_\Phi (\hat{\mu }_\psi ) -\,\delta -\frac{\log 2}{k_n}\right) + \gamma _{k_n}(\hat{\mu }_\psi , \delta )\\&\qquad +\,F_\Phi (\hat{\mu }_\psi )\cdot \hat{\mu }_\psi (\Sigma _I^+\times \Lambda {\setminus } R_{k_n}(\hat{\mu }_\psi , \delta )) \\&\quad =\,\gamma _{k_n}(\hat{\mu }_\psi , \delta ) + F_\Phi (\hat{\mu }_\psi ) - \hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta )\left( \delta + \frac{\log 2}{k_n}\right) \end{aligned} \end{aligned}$$

However if $\hat{\mu }_\psi (R_{k_n}(\hat{\mu }_\psi , \delta )) > \beta$ for $n > n(\delta )$ (for some integer $n(\delta ) \ge 1$), then it follows from the above and from the fact that: $\gamma _n(\hat{\mu }_\psi , \delta ) \rightarrow 0$, that

$$\begin{aligned} \begin{aligned}&\int _{\Sigma _I^+\times \Lambda } \frac{\log b_{k_n}((\omega , x), \tau , \hat{\mu }_\psi )}{k_n} d\hat{\mu }_\psi (\omega , x)&< F_\Phi (\hat{\mu }_\psi ) - \beta \left( \delta +\frac{\log 2}{k_n}\right) \\&+ \gamma _{k_n}(\hat{\mu }_\psi , \delta ) < F_\Phi (\hat{\mu }_\psi ) \end{aligned} \end{aligned}$$

But then, this would give contradiction with Theorem 2. Hence, for $\delta >0$ fixed there exists a sequence of positive numbers $\alpha _n \mathop {\rightarrow }\limits _{n \rightarrow \infty } 0$, such that the set $R_n(\hat{\mu }_\psi , \delta )$ of points $(\omega , x) \in \Sigma _I^+\times \Lambda$ for which $b_n((\omega , x), \tau , \hat{\mu }_\psi ) < \frac{1}{2} e^{n(F_\Phi (\hat{\mu }_\psi ) - \delta )}$, has $\hat{\mu }_\psi$-measure that satisfies:

$$\begin{aligned} \hat{\mu }_\psi (R_n(\hat{\mu }_\psi , \delta )) < \alpha _n, \ \text {for} \ n > n(\delta ) \end{aligned}$$

Let denote now the complement of the set $R_n(\hat{\mu }_\psi , \delta )$ in $\Sigma _I^+\times \Lambda$ by:

$$\begin{aligned} Q_n(\hat{\mu }_\psi , \delta ):=\Sigma _I^+ \times \Lambda {\setminus } R_n(\hat{\mu }_\psi , \delta ) \end{aligned}$$

From the $\Phi$-invariance of $\hat{\mu }_\psi$ on $\Sigma _I^+\times \Lambda$, and from the definition of $Q_n(\hat{\mu }_\psi , \delta )$, we obtain that

$$\begin{aligned} \hat{\mu }_\psi (\Phi ^n(Q_n(\hat{\mu }_\psi , \delta )) > 1-\alpha _n, \ n \ge n(\delta ) \end{aligned}$$

And from the definition of the set $\Phi ^{n}(Q_n(\hat{\mu }_\psi , \delta ))$, it follows that for any for point $(\eta ', y') \in \Phi ^n(Q_n(\hat{\mu }_\psi , \delta ))$, there exist at least $\frac{1}{2} e^{n(F_\Phi (\hat{\mu }_\psi ) - \delta )}$ indices $\underline{i}=(i_1, \ldots , i_n) \in I^n$, such that $y' \in \phi _{\underline{i}}(\Lambda )= \phi _{i_1}\circ \cdots \circ \phi _{i_n}(\Lambda )$. From above, the sequence $\hat{\mu }_\psi (R_n(\hat{\mu }_\psi , \delta ))$ converges to 0, so there exists an increasing sequence of integers $m_n \rightarrow \infty$ such that: $\hat{\mu }_\psi (R_{m_1}(\hat{\mu }_\psi , \delta )) < \frac{1}{2}, \ \hat{\mu }_\psi (R_{m_2}(\hat{\mu }_\psi )) < \frac{1}{2^2}, \ldots , \hat{\mu }_\psi (R_{m_n}(\hat{\mu }_\psi , \delta )) < \frac{1}{2^n}, \ldots .$ Employing the sequence $\{m_n\}_n$, define now the following measurable subsets of $\Lambda$,

$$\begin{aligned} \Lambda _n(\hat{\mu }_\psi , \delta ):= \pi _2\left( \mathop {\cap }\limits _{s \ge n} \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\right) , \end{aligned}$$

where $\pi _2: \Sigma _I^+\times \Lambda \rightarrow \Lambda$ is the canonical projection to the second coordinate. Moreover, denote the union of the Borel subsets in $\Lambda$ introduced above by,

$$\begin{aligned} \Lambda (\hat{\mu }_\psi , \delta ):= \mathop {\cup }\limits _{n \ge 1} \Lambda _n(\hat{\mu }_\psi , \delta ) = \pi _2\left( \mathop {\cup }\limits _{n \ge 1}\mathop {\cap }\limits _{s\ge n} \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\right) \end{aligned}$$

Firstly, notice that from the definition of the sequence of integers $\{m_n\}_{n\ge 1}$, we have

$$\begin{aligned} \hat{\mu }_\psi \left( \mathop {\cap }\limits _{s\ge n} \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\right)\ge & {} 1-\mathop {\sum }\limits _{s\ge n} \hat{\mu }_\psi \big (\Sigma _I^+\times \Lambda {\setminus } \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\big )\\\ge & {} 1 - \mathop {\sum }\limits _{s\ge n} \frac{1}{2^s} = 1- \frac{1}{2^{n-1}} \end{aligned}$$

Therefore by taking the union of these sets over all $n \ge 1$, recalling that $\mu _\psi = \pi _{2*}(\hat{\mu }_\psi )$, and observing that $\mu _\psi (\Lambda (\hat{\mu }_\psi , \delta )) = \hat{\mu }_\psi \big (\pi _2^{-1}(\Lambda (\hat{\mu }_\psi , \delta ))\big )\ge \hat{\mu }_\psi \big (\mathop {\cup }\limits _{n\ge 1} \mathop {\cap }\limits _{s\ge n} \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\big )$, we obtain that

$$\begin{aligned} \hat{\mu }_\psi \left( \mathop {\cup }\limits _{n\ge 1} \mathop {\cap }\limits _{s\ge n} \Phi ^{m_s}(Q_{m_s}(\hat{\mu }_\psi , \delta ))\right) = 1, \ \text {hence} \ \ \mu _\psi (\Lambda (\hat{\mu }_\psi , \delta )) = 1 \end{aligned}$$

(18)

We now investigate the influence of the number of roots on the Hausdorff dimension of the set $\Lambda (\hat{\mu }_\psi , \delta )$. Recall from above that, for any $(\eta ', y') \in \Phi ^n(Q_n(\hat{\mu }_\psi , \delta ))$, there exist at least $\frac{1}{2} e^{n(F_\Phi (\hat{\mu }_\psi ) - \delta )}$ indices $\underline{i}=(i_1, \ldots , i_n) \in I^n$, such that $y' \in \phi _{\underline{i}}(\Lambda )= \phi _{i_1}\circ \cdots \circ \phi _{i_n}(\Lambda )$. Hence the points in the projection $\pi _2(\Phi ^n(Q_n(\hat{\mu }_\psi , \delta )))$ are covered at least $\frac{1}{2} e^{n(F_\Phi (\hat{\mu }_\psi ) - \delta )}$ times by images of $\Lambda$, through compositions of n maps of type $\phi _i$. Now, $\mathcal {S}$ satisfies the condition that there exists $\kappa \in (0, 1)$ such that $|\phi _i'| < \kappa$ on $\Lambda$. It follows that, for any indices $i_1, \ldots , i_n \in I$, $\text {diam}(\phi _{i_1} \circ \cdots \circ \phi _{i_n}(\Lambda )) \le \kappa ^n$. Thus, every point in $\pi _2(\Phi ^n(Q_n(\hat{\mu }_\psi , \delta )))$ can be covered at least $\frac{1}{2} e^{n(F_\Phi (\hat{\mu }_\psi ) - \delta )}$ times with sets of diameter less than $\kappa ^n$. For $\alpha \ge 0$, let us denote now by $t(\alpha )$ the unique zero of the following pressure function with respect to the shift map $\sigma : \Sigma _I^+ \rightarrow \Sigma _I^+$,

$$\begin{aligned} t \rightarrow P_\sigma (t|\phi _{\omega _1}'(\sigma \omega )| - \alpha ) \end{aligned}$$

(19)

Take an arbitrary number $t' > t(F_\Phi (\hat{\mu }_\psi ) - \delta )$; we assume without loss of generality that $F_\Phi (\hat{\mu }_\psi ) > 0$ and that $\delta$ is small enough, so that $\delta < F_\Phi (\hat{\mu }_\psi )$. Let define the pressure function

$$\begin{aligned} p_\delta (s):= P( s|\phi _{\omega _1}'(\sigma \omega )| - F_\Phi (\hat{\mu }_\psi ) + \delta ), \ s \in \mathbb R \end{aligned}$$

From assumption above, $p_\delta (t') < 0$. So from the conformality of the contractions $\phi _i$, and by denoting in general $\phi _\eta := \phi _{\eta _1} \circ \cdots \circ \phi _{\eta _m}$ for $\eta = (\eta _1, \ldots , \eta _m) \in I^m, m \ge 1$, it follows that for n large:

$$\begin{aligned} \mathop {\sum }\limits _{|\omega | = n} |\phi _{\omega }'|^{t'} e^{-n(F_\Phi (\hat{\mu }_\psi ) - \delta )} \le e^{\frac{n\cdot p_\delta (t')}{4} } \end{aligned}$$

(20)

Now for any $s \ge n$, from the above definition of $Q_{m_s}(\hat{\mu }_\psi , \delta )$, it follows that any point in $\Lambda _n(\hat{\mu }_\psi , \delta )$ can be covered with at least $M_s:= \frac{1}{2} e^{m_s(F_\Phi (\hat{\mu }_\psi ) - \delta )}$ sets $\phi _\eta (V)$ for $|\eta | = m_s$, and every one of these sets $\phi _\eta (V)$ has diameter less than $\kappa ^{m_s}$. Denote the collection of the above sets $\phi _\eta (V)$ by $\mathcal {U}_s$, so $\mathcal {U}_s$ is a cover of $\Lambda _n(\hat{\mu }_\psi , \delta )$. We want now to perform extractions from this cover $\mathcal {U}_s$ of $\Lambda _n(\hat{\mu }_\psi , \delta )$ (by using its large multiplicity), in such a way that in the end we obtain a subcover which is minimal, from the point of view of the sum of diameters raised to power t. This will be the subcover which we shall use to estimate the Hausdorff dimension of the set $\Lambda _n(\hat{\mu }_\psi , \delta )$. We have that the maps $\phi _\eta$ are conformal, so we can apply the 5r-Covering Theorem (see [8]), where we consider 5U to denote the ball with the same center as U and 5 times the radius of U. One can then extract a subfamily $\mathcal {U}_s(1) \subset \mathcal {U}_s$, such that the sets $5U, U \in \mathcal {U}_s(1)$, cover $\Lambda _n(\hat{\mu }_\psi , \delta )$, and so that the sets in $\mathcal {U}_s(1)$ are mutually disjoint. From conformality we have that there exists x, r and a fixed constant C independent of U, such that $B(x, r) \subset U \subset B(x, Cr)$. We then eliminate this subfamily $\mathcal {U}_s(1)$. Since it was disjointed,   the multiplicity of the cover $\mathcal {U}_s$ of $\Lambda _n(\hat{\mu }_\psi , \delta )$ is still at least $M_s -1$. Therefore we can repeat this procedure and will extract a second subfamily $\mathcal {U}_s(2)$ in $\mathcal {U}_s {\setminus } \mathcal {U}_s(1)$, which is disjointed and such that $5U, U \in \mathcal {U}_s(2)$ cover the set $\Lambda _n(\hat{\mu }_\psi , \delta )$. After eliminating both $\mathcal {U}_s(1)$ and $\mathcal {U}_s(2)$ from $\mathcal {U}_s$, the multiplicity of the cover is at least $M_s - 2$. By induction, we obtain thus $M_s$ subfamilies $\mathcal {U}_s(j)$, which are disjointed and such that $5U, U \in \mathcal {U}_s(j)$, cover $\Lambda _n(\hat{\mu }_\psi , \delta )$. We then take, out of these subfamilies constructed above, the subfamily $\mathcal {U}_s(j_0)$ for which the expression $\mathop {\sum }\limits _{U \in \mathcal {U}_s(j_0)} (\text {diam} U)^{t'}$ is minimal. Then from (20), we obtain:

$$\begin{aligned} \mathop {\sum }\limits _{U \in \mathcal {U}_s(j_0)} (\text {diam} U)^{t'} \le \frac{1}{M_s} \mathop {\sum }\limits _{U \in \mathcal {U}_s} (\text {diam} U)^{t'} \le C e^{m_sp_\delta (t')/4} < 1, \end{aligned}$$

(21)

for some constant $C>0$, independent of s, n large. Since for any $s \ge n$, we can obtain such minimal covers $\mathcal {U}_s(j_0)$ for the set $\Lambda _n(\hat{\mu }_\psi , \delta )$ , and since $t'$ was chosen arbitrarily larger than $t(F_\Phi (\hat{\mu }_\psi )- \delta )$, it follows from (21) that:

$$\begin{aligned} HD(\Lambda _n(\hat{\mu }_\psi , \delta )) \le t(F_\Phi (\hat{\mu }_\psi ) - \delta ) \end{aligned}$$

Now recall the definition of $\Lambda (\hat{\mu }_\psi , \delta ) = \mathop {\cup }\limits _{n \ge 1} \Lambda _n(\hat{\mu }_\psi , \delta )$. From the last estimate, we infer that

$$\begin{aligned} HD(\Lambda (\hat{\mu }_\psi , \delta )) \le t(F_\Phi (\hat{\mu }_\psi ) - \delta ) \end{aligned}$$

Also from (18), $\mu _\psi (\Lambda (\hat{\mu }_\psi , \delta )) =1$. Define now the set $\Lambda (\psi ):= \mathop {\cap }\limits _{\delta >0} \Lambda (\hat{\mu }_\psi , \delta ) = \mathop {\cap }\limits _{n \ge 1} \Lambda (\hat{\mu }_\psi , \frac{1}{n})$. We have then that $\mu _\psi (\Lambda (\psi )) = 1$. Let us now remark that from definition (19) of the zero $t(\alpha )$, and from the continuity of the pressure function, we obtain that $t(F_\Phi (\hat{\mu }_\psi ) - \delta ) \rightarrow t(F_\Phi (\hat{\mu }_\psi ))$ when $\delta \rightarrow 0$. But from Theorem 2, we know that $\log o(\mathcal {S}, \psi ) = F_\Phi (\hat{\mu }_\psi )$. Hence, by taking the set $Z(\psi ):= \Lambda {\setminus } \Lambda (\psi ),$ we have $\mu _\psi (Z(\psi )) = 0$; thus from the definition of $HD(\mu _\psi )$, $HD(\mu _\psi ) \le HD(\Lambda {\setminus } Z(\psi )) \le t(\mathcal {S}, \psi )$. $\square$

3 Applications to Bernoulli Convolutions

Consider the random series $\mathop {\sum }\limits _{n \ge 0} \pm \lambda ^n$ for $\lambda \in (0, 1)$ where the $+, -$ signs are taken independently and with equal probability, and let us denote its distribution by $\nu _\lambda$. This is called a Bernoulli convolution, since it is in fact the infinite convolution of the atomic measures $\frac{1}{2}(\delta _{-\lambda ^n}+ \delta _{\lambda ^n})$, for $n \ge 0$ (for e.g. [3, 21]). The probability measure $\nu _\lambda$ can be written also as the self-similar measure associated to the probability vector $(\frac{1}{2}, \frac{1}{2})$ and to the iterated function system

$$\begin{aligned} \mathcal {S}_\lambda = \{S_{1}, S_{2}\}, \end{aligned}$$

where $S_{1}(x) = \lambda x-1, \ S_{2}(x) = \lambda x+1, \ x \in \mathbb R$. Hence, $\nu _\lambda$ satisfies the self-similarity relation:

$$\begin{aligned} \nu _\lambda = \frac{1}{2} \nu _\lambda \circ S_1^{-1} + \frac{1}{2} \nu _\lambda \circ S_2^{-1} \end{aligned}$$

The case $\lambda \in (0, \frac{1}{2})$ corresponds to $\mathcal {S}_\lambda$ having no overlaps, while the case when $\lambda \in [\frac{1}{2}, 1)$ corresponds to the more difficult situation of the iterated function system $\mathcal {S}_\lambda$ having overlaps. We assume in the sequel that $\lambda \in (\frac{1}{2}, 1)$, thus we are in the case when $\mathcal {S}_\lambda$ has overlaps. The limit set $\Lambda _\lambda$ is in this case the whole interval $I_\lambda = [-\frac{1}{1-\lambda }, \frac{1}{1-\lambda }]$. The measure $\nu _\lambda$ can be viewed as the projection $\pi _{\lambda *}\nu _{(\frac{1}{2}, \frac{1}{2})}$, where $\nu _{(\frac{1}{2}, \frac{1}{2})}$ is the Bernoulli measure on $\Sigma _2^+$ generated by the vector $(\frac{1}{2}, \frac{1}{2})$, and $\pi _\lambda : \Sigma _2^+ \rightarrow I_\lambda$ is the canonical coding map. It is well-known that $\nu _\lambda$ can be either singular or absolutely continuous. Several results on Bernoulli convolutions are in the paper by Peres, Schlag and Solomyak [15]. The case $\lambda > \frac{1}{2}$ attracted a lot of interest, starting with Erdös who proved in [3] that, when $\frac{1}{\lambda }$ is a Pisot number, then $\nu _\lambda$ is singular. Later Solomyak showed in [21] that the measure $\nu _\lambda$ is absolutely continuous for Lebesgue-a.e $\lambda \in [\frac{1}{2}, 1)$. If $\nu _\lambda$ is absolutely continuous, then $HD(\nu _\lambda ) = 1$. From the point of view of actual values of $\lambda$, Garsia proved in [5] that $\nu _\lambda$ is absolutely continuous when $\lambda ^{-1}$ is an algebraic integer in (1, 2), whose monic polynomial has other roots outside the unit circle and constant coefficient $\pm 2$; for example if $\lambda ^{-1} = 2^{\frac{1}{m}}, \ m \ge 2$, $\nu _\lambda$ is absolutely continuous. Przytycki and Urbański proved in [17] that, if $\lambda$ is the inverse of a Pisot number in (1, 2), then $HD(\nu _\lambda ) < 1$. In the special case when $\lambda = \frac{\sqrt{5} - 1}{2}$ (the reciprocal of the Pisot number $\frac{\sqrt{5} +1}{2}$), Alexander and Zagier [1] found precise estimates for $HD(\nu _\lambda )$, showing that $0.99557 < HD(\nu _\lambda ) < 0.99574$. Hochman showed recently in [6] that $HD(\nu _\lambda ) = 1$ for $\lambda$ outside a set of dimension zero.

For arbitrary $\lambda \in (\frac{1}{2}, 1)$, Theorem 4 below gives an upper estimate for $HD(\nu _\lambda )$, by using an expression involving $o(\mathcal {S}_\lambda )$; this allows to obtain bounds also for the overlap numbers $o(\mathcal {S}_\lambda )$. In particular, if $HD(\nu _\lambda ) = 1$ for some value $\lambda \in (\frac{1}{2}, 1)$, then $o(\mathcal {S}_\lambda ) \le 2\lambda$. In general, $1 \le o(\mathcal {S}_\lambda ) \le 2$, for any $\lambda \in (\frac{1}{2}, 1)$; we show that in fact, the overlap number $o(\mathcal {S}_\lambda )$ is never equal to 2 (even if, for $\lambda \rightarrow 1$ the overlaps become larger). For specific values of $\lambda$ (for e.g. $\lambda = 2^{-\frac{1}{m}}, m \ge 2$, or $\lambda = \frac{\sqrt{5} -1}{2}$), we obtain then more precise bounds for $o(\mathcal {S}_\lambda )$. First, for arbitrary $\lambda \in (\frac{1}{2}, 1)$, the measure $\nu _\lambda$ is supported on the limit set of $\mathcal {S}_\lambda$, which is the interval $I_\lambda = [-\frac{1}{1-\lambda }, \frac{1}{1-\lambda }]$; the coding map is $\pi _\lambda : \Sigma _2^+ \rightarrow I_\lambda$. Recall that for $x \in I_\lambda$ and $n\ge 2$, $\beta _n(x)$ denotes the number of n-chains $(\zeta _1, \ldots , \zeta _n) \in \{1, 2\}^n$ from points in $I_\lambda$ to x, i.e. $x \in \phi _{\zeta _1\ldots \zeta _n}\big ([-\frac{1}{1-\lambda }, \frac{1}{1-\lambda }]\big )$. From Corollary 2, in the formula for $o(\mathcal {S}_\lambda )$ we integrate $\log \beta _n$ with respect to the uniform Bernoulli measure $\nu _{(\frac{1}{2}, \frac{1}{2})}$.

Theorem 4

For all $\lambda \in (\frac{1}{2}, 1)$, the following relation is satisfied for the Bernoulli convolution $\nu _\lambda$:

$$\begin{aligned} HD(\nu _\lambda ) \ \le \ \frac{\log \frac{2}{o(\mathcal {S}_\lambda )}}{|\log \lambda |}, \end{aligned}$$

where $o(\mathcal {S}_\lambda )$ denotes the overlap number of $\mathcal {S}_\lambda$, which can be computed as:

$$\begin{aligned} o(\mathcal {S}_\lambda ) = \exp \left( \mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n} \int _{\Sigma _2^+} \log \beta _n(\pi _\lambda \omega ) \ d\nu _{(\frac{1}{2}, \frac{1}{2})}(\omega )\right) \end{aligned}$$

And from the above, $o(\mathcal {S}_\lambda ) \le 2 \lambda ^{HD(\nu _\lambda )}$.

Proof

From Theorem 1, in our case the measure $\nu _\lambda$ can be written as $\pi _{\lambda *}\nu _{(\frac{1}{2}, \frac{1}{2})}$ and it is equal to the $\pi _2$-projection of an equilibrium state $\hat{\mu }_\psi$ on $\Sigma _2^+ \times I_\lambda$. Therefore, from Corollary 2,

$$\begin{aligned} o(\mathcal {S}_\lambda ) = \exp \left( \mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n} \int _{\Sigma _2^+} \log \beta _n(\pi _\lambda \omega ) \ d\nu _{(\frac{1}{2}, \frac{1}{2})}(\omega )\right) \end{aligned}$$

$\mathcal {S}_\lambda$ is a system of similarities, thus from Theorem 3, $HD(\nu _\lambda )$ is bounded above by the unique zero of the pressure function with respect to $\sigma : \Sigma _2^+ \rightarrow \Sigma _2^+$:

$$\begin{aligned} t \rightarrow P_\sigma (t \log \lambda - o(\mathcal {S}_\lambda )) = t \log \lambda + \log 2 - \log o(\mathcal {S}_\lambda ) \end{aligned}$$

Hence it follows that $HD(\nu _\lambda ) \le \ \frac{\log \frac{2}{o(\mathcal {S}_\lambda )}}{|\log \lambda |}$, and the corresponding bound for $o(\mathcal {S}_\lambda )$. $\square$

For any $\lambda \in (\frac{1}{2}, 1)$, the number of overlaps between images $S_{i_1\ldots i_n}(I_\lambda )$ is less than $2^n$, so $1 \le o(\mathcal {S}_\lambda ) \le 2$. In fact, it turns out that the overlap number of $\mathcal {S}_\lambda$ is always strictly less than 2:

Corollary 3

In the above setting, it follows that for all parameters $\lambda \in (\frac{1}{2}, 1)$,

$$\begin{aligned} o(\mathcal {S}_\lambda ) < 2 \end{aligned}$$

Proof

If $o(\mathcal {S}_\lambda ) = 2$, then from Theorem 4, it would follow that $\lambda = 1$. Hence contradiction.

For a large set of values of $\lambda$, by using Theorem 4 and the above mentioned results of [1, 5, 21], we can obtain more precise estimates for the overlap number:

Corollary 4

• (a) For Lebesgue-almost all parameters $\lambda$ in $(\frac{1}{2}, 1)$, we have

  $$\begin{aligned} o(\mathcal {S}_\lambda ) \le 2\lambda \end{aligned}$$

  This happens for example when $\lambda ^{-1}$ is an algebraic number whose monic polynomial has other roots outside the unit circle and constant coefficient $\pm 2$. In particular, if $\lambda = 2^{-\frac{1}{m}}$ for $m \ge 2$, then

  $$\begin{aligned} o(\mathcal {S}_\lambda ) \le 2^{\frac{m-1}{m}} \end{aligned}$$

• (b) In case $\lambda = \frac{\sqrt{5} -1}{2}$, then $o(\mathcal {S}_\lambda ) \le 2 \lambda ^{0.99557} < 1.25$.

Let now p arbitrary in (0, 1) and denote by $\nu _{(p, 1-p)}$ the Bernoulli measure on $\Sigma _2^+$ determined by the vector $(p, 1-p)$. For $\lambda \in (\frac{1}{2}, 1)$, one defines the biased Bernoulli convolution $\nu _{\lambda , p}$ (see for e.g. [16]), where $\nu _{\lambda , p}$ is the $\pi _\lambda$-projection of $\nu _{(p, 1-p)}$ onto the limit set $I_\lambda = [-\frac{1}{1-\lambda }, \frac{1}{1-\lambda }]$. We have as above the associated lift map $\Phi _\lambda : \Sigma _2^+\times I_\lambda \rightarrow \Sigma _2^+\times I_\lambda$. From the discussion before Theorem 1, there exists a $\Phi _\lambda$-invariant equilibrium measure $\hat{\nu }_{\lambda , p}$ on $\Sigma _2^+\times I_\lambda$, such that $\pi _{2*}\hat{\nu }_{\lambda , p} = \nu _{\lambda , p}$. For integers $0 < k < n$, denote by W(x, n, k) the set of n-chains $(i_1, \ldots , i_n) \in \{1, 2\}^n$ from points in $I_\lambda$ to x, having exactly k indices $i_j$ equal to 1. From (15), for any $x\in I_\lambda$, $\tau >0$ and $n \ge 2$, we have

$$\begin{aligned} \beta _n\left( x, \tau \left| \log \frac{p}{1-p}\right| , (p, 1-p)\right) = \mathop {\sum }\limits _{k, \ |\frac{k}{n} - p| < \tau }\text {Card} \ W(x, n, k) \end{aligned}$$

Thus, for any parameter $\lambda \in (\frac{1}{2}, 1)$, it follows from Theorem 3 and Corollary 2 that:

Corollary 5

For all $\lambda \in (\frac{1}{2}, 1)$ and $p \in (0, 1)$, the biased Bernoulli convolution $\nu _{\lambda , p}$ satisfies:

$$\begin{aligned} HD(\nu _{\lambda , p}) \ \le \ \frac{\log \frac{2}{o(\mathcal {S}_{\lambda }, \hat{\nu }_{\lambda , p})}}{|\log \lambda |}, \end{aligned}$$

where $o(\mathcal {S}_\lambda , \hat{\nu }_{\lambda , p})$ denotes the overlap number of $\mathcal {S}_\lambda$ with respect to $\hat{\nu }_{\lambda , p}$, which can be computed by:

$$\begin{aligned} o(\mathcal {S}_\lambda , \hat{\nu }_{\lambda , p}) = \exp \left( \mathop {\lim }\limits _{\tau \rightarrow 0} \mathop {\lim }\limits _{n \rightarrow \infty } \frac{1}{n} \int _{\Sigma _2^+} \log \mathop {\sum }\limits _{|\frac{k}{n} - p| < \tau } \text {Card} \ W(\pi _\lambda \omega , n, k) \ d\nu _{(p, 1-p)}(\omega )\right) . \end{aligned}$$