Renormalization of Symmetric Bimodal Maps with Low Smoothness

Abstract

This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space $C ^{1+Lip}$ symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of $C^{1+Lip}$ symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps.

1 Introduction

Renormalization is a technique to analyze maps having the property that the first return map to a small part of the phase space resembles the original map itself. Period doubling renormalization operator was introduced by Feigenbaum [1, 2] and by Coullet and Tresser [3], to study asymptotic small scale geometry of the attractor of one dimensional systems which are at the transition from simple to chaotic dynamics. Renormalization is a method to study microscopic geometrical properties of attractors. The geometric rigidity of the attractors is the center of attention in one dimensional theory. The smoothness of the maps plays a crucial role for rigidity. From the last four decades, a lot of mathematical theory have been developed for the renormalization theory in low-dimensional dynamics. Especially, Sullivan [4] showed the convergence of renormalizations. Moreover, all limits of renormalization are quadratic-like maps with a definite modulus. Hu [5] proved that the real polynomial map having the periodic points of all power of 2 is infinitely renormalizable. Further, McMullen [6] proved the exponential convergence towards the limit set of renormalization. Also, Martens [7] showed that the renormalization operator acting on the space of smooth unimodal maps with critical exponent greater than one, has periodic points of any combinatorial type. Lyubich considered the renormalization with bounded combinatorics in [8]. Then, the hyperbolicity of the renormalization fixed point in the space of $C^{2+\alpha }\, (\alpha > 0)$ unimodal maps, was shown by Davie [9]. Using the results of [8]; de Faria et al. [10] extended to more general types of renormalization in the space $C^r,$ provided $r \ge 2+\alpha$ with $\alpha$ close to one. Later, Chandramouli et al. [11], proved that the period doubling renormalization converges to the analytic generic fixed point proving it to be globally unique in a class $C^{2+|\cdot |}$ which is bigger than $C^{2+\alpha }$ (for any positive $\alpha \le 1$). Furthermore, they showed that the uniqueness is lost below $C^2$ space and other asymptotic behavior encountered. Recently, Kozlovski and van Strien [12] proved the existence of a period doubling infinitely renormalizable asymmetric unimodal map with non universal scaling laws.

In the context of circle diffeomorphisms, Herman [13] proved the rigidity result, using real variable techniques. Yoccoz [14] proved the other fundamental rigidity results by using conformal surgery, where Herman’s theorem holds in the real-analytic category. Further, Khanin and Sinai [15] gave a proof of Herman’s theorem which is based on the thermodynamic formalism and ergodic properties for the corresponding random variables. Later, Yampolsky [16] proved the rigidity of circle map with a critical point. Furthermore, the rigidity theory for circle maps with break type singularities have been developed by Khanin et al. [17,18,19,20], Cunha and Smania [21], Akhadkulov et al. [22]. In the context of interval maps, the rigidity phenomena is understood for $C^{2+\alpha } \, (\alpha >0)$ smooth maps. Further, de Melo and Pinto [23] proved the rigidity of $C^2$ infinitely renormalizable unimodal maps with bounded combinatorial type. The measure-theoretical properties of real family of unimodal maps are studied by Lyubich et al. [24] proved that almost any real quadratic map has either an attracting cycle or an absolutely continuous invariant measure. Further, Avila et al. [25] extended these result for any non-trivial real analytic family of quasiquadratic maps. Bruin et al. [26] showed that almost every unicritical polynomial with even critical order greater than or equal to 2, admits a physical measure, which is either supported on an attracting periodic orbit, or is absolutely continuous, or is supported on the postcritical set. Further, Moreira and Smania [27] showed the rigidity of infinitely renormalizable Fibonacci unimodal maps with even critical order and having negative Schwarzian derivative. Bruin and Todd [28] proved the existence of wild attractor for a countably piecewise linear infinitely renormalizable Fibonacci unimodal map with infinite critical order.

In the context of two dimensional maps, de Carvalho et al. [29] showed the non-rigidity of Cantor attractors of Hénon-like maps. Further, Hazard et al. [30] discussed the unbounded geometry of Cantor attractor of strongly dissipative infinitely renormalizable Hénon-like map with stationary combinatorics. In case of Lorenz maps, Martens and Winckler [31, 32] studied the hyperbolicity of Lorenz renormalization and also proved the non-existence of physical measures for Lorenz maps which are infinitely renormalizable.

With a relatively complete understanding of the period doubling renormalization of unimodal maps, recent research in dynamical systems has either focused on more complicated maps of the real line or other low dimensional maps. Jonker and Rand [33], and van Strien [34] used renormalization as a natural vehicle to decompose the non-wandering set in a hierarchical manner, for unimodal maps. The multimodal maps are interesting as generalizations of unimodal maps, as well as for their applications. For example, in the case of bimodal maps, they are essential to understand the non-invertible circle maps which have been used extensively to model the transitions to chaos in two frequency systems [35]. Mackay and van Zeijts [36] explained the period doubling renormalization of two parameter families of bimodal maps in the term of a horseshoe with a Cantor set of two dimensional unstable manifold. Also, they calculated the periodic points of renormalization up to period five. Veitch [37] presented some work on topological renormalization of $C^0$ bimodal maps with zero and positive entropy. Further, Smania developed a combinatorial theory for certain kind of multimodal maps and proved that for the same combinatorial type the renormalizations of infinitely renormalizable smooth multimodal maps are exponentially close [38, 39]. Later, Smania [40] proved the hyperbolicity of renormalization for real analytic multimodal maps with bounded combinatorics.

In this work, we focus on the construction of renormalization fixed point for the family of symmetric bimodal maps with low smoothness (i.e., below $C^2$ space). First, we show that there exists a sequence of affine pieces which are nested and shrinking down to the critical points of the bimodal map corresponding to a pair of proper scaling data $s^* = (s_l^*, s_r^*)$. This helps us to show the existence of a fixed point $f_{s^*}$ of the renormalization operator defined on the space of piece-wise affine infinitely renormalizable maps, which is denoted by W,  corresponding to a pair of proper scaling data $s^*$. This gives us the following result.

Theorem 1

There exists a map $f_{s^*} \in W,$ where $s^* = (s_l^*, s_r^*)$ is characterized by

$$\begin{aligned} Rf_{s^*} = f_{s^*}. \end{aligned}$$

In particular, $W = \{f_{s^*}\}.$

Here, the renormalization operator R is a pair of period tripling renormalization operators $R^l$ and $R^r$ which are defined on piecewise affine period tripling infinitely renormalizable maps corresponding to a proper scaling data $s_l$ and $s_r$, respectively.

The proof of theorem 1 mainly relies on the Propositions 1 and 2 presented in Sect. 2. We use Mathematica for some computational work to prove these propositions.

In the next Sect. 3, we explain the extension of the renormalization fixed point $f_{s^*}$ to a $C^{1+Lip}$ symmetric bimodal map $g_{s^*}$. Then, we have the following theorem,

Theorem 2

There exists an infinitely renormalizable $C^{1+Lip}$ symmetric bimodal map $g_{s^*},$ which is not $C^2$ map, such that

$$\begin{aligned} R g_{s^*} = g_{s^*} . \end{aligned}$$

In Sect. 4, we describe the topological entropy of renormalization defined on the space of $C^{1+Lip}$ symmetric bimodal maps. Then we obtain the following theorem,

Theorem 3

The renormalization operator R acting on the space of $C^{1+Lip}$ symmetric bimodal maps has unbounded topological entropy.

Furthermore, we discuss the existence of another fixed point of renormalization by considering the small perturbation on the scaling data. Then, we get the following result,

Theorem 4

There exists a continuum of fixed points of the renormalization operator acting on $C^{1+Lip}$ symmetric bimodal maps.

Consequently, this result leads to the non-rigidity of the Cantor attractors of infinitely renormalizable symmetric bimodal maps, whose smoothness is below $C^2$.

We recall some basic definitions. Let $I = [0, 1]$ be a closed interval.

A point $c \in I$ is said to be a critical point of a $C^1$ map $u : I \rightarrow I$ if $Du(c)=0$.

The critical point c is called non-flat critical point of order k if u is $C^{k+1}$ in a neighborhood of c and $Du(c) = D^2u(c) = \cdots = D^{k-1}u(c) = 0$ and $D^ku(c) \ne 0.$

Note that $D^ ku(c)$ stands for $k^{th}-$derivative of u at c.

A unimodal map ${\mathfrak {u}}: I \rightarrow I ,$ which is a $C ^1$ map having a unique non-flat critical point c, is called period tripling renormalizable map if there exists a proper subinterval $J \subset I$ with $c \in J$ such that

(1) J,  ${\mathfrak {u}}(J)$ and ${\mathfrak {u}}^2(J)$ are pairwise disjoint,

(2) ${\mathfrak {u}}^3(J) \subset J.$

Then ${\mathfrak {u}}^3: J \rightarrow J$ is called a pre-renormalization of ${\mathfrak {u}}.$

Where, $u^n$ denotes n fold composition of u with itself.

Let ${\mathscr {U}}$ be the collection of unimodal maps and ${\mathscr {U}}_\infty ( \subset {\mathscr {U}})$ be the collection of period tripling infinitely renormalizable unimodal maps.

An interval map f is piece-wise monotone if there exists a partition of I into finitely many subintervals on each of which the restriction of f is continuous and strictly monotonic.

A map f is called a bimodal map if three is the minimal number of such subintervals.

Definition 1

Let $f : I \rightarrow I$ be a map with two subsets $J_l$ and $J_r$ such that If $f|_{J_l}$ and $f|_{J_r}$ are unimodal maps which are concave up and concave down respectively, their join, denoted by $f|_{J_l} \oplus f|_{J_r}$, is a bimodal map whose graph is obtained by joining $\big (max(J_l), f(max(J_l))\big )$ and $\big (min(J_r), f(min(J_r))\big )$ by a $C^{1+Lip}$ curve.

The notation stands for the interior of I.

Definition 2

A bimodal map $b : I \rightarrow I ,$ is a $C^1$ map having two critical points $c_l$ and $c_r,$ which is said to be renormalizable if there exists two disjoint intervals $I_l$ containing $c_l$ and $I_r$ containing $c_r$ such that

• (i) $b^i(I_l) \cap b^j(I_l) = \emptyset ,$ for each $i \ne j$ and $i,j \in \{0,1,2\},$

  $b^i(I_r) \cap b^j(I_r) = \emptyset ,$ for each $i \ne j$ and $i,j \in \{0,1,2\},$

• (ii) $b^3(I_l) \subset I_l$ and $b^3(I_r) \subset I_r,$

• (iii) The unimodal maps ${\hat{b}}_{l} :[0,\, b(0)] \rightarrow [0, \,b(0)]$ and ${\hat{b}}_{r} : [b(1), 1] \rightarrow [b(1), 1]$ are joined to generate a bimodal map ${\hat{b}}_{l} \oplus {\hat{b}}_{r}$. The unimodal maps ${\hat{b}}_{l}$ and ${\hat{b}}_{r}$ are defined as

  $$\begin{aligned} {\hat{b}}_{l}(x) = h_1^{-1}b^3h_1(x) \end{aligned}$$

  and

  $$\begin{aligned} {\hat{b}}_{r}(x) = h_2^{-1}b^3h_2(x) \end{aligned}$$

  where $h_1 : [0, \, b(0)] \rightarrow I_l$ and $h_2 : [b(1), 1] \rightarrow I_r$ are the affine orientation reversing homeomorphisms.

The renormalization of a bimodal map is illustrated in Fig. 1.

Fig. 1

Renormalization of a bimodal map

In the next section, we construct the renormalization operator defined on the space of piece-wise affine maps which are infinitely renormalizable maps.

2 Piece-Wise Affine Renormalizable Maps

A symmetric bimodal map $b: [0,1] \rightarrow [0,1]$ of the form $b(x) = a_3x^3+a_2 x^2+a_1x+a_0,$ for $a_3<0,$ is a $C^1$ map with the following conditions

• $b(0)= 1-b(1),$

• $b(\frac{1}{2})= \frac{1}{2},$

• let $c_l$ and $c_r$ be the two critical points of b(x) , then $b(c_l) = 0$ and $b(c_r)=1.$

Let us consider a one parameter family of symmetric bimodal maps ${\mathscr {B}}_c : [0, 1] \rightarrow [0, 1]$ which are increasing on the interval between the critical points and decreasing elsewhere. then, we obtained a family of bimodal maps as

$$\begin{aligned} {\mathscr {B}}_c(x)&= \left\{ \begin{array}{ll} 1-\frac{1 - 6 c + 9 c^2 - 4 c^3 + 6 c x - 6 c^2 x - 3 x^2 + 2 x^3}{(1 - 2 c)^3}, \,\,\, \, \, {\text {if}} \, c \in \left[ 0, \frac{1}{4}\right] \\ 1-\frac{4c^3 - 3 c^2 + 6 c x - 6 c^2 x - 3 x^2 + 2 x^3}{( 2 c-1)^3}, \,\,\, \, \,\,\,\,\,\,\, \, {\text {if}} \, c \in \left[ \frac{3}{4}, 1\right] \end{array} \right. \nonumber \\&\equiv \left\{ \begin{array}{ll} b_{c}(x), \,\,\, \, \, {\text {if}} \, c \in \left[ 0, \frac{1}{4}\right] \\ {\tilde{b}}_{c}(x), \,\,\, \, \, {\text {if}} \, c \in \left[ \frac{3}{4}, 1\right] \end{array} \right. \end{aligned}$$

(1)

Note that the bimodal maps $b_c$ and ${\tilde{b}}_c$ are identical maps.

Let us define an open set

$$\begin{aligned} \varDelta ^3 = \left\{ (s_0,s_1,s_2) \in {\mathbb {R}}^3 \, : \, s_0,s_1,s_2 > 0, \, \sum \limits _{i = 0}^{2} s_i < 1 \right\} . \end{aligned}$$

Each element $(s_0,s_1,s_2)$ of $\varDelta ^3$ is called a scaling tri-factor. A pair of scaling tri-factors $(s_{0,l},s_{1,l},s_{2,l})$ and $(s_{0,r},s_{1,r},s_{2,r})$ induces two sets of affine maps $(F_{0,l},F_{1,l},F_{2,l})$ and $(F_{0,r},F_{1,r},F_{2,r})$ respectively. For each $i = 0,1,2,$

$$\begin{aligned} F_{i,l} : I_L = [0 , b_c(0)] \longrightarrow I_L \end{aligned}$$

are defined as

$$\begin{aligned} F_{0,l}(t)&= b_c(0) - s_{0,l} \cdot t, \\ F_{1,l}(t)&= b_c^2(0) - s_{1,l} \cdot t, \\ F_{2,l}(t)&= s_{2,l} \cdot t \end{aligned}$$

and

$$\begin{aligned} F_{i,r} : I_R = [ {\tilde{b}}_c(1), 1] \longrightarrow I_R \end{aligned}$$

are defined as

$$\begin{aligned} F_{0,r}(t)&= {\tilde{b}}_c(1) + s_{0,r} \cdot (1-t), \\ F_{1,r}(t)&= {\tilde{b}}_c^2(1) + s_{1,r} \cdot (1-t), \\ F_{2,r}(t)&= 1-s_{2,r} \cdot (1-t). \end{aligned}$$

Note that for $c \in [0,\frac{3-\sqrt{3}}{6}].$

The functions $s_l : {\mathbb {N}} \rightarrow \varDelta ^3$ and $s_r : {\mathbb {N}} \rightarrow \varDelta ^3$ are said to be a scaling data. We set scaling tri-factors

$s_l(n) = (s_{0,l}(n),s_{1,l}(n), s_{2,l}(n)) \in \varDelta ^3$ and $s_r(n) = (s_{0,r}(n),s_{1,r}(n), s_{2,r}(n)) \in \varDelta ^3,$

so that $s_l(n)$ and $s_r(n)$ induce the triplets of affine maps $(F_{0,l}(n)(t) , F_{1,l}(n)(t) , F_{2,l}(n)(t))$ and $(F_{0,r}(n)(t) , F_{1,r}(n)(t) , F_{2,r}(n)(t))$ as described above.

For $i=0,1,2,$ let us define the intervals

$$\begin{aligned} I_{i,l}^{n} = F_{1,l}(1)\circ F_{1,l}(2)\circ F_{1,l}(3) \circ .....\circ F_{1,l}(n-1)\circ F_{i,l}(n)([0,b_c(0)]). \end{aligned}$$

Also,

$$\begin{aligned} I_{i,r}^{n} = F_{1,r}(1)\circ F_{1,r}(2)\circ F_{1,r}(3) \circ .....\circ F_{1,r}(n-1)\circ F_{i,r}(n)([{\tilde{b}}_c(1),1]). \end{aligned}$$

Definition 3

A scaling data $s_j \equiv \{s_j(n)\},$ for $j = l, r,$ is said to be proper if, for each $n \in {\mathbb {N}},$

$$\begin{aligned} d(s_j(n),\partial \varDelta ^3) \ge \epsilon , \,\,\,\,\, \text {for some} \,\,\, \epsilon > 0. \end{aligned}$$

Where $d(s_j(n),\partial \varDelta ^3)$ stands for the Euclidean distance between $s_j(n)$ and the closest boundary point of $\varDelta ^3.$

A pair of proper scaling data $s_l : {\mathbb {N}} \rightarrow \varDelta ^3$ and $s_r : {\mathbb {N}} \rightarrow \varDelta ^3 ,$ which is denoted by $s = (s_l, s_r),$ induce the sets $D_{s_l} = \bigcup \limits _{n \ge 1} (I_{0,l}^{n} \cup I_{2,l}^{n})$ and $D_{s_r} = \bigcup \limits _{n \ge 1} (I_{0,r}^{n} \cup I_{2,r}^{n}),$ respectively. Consider a map

$$\begin{aligned} f_{s} : D_{s_l} \cup D_{s_r} \rightarrow [0,1] \end{aligned}$$

defined as

$$\begin{aligned} f_s(x) = \left\{ \begin{array}{ll} f_{s_l}(x), \,\;\, \; \, {\text {if}} \; x \in D_{s_l} \\ f_{s_r}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_r} \end{array} \right. \end{aligned}$$

where $f_{s_l}|_{I_{0,l}^{n}}$ and $f_{s_l}|_{I_{2,l}^{n}}$ are the affine extensions of $b_c|_{\partial I_{0,l}^{n}}$ and $b_c|_{\partial I_{2,l}^{n}}$ respectively. Similarly, $f_{s_r}|_{I_{0,r}^{n}}$ and $f_{s_r}|_{I_{2,r}^{n}}$ are the affine extensions of $b_c|_{\partial I_{0,r}^{n}}$ and $b_c|_{\partial I_{2,r}^{n}}$ respectively. These affine extensions are shown in Fig. 2.

Fig. 2

Piece-wise affine extension

The end points of the intervals at each level are labeled by

$$\begin{aligned} y_0 = 0,\; z_0 = b_c(0),\; I_{1,l}^0 = I_L = [0,b_c(0)] \end{aligned}$$

and for $n \ge 1$

$$\begin{aligned} x_n= & {} \partial I_{0,l}^n \backslash \partial I_{1,l}^{n-1} \\ y_{2n-1}= & {} max\{ \partial I_{1,l}^{2n-1} \} \\ y_{2n}= & {} min\{ \partial I_{1,l}^{2n} \} \\ z_{2n-1}= & {} min\{ \partial I_{1,l}^{2n-1} \} \\ z_{2n}= & {} max\{ \partial I_{1,l}^{2n} \} \\ w_n= & {} \partial I_{2,l}^n \backslash \partial I_{1,l}^{n-1}, \end{aligned}$$

where $\partial I$ stands for the boundary of I. These points are illustrated in Fig. 3.

Fig. 3

Formation of interval $I_{1,l}^{n-1}$ into three sub-intervals $I_{2,l}^{n},$ $I_{1,l}^{n}$ and $I_{0,l}^{n}$

Also, the end points of the intervals at each level are labeled by

$$\begin{aligned} z'_0 = {\tilde{b}}_c(1), \; y'_0 = 1, \; I_{1,r}^0 = I_R = [{\tilde{b}}_c(1),1] \end{aligned}$$

and for $n \ge 1$

$$\begin{aligned} x'_n= & {} \partial I_{0,r}^n \backslash \partial I_{1,r}^{n-1} \\ y'_{2n-1}= & {} min\{ \partial I_{1,r}^{2n-1} \} \\ y'_{2n}= & {} max\{ \partial I_{1,r}^{2n} \} \\ z'_{2n-1}= & {} max\{ \partial I_{1,r}^{2n-1} \} \\ z'_{2n}= & {} min\{ \partial I_{1,r}^{2n} \} \\ w'_n= & {} \partial I_{2,r}^n \backslash \partial I_{1,r}^{n-1}. \end{aligned}$$

These points are illustrated in Fig. 4.

Fig. 4

Formation of interval $I_{1,r}^{n-1}$ into three sub-intervals $I_{0,r}^{n},$ $I_{1,r}^{n}$ and $I_{2,r}^{n}$

Definition 4

For a given pair of proper scaling data $s_l, s_r : {\mathbb {N}} \rightarrow \varDelta ^3,$ a map $f_{s}$ is said to be infinitely renormalizable if for $n \ge 1,$

• 1(i) $[0,f_{s_l}(y_{n})]$ is the maximal domain containing 0 on which $f_{s_l}^{3^n-1}$ is defined affinely, $[f_{s_l}^2(y_{n}), f_{s_l}(0)]$ is the maximal domain containing $f_{s_l}(0)$ on which $f_{s_l}^{3^n-2}$ is defined affinely,

• (ii) $[f_{s_r}(y'_{n}),1]$ is the maximal domain containing 1 on which $f_{s_r}^{3^n-1}$ is defined affinely and $[ f_{s_r}(1),f_{s_r}^2(y'_{n})]$ is the maximal domain containing $f_{s_r}(1)$ on which $f_{s_r}^{3^n-2}$ is defined affinely,

• 2(i) $f_{s_l}^{3^n-1}([0,f_{s_l}(y_{n})]) \;\;\;\;\;\;\;\; = I_{1,l}^n,$

• (ii) $f_{s_l}^{3^n-2}([f_{s_l}^2(y_{n}), f_{s_l}(0)]) = I_{1,l}^n ,$

• (iii) $f_{s_r}^{3^n-1}([f_{s_r}(y'_{n}),1]) \;\;\;\;\;\;\;\; = I_{1,r}^n,$

• (iv) $f_{s_r}^{3^n-2}([f_{s_r}(1),f_{s_r}^2(y'_{n})]) = I_{1,r}^n.$

Define $W = \{ f_s : f_s \; \text { is infinitely renormalizable map} \}.$

Further using Definition 4, we write $W_l = \{ f_{s_l} : f_{s_l} \text {\; satisfies 1(i), 2(i) and 2(ii)} \}$

and $W_r = \{ f_{s_r} : f_{s_r} \text {\; satisfies 1(ii), 2(iii) and 2(iv)} \}.$

Note that $W_l$ and $W_r$ be the collection of the piece-wise affine period tripling infinitely renormalizable maps $f_{s_l}$ on $I_L$ and $f_{s_r}$ on $I_R,$ respectively.

The combinatorics for renormalization of $f_{s_l}$ and $f_{s_r}$ are shown in the following Fig. 5a, b.

Fig. 5

The combinatorics: a corresponding to $f_{s_l},$ ($I_{1,l}^n \rightarrow I_{2,l}^n \rightarrow I_{0,l}^n \rightarrow I_{1,l}^n)$ and b corresponding to $f_{s_r},$ ($I_{1,r}^n \rightarrow I_{2,r}^n \rightarrow I_{0,r}^n \rightarrow I_{1,r}^n)$

2.1 Renormalization on $I_L = [0,b_c(0)]$

Let $f_{s_l} \in W_l$ be given by the proper scaling data $s_l : {\mathbb {N}} \rightarrow \varDelta ^3$ and define

$$\begin{aligned} {{\tilde{I}}}_{1,l}^n = [b_c^{2}( y_{n}),b_c(0)] = [ f_{s_l}^{2}( y_{n}),f_{s_l}(0)] , \end{aligned}$$

and

$$\begin{aligned} {{\hat{I}}}_{1,l}^n = [0,b_c(y_{n})] = [0,f_{s_l}(y_{n})]. \end{aligned}$$

Let

$$\begin{aligned} h_{s_l,n} : [0,b_c(0)] \rightarrow I_{1,l}^n \end{aligned}$$

be defined by

$$\begin{aligned} h_{s_l,n} = F_{1,l}(1)\circ F_{1,l}(2)\circ F_{1,l}(3) \circ .....\circ F_{1,l}(n) \end{aligned}$$

Furthermore, let

$$\begin{aligned} {{\tilde{h}}}_{s_l,n} : [0,b_c(0)] \rightarrow {{\tilde{I}}}_{1,l}^n \;\; \text {and} \;\; {{\hat{h}}}_{s_l,n} : [0,b_c(0)] \rightarrow {{\hat{I}}}_{1,l}^n \end{aligned}$$

be the affine orientation preserving homeomorphisms. Then define

$$\begin{aligned} R_n^lf_{s_l}: h_{s_l,n}^{-1}(D_{s_l} \cap I_{1,l}^n) \rightarrow [0,b_c(0)] \end{aligned}$$

by

$$\begin{aligned} R_n^lf_{s_l}(x) = \left\{ \begin{array}{ll} R_n^{l-}f_{s_l}(x), \,\,\, \, \, {\text {if}} \; x \in h_{s_l,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{0,l}^m) \\ R_n^{l+}f_{s_l}(x), \,\,\, \, \, {\text {if}} \; x \in h_{s_l,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{2,l}^m) \end{array} \right. \end{aligned}$$

where,

$$\begin{aligned} R_n^{l-}f_{s_l}: h_{s_l,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{0,l}^m) \rightarrow [0,b_c(0)] \end{aligned}$$

   and

$$\begin{aligned} R_n^{l+}f_{s_l}: h_{s_l,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{2,l}^m) \rightarrow [0,b_c(0)] \end{aligned}$$

are defined by

$$\begin{aligned} R_n^{l-}f_{s_l}(x)= & {} {{\tilde{h}}}_{s_l,n}^{-1} \circ f_{s_l}^{2} \circ h_{s_l,n}(x)\\ R_n^{l+}f_{s_l}(x)= & {} {{\hat{h}}}_{s_l,n}^{-1} \circ f_{s_l} \circ h_{s_l,n}(x), \end{aligned}$$

which are illustrated in Fig. 6.

Fig. 6

Illustration of operators $R_n^{l-}$ and $R_n^{l+}$

Let $\sigma : (\varDelta ^3)^{{\mathbb {N}}} \rightarrow (\varDelta ^3)^{{\mathbb {N}}}$ be the shift map defined as

$$\begin{aligned} \sigma (s_l(1)s_l(2)s_l(3)s_l(4)....) = (s_l(2)s_l(3)s_l(4)....), \end{aligned}$$

where $s_l(i) \in \varDelta ^3$ for all $i \in {\mathbb {N}}.$

Note that the operator $R_n^l$ normalize the affine pieces $f_{s_l}^2(\mathop {\cup }\limits _{m \ge n+1}I_{0,l}^m)$ and $f_{s_l}(\mathop {\cup }\limits _{m \ge n+1}I_{2,l}^m)$ to $I_L$ with the help of affine homeomorphism ${{\tilde{h}}}_{s_l,n}^{-1}$ and ${{\hat{h}}}_{s_l,n}^{-1},$ respectively.

This implies, $R_n^lf_{s_l}$ is a piecewise affine map associated with the scaling data

$(s_l(n+1)s_l(n+2)s_l(n+3)\ldots ).$ Thus,

$$\begin{aligned} R_n^lf_{s_l} = f_{s_l(n+1)s_l(n+2)s_l(n+3)\ldots } . \end{aligned}$$

The above explanation leads the following lemma.

Lemma 1

Let $s_l : {\mathbb {N}} \rightarrow \varDelta ^3$ be proper scaling data such that $f_{s_l}$ is infinitely renormalizable. Then

$$\begin{aligned} R_n^l{f_{s_l}} = f_{\sigma ^{n}(s_l)}. \end{aligned}$$

$\square$

Let $f_{s_l}$ be infinitely renormalization, then for $n \ge 0,$ we have

$$\begin{aligned} f_{s_l}^{3^n}: D_{s_l} \cap I_{1,l}^n \rightarrow I_{1,l}^n \end{aligned}$$

is well defined.

Define the renormalization $R^l : W_l\rightarrow W_l$ by

$$\begin{aligned} R^lf_{s_l} = h_{s_l,1}^{-1} \circ f_{s_l}^3 \circ h_{s_l,1} . \end{aligned}$$

The maps $f_{s_l}^{3^n-2} : {{\tilde{I}}}_{1,l}^n \rightarrow I_{1,l}^n$ and $f_{s_l}^{3^n-1} : {{\hat{I}}}_{1,l}^n \rightarrow I_{1,l}^n$ are the affine homeomorphisms whenever $f_{s_l} \in W_l$.

One can observe that, for each $n \in {\mathbb {N}},$ $\mathop {\cup }\limits _{m \ge n+1}I_{0,l}^m \subset I_{1,l}^n$ and $\mathop {\cup }\limits _{m \ge n+1}I_{2,l}^m \subset I_{1,l}^n.$

By the definition of $R_n^l,$ the operator $R_n^l$ is just normalizing the affine pieces, which are contained in $I_{1,l}^n,$ to $I_L.$ Also, $I_{1,l}^n$ are the renormalization intervals corresponding to $n^{th}$ renormalization operator $(R^l)^n.$ Then, we have the following lemma,

Lemma 2

We have $(R^l)^n{f_{s_l}} : D_{\sigma ^{n}(s_l)} \rightarrow [0,b_c(0)]$ and $(R^l)^n{f_{s_l}} = R_n^l{f_{s_l}}.$ $\square$

Using Lemmas 1 and 2, now we are in a position to state the following proposition:

Proposition 1

There exists a map $f_{s_l^*} \in W_l,$ where $s_l^*$ is characterized by

$$\begin{aligned} R^lf_{s_l^*} = f_{s_l^*}. \end{aligned}$$

Fig. 7

Length of intervals

Proof

Consider $s_l : {\mathbb {N}} \rightarrow \varDelta ^3$ be proper scaling data such that $f_{s_l}$ is an infinitely renormalizable. Let $c_n$ be the critical point of $f_{\sigma ^n(s_l)}.$ Then

we have the following scaling ratios which are illustrated in Fig. 7

$$\begin{aligned} s_{0,l}(n)&= \frac{b_{c_n}(0)-b_{c_n}^4(0)}{b_{c_n}(0)} \end{aligned}$$

(2)

$$\begin{aligned} s_{1,l}(n)&= \frac{b_{c_n}^2(0)-b_{c_n}^5(0)}{b_{c_n}(0)} \end{aligned}$$

(3)

$$\begin{aligned} s_{2,l}(n)&= \frac{b_{c_n}^3(0)}{b_{c_n}(0)} \end{aligned}$$

(4)

$$\begin{aligned} c_{n+1}&= \frac{b_{c_n}^2(0)-c_n}{s_{1,l}(n)} \equiv {\mathscr {R}}(c_n). \end{aligned}$$

(5)

Since $(s_{0,l}(n), s_{1,l}(n), s_{2,l}(n)) \in \varDelta ^3,$ this implies the following conditions

$$\begin{aligned} s_{0,l}(n),\; s_{1,l}(n), \; s_{2,l}(n)&> 0 \end{aligned}$$

(6)

$$\begin{aligned} s_{0,l}(n) +s_{1,l}(n) +s_{2,l}(n)&< 1 \end{aligned}$$

(7)

As the intervals $I_{i,l}^n,$ for $i = 0,1,2,$ are mutually disjoint, we denote the gap ratios as $g_{0,l}^n$ and $g_{1,l}^n$ which are in between $I_{0,l}^n \, \& \, I_{1,l}^n$ and $I_{1,l}^n \, \& \, I_{2,l}^n$ respectively. The gap ratios are defined as, for $n \in {\mathbb {N}},$

$$\begin{aligned} g_{0,l}^n&= \frac{b_{c_n}^4(0)-b_{c_n}^2(0)}{b_{c_n}(0)} \equiv G_{0,l}(c_n) >0 \end{aligned}$$

(8)

$$\begin{aligned} g_{1,l}^n&= \frac{b_{c_n}^5(0)-b_{c_n}^3(0)}{b_{c_n}(0)} \equiv G_{1,l}(c_n) > 0 \end{aligned}$$

(9)

$$\begin{aligned} 0&< c_n < \frac{3-\sqrt{3}}{6} \end{aligned}$$

(10)

We use Mathematica for solving the Eqs. (2), (3) and (4), then we get the expressions for $s_{0,l}(n),\;s_{1,l}(n)$ and $\;s_{2,l}(n).$

Let $s_{i,l}(n) \equiv S_{i,l}(c_n)$ for $i = 0,1,2.$ The graphs of $S_{i,l}(c)$ are shown in Figs.  8a, b and 9a.

Fig. 8

a and b shows the graph of $S_{0,l}(c),$ and $S_{1,l}( c)$ respectively

Fig. 9

a and b shows the graph of $S_{2,l}(c),$ and $(S_{0,l}+S_{1,l}+S_{2,l})(c)$ respectively

Note that the conditions (6), (8) and (9) give the condition (7)

$$\begin{aligned} 0<\sum _{i=0}^{2} s_{i,l}(n) < 1. \end{aligned}$$

The conditions  (6) together with  (8) to  (10) define the feasible domain $F_d^l$ is to be:

$$\begin{aligned} F_d^l = \Big \{ \; c \in \left( 0,\, \frac{3-\sqrt{3}}{6}\right) \, : \; S_{i,l}(c)> 0 \; \text {for} \, i=0,1,2,&G_{0,l}(c)>0, G_{1,l}(c) >0 \Big \} . \end{aligned}$$

(11)

To compute the feasible domain $F_d^l,$ we need to find subinterval(s) of $\left( 0, \frac{3-\sqrt{3}}{6}\right)$ which satisfies the conditions of (11). By using Mathematica software, we employ the following command to obtain the feasible domain

$$\begin{aligned} \text {N[Reduce}[\bigg \{S_{0,l}(c)> 0,&S_{1,l}(c)> 0, S_{2,l}(c)> 0, G_{0,l}(c)> 0, G_{1,l}(c) > 0, 0<c<\frac{3-\sqrt{3}}{6} \bigg \}, c]]. \end{aligned}$$

This yields:

$$\begin{aligned} F_d^l = (0.188816...,\; 0.194271...) \cup (0.194271...,\; 0.199413...) \equiv F_{d_1}^l \cup F_{d_2}^l. \end{aligned}$$

From the Eq. (5), the graphs of ${\mathscr {R}}(c)$ are plotted in the sub-domains $F_{d_1}^l$ and $F_{d_2}^l$ of $F_d^l$ which are shown in Fig. 10.

Fig. 10

The graph of ${\mathscr {R}} : F_d^l \rightarrow {\mathbb {R}}$ and the diagonal ${\mathscr {R}}(c) = c$

The map ${\mathscr {R}} : F_d^l \rightarrow {\mathbb {R}}$ is expanding in the neighborhood of fixed point $c_l^*$ which is illustrated in Fig. 10b. By Mathematica computations, we get an unstable fixed points $c_l^* = 0.196693...$ in $F_d^l$ such that

$$\begin{aligned} {\mathscr {R}}(c_l^*) = c_l^* \end{aligned}$$

corresponds to an infinitely renormalizable maps $f_{s_l^*}.$ We observe that the map $f_{{s_l}^*}$ corresponding to $c_l^*$ has the following property

$$\begin{aligned} \{c_l^*\} = \bigcap \limits _{n \ge 1} I_{1,l}^n. \end{aligned}$$

In other words, consider the scaling data ${s_l}^* : {\mathbb {N}} \rightarrow \varDelta ^3$ with

$$\begin{aligned} {s_l}^*(n)&= (s_{0,l}^*(n), s_{1,l}^*(n), s_{2,l}^*(n) ) \\&=\left( \frac{b_{c_l^*}(0)-b_{c_l^*}^4(0)}{b_{c_l^*}(0)}, \; \frac{b_{c_l^*}^2(0)-b_{c_l^*}^5(0)}{b_{c_l^*}(0)} , \; \frac{b_{c_l^*}^3(0)}{b_{c_l^*}(0)}\right) . \end{aligned}$$

Then $\sigma ({s_l^*} ) = s_l^*$ and using Lemma 1 we have

$$\begin{aligned} R^lf_{s_l^*} = f_{s_l^*}. \end{aligned}$$

$\square$

2.2 Renormalization on $I_R = [{\tilde{b}}_c(1), 1]$

In Sect. 2.1, the bimodal map $b_c(x)$ has two critical points $c \in I_L$ and $1-c \in I_R$ and we define the piece-wise renormalization on $I_L.$ In similar fashion, to define the renormalization on $I_R$ with $c \in I_R$, from Eq. 1, we consider

$$\begin{aligned} {\tilde{b}}_c(x) = 1-\frac{4c^3 - 3 c^2 + 6 c x - 6 c^2 x - 3 x^2 + 2 x^3}{( 2 c-1)^3} \end{aligned}$$

where $x \in [0,1]$ and $c \in [ \frac{3}{4},1].$

Note that for $c \in [\frac{3+\sqrt{3}}{6}, 1].$

Let $f_{s_r} \in W_r$ be given by the proper scaling data $s_r : {\mathbb {N}} \rightarrow \varDelta ^3$ and define

$$\begin{aligned} {{\tilde{I}}}_{1,r}^n = [{\tilde{b}}_c(1),{\tilde{b}}_c^{2}( y'_{n})] = [f_{s_r}(1), f_{s_r}^{2}( y'_{n})] , \end{aligned}$$

and

$$\begin{aligned} {{\hat{I}}}_{1,r}^n = [{\tilde{b}}_c(y'_{n}),1] = [f_{s_r}(y'_{n}),1]. \end{aligned}$$

Let

$$\begin{aligned} h_{s_r,n} : [{\tilde{b}}_c(1),1] \rightarrow I_{1,r}^n \end{aligned}$$

be defined by

$$\begin{aligned} h_{s_r,n} = F_{1,r}(1)\circ F_{1,r}(2)\circ F_{1,r}(3) \circ .....\circ F_{1,r}(n). \end{aligned}$$

Furthermore, let

$$\begin{aligned} {{\tilde{h}}}_{s_r,n} : [{\tilde{b}}_c(1),1] \rightarrow {{\tilde{I}}}_{1,r}^n \;\; \text {and} \;\; {{\hat{h}}}_{s_r,n} : [{\tilde{b}}_c(1),1] \rightarrow {{\hat{I}}}_{1,r}^n \end{aligned}$$

be the affine orientation preserving homeomorphisms. Then define

$$\begin{aligned} R_n^rf_{s_r}: h_{s_r,n}^{-1}(D_{s_r} \cap I_{1,r}^n) \rightarrow [{\tilde{b}}_c(1),1] \end{aligned}$$

by

$$\begin{aligned} R_n^rf_{s_r}(x) = \left\{ \begin{array}{ll} R_n^{r-}f_{s_r}(x), \,\,\, \, \, {\text {if}} \, x \in h_{s_r,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{0,r}^m) \\ R_n^{r+}f_{s_r}(x), \,\,\, \, \, {\text {if}} \, x \in h_{s_r,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{2,r}^m) \end{array} \right. \end{aligned}$$

where,

$$\begin{aligned} R_n^{r-}f_{s_r}: h_{s_r,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{0,r}^m) \rightarrow [{\tilde{b}}_c(1),1] \end{aligned}$$

   and

$$\begin{aligned} R_n^{r+}f_{s_r}: h_{s_r,n}^{-1}(\mathop {\cup }\limits _{m \ge n+1}I_{2,r}^n) \rightarrow [{\tilde{b}}_c(1),1] \end{aligned}$$

are defined by

$$\begin{aligned} R_n^{r-}f_{s_r}(x)= & {} {{\tilde{h}}}_{s_r,n}^{-1} \circ f_{s_r}^{2} \circ h_{s_r,n}(x)\\ R_n^{r+}f_{s_r}(x)= & {} {{\hat{h}}}_{s_r,n}^{-1} \circ f_{s_r} \circ h_{s_r,n}(x), \end{aligned}$$

which are illustrated in Fig. 11.

Fig. 11

Illustration of operators $R_n^{r-}$ and $R_n^{r+}$

Let $\sigma : (\varDelta ^3)^{{\mathbb {N}}} \rightarrow (\varDelta ^3)^{{\mathbb {N}}}$ be the shift map which is defined as

$$\begin{aligned} \sigma (s_r(1)s_r(2)s_r(3)s_r(4)....) = (s_r(2)s_r(3)s_r(4)....), \end{aligned}$$

where $s_r(i) \in \varDelta ^3$ for all $i \in {\mathbb {N}}.$

Lemma 3

Let $s_r : {\mathbb {N}} \rightarrow \varDelta ^3$ be proper scaling data such that $f_{s_r}$ is infinitely renormalizable. Then

$$\begin{aligned} R_n^r{f_{s_r}} = f_{\sigma ^{n}(s_r)}. \end{aligned}$$

$\square$

Let $f_{s_r}$ be infinitely renormalization, then for $n \ge 0,$ we have

$$\begin{aligned} f_{s_r}^{3^n}: D_{s_r} \cap I_{1,r}^n \rightarrow I_{1,r}^n \end{aligned}$$

is well defined.

Define the renormalization $R^r : W_r \rightarrow W_r$ by

$$\begin{aligned} R^rf_{s_r} = h_{s_r,1}^{-1} \circ f_{s_r}^3 \circ h_{s_r,1} . \end{aligned}$$

The maps $f_{s_r}^{3^n-2} : {{\tilde{I}}}_{1,r}^n \rightarrow I_{1,r}^n$ and $f_{s_r}^{3^n-1} : {{\hat{I}}}_{1,r}^n \rightarrow I_{1,r}^n$ are the affine homeomorphisms whenever $f_{s_r} \in W_r$. Then we have:

Lemma 4

We have $(R^r)^n{f_{s_r}} : D_{\sigma ^{n}(s_r)} \rightarrow [{\tilde{b}}_c(1),1]$ and $(R^r)^n{f_{s_r}} = R_n^r{f_{s_r}}.$ $\square$

From the above Lemmas 3 and 4, consequently we get

Proposition 2

There exists a map $f_{s_r^*} \in W_r,$ where $s_r^*$ is characterized by

$$\begin{aligned} R^rf_{s_r^*} = f_{s_r^*}. \end{aligned}$$

Fig. 12

Length of intervals

Proof

Consider $s_r : {\mathbb {N}} \rightarrow \varDelta ^3$ be proper scaling data such that $f_{s_r}$ is an infinitely renormalizable. Let $c_n$ be the critical point of $f_{\sigma ^n(s_r)}.$ Then

from Fig. 12, we have the following scaling ratios

$$\begin{aligned} s_{0,r}(n)&= \frac{{\tilde{b}}_{c_n}^4(1)-{\tilde{b}}_{c_n}(1)}{1-{\tilde{b}}_{c_n}(1)} \end{aligned}$$

(12)

$$\begin{aligned} s_{1,r}(n)&= \frac{{\tilde{b}}_{c_n}^5(1)-{\tilde{b}}_{c_n}^2(1)}{1-{\tilde{b}}_{c_n}(1)} \end{aligned}$$

(13)

$$\begin{aligned} s_{2,r}(n)&= \frac{1-{\tilde{b}}_{c_n}^3(1)}{1-{\tilde{b}}_{c_n}(1)} \end{aligned}$$

(14)

$$\begin{aligned} c_{n+1}&= 1-\frac{c_n-{\tilde{b}}_{c_n}^2(1)}{s_{1,r}(n)} \equiv {\mathscr {R}}(c_n). \end{aligned}$$

(15)

Use the same argument as was given in Sect. 2.1, one can compute feasible domain $F_d^r.$ Finally, we get

$$\begin{aligned} F_d^r = (0.800587...,\; 0.805729...) \cup (0.805729...,\; 0.811184...) \equiv F_{d_1}^r \cup F_{d_2}^r. \end{aligned}$$

From the Eq. (15), the graphs of ${\mathscr {R}}(c)$ are plotted in the sub-domains $F_{d_1}^r$ and $F_{d_2}^r$ of $F_d^r$ which are shown in Fig. 13.

Fig. 13

The graph of ${\mathscr {R}} : F_d^r \rightarrow {\mathbb {R}}$ and the diagonal ${\mathscr {R}}(c) = c$

The map ${\mathscr {R}} : F_d^r \rightarrow {\mathbb {R}}$ is expanding in the neighborhood of fixed point $c_r^*$ which is illustrated in Fig. 13b. By Mathematica computations, we get an unstable fixed points $c_r^* = 0.803307...$ in $F_d^r$ such that

$$\begin{aligned} {\mathscr {R}}(c_r^*) = c_r^* \end{aligned}$$

corresponds to an infinitely renormalizable maps $f_{s_r^*}.$ We observe that the map $f_{{s_r}^*}$ corresponding to $c_r^*$ has the following property

$$\begin{aligned} \{c_r^*\} = \bigcap \limits _{n \ge 1} I_{1,r}^n. \end{aligned}$$

In other words, consider the scaling data ${s_r}^* : {\mathbb {N}} \rightarrow \varDelta ^3$ with

$$\begin{aligned} {s_r}^*(n)&= (s_{0,r}^*(n), s_{1,r}^*(n), s_{2,r}^*(n) ) \\&=\left( \frac{{\tilde{b}}_{c_r^*}^4(1)-{\tilde{b}}_{c_r^*}(1)}{1-{\tilde{b}}_{c_r^*}(1)}, \; \frac{{\tilde{b}}_{c_r^*}^5(1)-{\tilde{b}}_{c_r^*}^2(1)}{1-{\tilde{b}}_{c_r^*}(1)} , \; \frac{1-{\tilde{b}}_{c_r^*}^3(1)}{1-{\tilde{b}}_{c_r^*}(1)}\right) . \end{aligned}$$

Then $\sigma ({s_r^*} ) = s_r^*$ and using Lemma 3 we have

$$\begin{aligned} R^rf_{s_r^*} = f_{s_r^*}. \end{aligned}$$

$\square$

For a given pair of proper scaling data $s = (s_l, s_r),$ we defined a map

$$\begin{aligned} f_{s} : D_{s_l} \cup D_{s_r} \rightarrow [0,1] \end{aligned}$$

as

$$\begin{aligned} f_s(x) = \left\{ \begin{array}{ll} f_{s_l}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_l} \\ f_{s_r}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_r} \end{array} \right. \end{aligned}$$

Then, the renormalization of $f_s$ is defined as

$$\begin{aligned} Rf_{s}(x) = \left\{ \begin{array}{ll} R^lf_{s_l}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_l} \\ R^rf_{s_r}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_r} \end{array} \right. \end{aligned}$$

From Propositions 1 and 2, we conclude that the period tripling infinitely renormalizable maps $f_{s_l^*}$ and $f_{s_r^*}$ are fixed points of $R^l$ and $R^r$ corresponding to the proper scaling data ${s_l^*}$ and ${s_r^*},$ respectively. Then, for a given pair of scaling data $s^* = (s_l^*, s_r^*),$ we have

$$\begin{aligned} Rf_{s^*}(x)&= \left\{ \begin{array}{ll} R^lf_{s_l^*}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_l^*} \\ R^rf_{s_r^*}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_r^*} \end{array} \right. \\&= \left\{ \begin{array}{ll} f_{s_l^*}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_l^*} \\ f_{s_r^*}(x), \;\;\; \; \; {\text {if}} \; x \in D_{s_r^*} \end{array} \right. \\&=f_{s^*}(x) \end{aligned}$$

This will give us the following theorem,

Theorem 1

There exists a map $f_{s^*} \in W,$ where $s^* = (s_l^*, s_r^*)$ is characterized by

$$\begin{aligned} Rf_{s^*} = f_{s^*}. \end{aligned}$$

In particular, $W = \{f_{s^*}\}.$

Remark 1

The constructed map $f_{s^*}$ with a pair of proper scaling data $s^* = (s_l^*, s_r^*)$ holds the following conditions,

• (i) $s_{2,l}^* \le (s_{1,l}^*)^2$

• (ii) $s_{2,r}^* \le (s_{1,r}^*)^2$

Note that for $i \in \{0,1,2\},$ the scaling ratios $s_{i,l}(n)$ are the expressions in the terms of $c_n$ which are described in Eqs. (2)–(4). Therefore, one can easily compute $s_{0,l}^*,$ $s_{1,l}^*$ and $s_{2,l}^*$ by substituting $c_n = c_l^*$ in the respective expressions. Then,

$$\begin{aligned} s_{2,l}^* = s_{2,l}(n)\big |_{c_n = c_l^*} \le \left( s_{1,l}(n)\big |_{c_n = c_l^*}\right) ^2 = \left( s_{1,l}^*\right) ^2 . \end{aligned}$$

Similarly,

$$\begin{aligned} s_{2,r}^* = s_{2,r}(n)\big |_{c_n = c_r^*} \le \left( s_{1,r}(n)\big |_{c_n = c_r^*}\right) ^2 = \left( s_{1,r}^*\right) ^2. \end{aligned}$$

Remark 2

The invariant Cantor set of the map $f_{s^*}$, namely $\varLambda _{s^*},$ is next in complexity to the invariant doubling Cantor set, namely $\varLambda _{\sigma ^*},$ of piece-wise affine period doubling infinitely renormalizable map $f_{\sigma ^*}$ [11] in the following sense,

• (i) like the both Cantor sets $\varLambda _{s^*}$ and $\varLambda _{\sigma ^*},$ on each scale and everywhere the same scaling ratio $s^*$ and $\sigma ^*$ are used respectively,

• (ii) but unlike the doubling Cantor set $\varLambda _{\sigma ^*},$ there are now a pair of three different ratios at each scale corresponding to $s^*.$

Furthermore, the geometry of the invariant Cantor set of $f_{s^*}$ is different from the geometry of the invariant Cantor set of piece-wise affine period tripling renormalizable map because the Cantor set of $f_{s^*}$ has $2-$copy of Cantor set of [41].

3 $C^{1+Lip}$ Extension of $f_{s^*}$

In Sect. 2, we have constructed a piece-wise affine infinitely renormalizable map $f_{s^*}$ corresponding to the pair of scaling data $s^* = (s_l^*,\; s_r^*).$ Let us define a pair of scaling functions

$$\begin{aligned}&S_l: [0,\;b_{c_l^*}(0)]^2 \rightarrow [0,\;b_{c_l^*}(0)]^2\\&S_r: [{\tilde{b}}_{c_r^*}(1),\;1]^2 \rightarrow [{\tilde{b}}_{c_r^*}(1),\;1]^2 \end{aligned}$$

as

$$\begin{aligned} \quad S_l\left( \begin{array}{ll} x \\ y \end{array} \right) = \left( \begin{array}{ll} b_{c_l^*}^2(0)-s_{1,l}^* \cdot x \\ s_{2,l}^*\cdot y \end{array} \right) ;&\;\;\;\;\;\;\;\; S_r\left( \begin{array}{ll} x \\ y \end{array} \right) = \left( \begin{array}{ll} {\tilde{b}}_{c_r^*}^2(1)+s_{1,r}^* \cdot (1-x) \\ 1-s_{2,r}^*\cdot (1-y) \end{array} \right) . \end{aligned}$$

Let G be the graph of $g_{s^*}$ which is an extension of $f_{s^*}$ where $f_{s^*} : D_{s_l^*}\cup D_{s_r^*} \rightarrow [0,\;1].$ Let $G_l^1$ and $G_l^2$ are the graphs of $g_{s^*}|_{[y_1,\;z_0]}$ which is an $C^{1+Lip}$ extension of $f_{s^*}$ on $D_{s_l^*} \cap {[y_1,\;z_0]}$ and $g_{s^*}|_{[y_0,\;z_1]}$ which is an $C^{1+Lip}$ extension of $f_{s^*}$ on $D_{s_l^*} \cap [y_0,\;z_1]$ respectively. Also, $G_r^1$ and $G_r^2$ are the graphs of $g_{s^*}|_{[z_0',\;y_1']}$ which is an $C^{1+Lip}$ extension of $f_{s^*}$ on $D_{s_r^*} \cap {[z_0',\;y_1']}$ and $g_{s^*}|_{[z_1',\;y_0']}$ which is an $C^{1+Lip}$ extension of $f_{s^*}$ on $D_{s_r^*} \cap {[z_1',\;y_0']}$ respectively which are shown in Fig. 14. Also, note that $G_r^1$ and $G_r^2$ are the reflections of $G_l^1$ and $G_l^2$ across the point $\left( \frac{1}{2},\frac{1}{2}\right)$ respectively. Define

$$\begin{aligned} G_l = \cup _{n \ge 0} S_l^n(G_l^1 \cup G_l^2) \;\; \text {and} \;\; G_r = \cup _{n \ge 0} S_r^n(G_r^1 \cup G_r^2). \end{aligned}$$

Then, $G_l$ is the graph of a unimodal map $g_{s_l^*}$ which extends $f_{s_l^*}$ and $G_r$ is the graph of a unimodal map $g_{s_r^*}$ which extends $f_{s_r^*}$. Consequently, G is the graph of $g_{s^*} = g_{s_l^*} \oplus g_{s_r^*}.$ We claim that $g_{s^*}$ is a $C^{1+Lip}$ symmetric bimodal map.

Let $B_l^0 = [0,\;b_{c_l^*}(0)] \times [0,\;b_{c_l^*}(0)]$ and $B_r^0 = [{\tilde{b}}_{c_r^*}(1),\; 1] \times [{\tilde{b}}_{c_r^*}(1),\;1].$

For $n \in {\mathbb {N}},$ define

$$\begin{aligned} B_l^n = S_l^n(B_l^0) \;\;\;\;\;\;\; \text {and} \;\;\;\;\;\;\; B_r^n = S_r^n(B_r^0) \end{aligned}$$

as

$B_l^n= \left\{ \begin{array}{ll} \;[z_n,\;y_n] \times [0,\; {\hat{y}}_{n}], \;\;\; {\text {if}}\; n \; \text {is odd} \\ \ [y_n,\;z_n] \times [0,\;{\hat{y}}_{n}], \;\;\; {\text {if}}\; n \; \text {is even} \end{array} \right.$

and

$B_r^n= \left\{ \begin{array}{ll} \;[y_n',\;z_n'] \times [\hat{y'}_{n},\;1], \;\;\; {\text {if}}\; n \; \text {is odd} \\ \ [z_n',\;y_n'] \times [\hat{y'}_{n},\;1], \;\;\; {\text {if}}\; n \; \text {is even}. \end{array} \right.$

Let $p_l^n$ and $p_r^n$ be the points on the graph of the bimodal map $b_{{c}_l^*}(x)$ and $b_{{c}_r^*}(x)$ respectively. For all $n \in {\mathbb {N}},$ $p_l^n$ and $p_r^n$ are defined as

$p_l^n= \left\{ \begin{array}{ll} \left( \begin{array}{ll} y_{\frac{n+1}{2}} \\ {\hat{y}}_{\frac{n+1}{2}} \end{array} \right) , \;\;\; {\text {if}}\; n \; \text {is odd} \\ \left( \begin{array}{ll} z_{\frac{n}{2}} \\ {\hat{z}}_{\frac{n}{2}} \end{array} \right) , \;\;\; \;\;\;\; \;\;{\text {if}}\; n \; \text {is even} \end{array} \right.$

$p_r^n= \left\{ \begin{array}{ll} \left( \begin{array}{ll} y'_{\frac{n+1}{2}} \\ \hat{y'}_{\frac{n+1}{2}} \end{array} \right) , \;\;\; {\text {if}}\; n \; \text {is odd} \\ \left( \begin{array}{ll} z'_{\frac{n}{2}} \\ \hat{z'}_{\frac{n}{2}} \end{array} \right) , \;\;\; \;\;\;\; \;\;{\text {if}}\; n \; \text {is even} \end{array} \right.$

where ${\hat{y}}_{n} = b_{c_l^*}(y_n),$ ${\hat{z}}_{n} = b_{c_l^*}(z_n),$ $\hat{y'}_{n} = {\tilde{b}}_{c_r^*}(y'_n)$ and $\hat{z'}_{n} = {\tilde{b}}_{c_r^*}(z'_n).$

Fig. 14

Extension of $f_{s^*}$

Then the above construction will lead to following proposition,

Proposition 3

G is the graph of $g_{s^*}$ which is a $C^1$ extension of $f_{s^*}.$

Proof

Since $G_l^1$ and $G_l^2$ are the graph of $f_{s_l^*}|_{[y_1, z_{0}]}$ and $f_{s_l^*}|_{[y_0, z_{1}]},$ respectively, and $G_r^1$ and $G_r^2$ are the graph of $f_{s_r^*}|_{[z_0', y_1']}$ and $f_{s_r^*}|_{[z_1', y_0']},$ respectively, we obtain $G_l^{2n+1} = S_l^n (G_l^1)$ and $G_l^{2n+2} = S_l^n (G_l^2)$ for each $n \in {\mathbb {N}}.$ Note that $G_l^n$ is the graph of a $C^1$ function defined

$$\begin{aligned} \text {on} \;\;&[z_{\frac{n-1}{2}},\; y_{\frac{n+1}{2}}] \;\;\; {\text {if}} \;\; n \in 4{\mathbb {N}}-1, \\ \text {on}\;\;&[z_{\frac{n}{2}},\; y_{\frac{n}{2}-1}] \;\;\;\;\;{\text {if}} \;\; n \in 4{\mathbb {N}}, \\ \text {on}\;\;&[y_{\frac{n+1}{2}},\; z_{\frac{n-1}{2}}] \;\;\;{\text {if}}\;\; n \in 4{\mathbb {N}}+1, \\ \text {and on}\;\;&[y_{\frac{n}{2}-1},\; z_{\frac{n}{2}}]\;\;\;\;\; {\text {if}}\;\; n \in 4{\mathbb {N}}+2. \end{aligned}$$

Also, we have $G_r^{2n+1} = S_r^n (G_r^1)$ and $G_r^{2n+2} = S_r^n (G_r^2)$ for each $n \in {\mathbb {N}}.$ Note that $G_r^n$ is the graph of a $C^1$ function defined

$$\begin{aligned} \text {on} \;\;&[y_{\frac{n+1}{2}}',\; z_{\frac{n-1}{2}}'] \;\;\; {\text {if}} \;\; n \in 4{\mathbb {N}}-1, \\ \text {on}\;\;&[y_{\frac{n}{2}-1}',\; z_{\frac{n}{2}}'] \;\;\;\;\;{\text {if}} \;\; n \in 4{\mathbb {N}}, \\ \text {on}\;\;&[z_{\frac{n-1}{2}}',\; y_{\frac{n+1}{2}}'] \;\;\;{\text {if}}\;\; n \in 4{\mathbb {N}}+1, \\ \text {and on}\;\;&[z_{\frac{n}{2}}',\; y_{\frac{n}{2}-1}']\;\;\;\;\; {\text {if}}\;\; n \in 4{\mathbb {N}}+2. \end{aligned}$$

To prove the proposition, we have to check continuous differentiability at the points $p_l^n$ and $p_r^n.$ Consider the neighborhoods $(y_1-\epsilon ,y_1+\epsilon )$ around $y_1$ and $(z_1-\epsilon ,z_1+\epsilon )$ around $z_1$, the slopes are given by an affine pieces of $f_{s_l^*}$ on the subintervals $(y_1-\epsilon ,y_1)$ and $(z_1,z_1+\epsilon )$ and the slopes are given by the chosen $C^1$ extension on $(y_1,y_1+\epsilon )$ and $(z_1-\epsilon ,z_1).$ This implies, $G_l^1$ and $G_l^2$ are $C^1$ at $p_l^1$ and $p_l^2,$ respectively.

Let $\gamma _1 \subset G_l$ be the graph over the interval $(y_1-\epsilon ,y_1+\epsilon )$ and $\gamma _2 \subset G_l$ be the graph over the interval $(z_1-\epsilon ,z_1+\epsilon ),$

then the graph $G_l$ locally around $p_l^n$ is equal to $\left\{ \begin{array}{ll} S_l^{\frac{n-1}{2}}{(\gamma _1)} \;\;\; {\text {if}}\; n \; \text {is odd} \\ S_l^{\frac{n-2}{2}}{(\gamma _2)} \;\;\; {\text {if}}\; n \; \text {is even} \end{array} \right.$. This implies, for $n \in {\mathbb {N}},$ $G_l^{2n-1}$ is $C^1$ at $p_l^{2n-1}$ and $G_l^{2n}$ is $C^1$ at $p_l^{2n}.$

Hence $G_l$ is a graph of a $C^1$ function on $[0,b_{c_l^*}(0)] \setminus \{c_l^*\}.$

We note that the horizontal contraction of $S_l$ is smaller than the vertical contraction. This implies that the slope of $G_l^n$ tends to zero when n is large. Therefore, $G_l$ is the graph of a $C^1$ function $g_{s_l^*}$ on $[0,b_{c_l^*}(0)].$

In similar way, one can prove that $G_r$ is the graph of a $C^1$ function $g_{s_r^*}$ on $[\tilde{b}_{c_r^*} (1),1].$ Therefore, $G = G_l \oplus G_r$ is the graph of a $C^1$ bimodal map $g_{s^*} =g_{s_l^*} \oplus g_{s_r^*}$ which is a $C^1$ extension of $f_{s^*}.$ $\square$

Proposition 4

Let $g_{s^*}$ be the function whose graph is G then $g_{s^*}$ is a $C^{1+Lip}$ symmetric bimodal map.

Proof

As the function $g_{s^*}$ is a $C^1$ extension of $f_{s^*}.$ We have to show that, for $i \in \{l, \;r\},$ $G_i^n$ is the graph of a $C^{1+Lip}$ function

$$\begin{aligned} g_{s_i^*}^n: Dom(G_i^n) \rightarrow [0,1] \end{aligned}$$

with an uniform Lipschitz bound.

That is, for $n \ge 1,$

$$\begin{aligned} Lip((g_{s_i^*}^{n+1})') \le Lip((g_{s_i^*}^n)') \end{aligned}$$

let us assume that $g_{s_l^*}^n$ is $C^{1+Lip}$ with Lipschitz constant $\lambda _n$ for its derivatives. We show that $\lambda _{n+1} \le \lambda _{n}$.

For given $\left( \begin{array}{ll} u \\ v \end{array} \right)$ on the graph of $g_{s_l^*}^n,$ there is $\left( \begin{array}{ll} {\tilde{u}} \\ {\tilde{v}} \end{array} \right) = S_l\left( \begin{array}{ll} u \\ v \end{array} \right)$ on the graph of $g_{s_l^*}^{n+1}$, this implies

$$\begin{aligned} g_{s_l^*}^{n+1}({\tilde{u}}) = s_{2,l}^*\cdot g_{s_l^*}^n(u) \end{aligned}$$

Since $u = \frac{b_{c_l^*}^2(0)-{\tilde{u}}}{{s_{1,l}^*}},$ we have

$$\begin{aligned} g_{s_l^*}^{n+1}({\tilde{u}}) = s_{2,l}^*\cdot g_{s_l^*}^n\left( \frac{b_{c_l^*}^2(0)-{\tilde{u}}}{{s_{1,l}^*}}\right) \end{aligned}$$

Differentiate both sides with respect to ${\tilde{u}}$, we get

$$\begin{aligned} \left( g_{s_l^*}^{n+1}\right) '({\tilde{u}}) = -\frac{s_{2,l}^*}{s_{1,l}^*} \cdot \left( g_{s_l^*}^n\right) '\left( \frac{b_{c_l^*}^2(0)-{\tilde{u}}}{{s_{1,l}^*}}\right) \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \left( g_{s_l^*}^{n+1}\right) '({\tilde{u}}_1) - \left( g_{s_l^*}^{n+1}\right) '({\tilde{u}}_2)\right|&= \left| \frac{s_{2,l}^*}{s_{1,l}^*}\right| \cdot \left| \left( g_{s_l^*}^n\right) '\left( \frac{b_{c_l^*}^2(0)-{\tilde{u}}_1}{{s_{1,l}^*}}\right) -\left( g_{s_l^*}^n\right) '\left( \frac{b_{c_l^*}^2(0)-{\tilde{u}}_2}{{s_{1,l}^*}}\right) \right| \\&\le \frac{s_{2,l}^*}{({s_{1,l}^*})^2} \cdot \lambda \left( g_{s_l^*}^n\right) '|{\tilde{u}}_1-{\tilde{u}}_2| \end{aligned}$$

From Remark 1, we have $({s_{1,l}^*})^2 \ge {s_{2,l}^*}.$ Then,

$$\begin{aligned} \lambda (g_{s_l^*}^{n+1})' \le \lambda (g_{s_l^*}^{n})' \le \lambda (g_{s_l^*}^{1})'. \end{aligned}$$

Similarly, one can show that

$$\begin{aligned} \lambda (g_{s_r^*}^{n+1})' \le \lambda (g_{s_r^*}^{n})' \le \lambda (g_{s_r^*}^{1})'. \end{aligned}$$

Therefore, choose $\lambda = max\{ \lambda (g_{s_l^*}^{1})',\; \lambda (g_{s_r^*}^{1})' \}$ is the uniform Lipschitz bound. This completes the proof. $\square$

Note that for a given pair of proper scaling data $s^* = (s_l^*,s_r^*),$ the piece-wise affine map $f_{s^*}$ is infinitely renormalizable and $g_{s^*}$ is a $C^{1+Lip}$ extension of $f_{s^*}$. This implies $g_{s^*}$ is also renormalizable map. Further, we observe that $R g_{s^*}$ is an extension of $Rf_{s^*}.$ Therefore $R g_{s^*}$ is renormalizable. Hence, $g_{s^*}$ is infinitely renormalizable map which is not a $C^2$ map. Then we have the following theorem,

Theorem 2

There exists an infinitely renormalizable $C^{1+Lip}$ symmetric bimodal map $g_{s^*}$ such that

$$\begin{aligned} R g_{s^*} = g_{s^*} . \end{aligned}$$

$\square$

4 Topological Entropy of Renormalization

In this section, we calculate the topological entropy of the renormalization operator defined on the space of $C^{1+Lip}$ bimodal maps.

Let us consider three pairs of $C^{1+Lip}$ maps $\phi _i : [0,z_1] \cup [y_1,b_{c_l^*}(0)] \rightarrow [0,b_{c_l^*}(0)]$ and $\psi _i : [{\tilde{b}}_{c_r^*}(1),y_1'] \cup [z_1',1] \rightarrow [{\tilde{b}}_{c_r^*}(1),1],$ for $i = 0,1,2,$ which extend $f_{s^*}.$ Because of symmetricity, $\psi _i(x) = 1-\phi _i(1-x).$ For a sequence $\alpha = \{\alpha _n\}_{n \ge 1} \in \varSigma _3,$

where $\varSigma _3 = \{ \{x_n\}_{n \ge 1} : x_n \in \{0,\;1,\;2\} \}$ is called full 3-Shift.

Now define

$$\begin{aligned} G_l^n(\alpha ) = S_l^n(graph \; \phi _{{\alpha }_n}) \;\;\;\; \text {and} \;\;\;\; G_r^n(\alpha ) = S_r^n(graph \; \psi _{{\alpha }_n}), \end{aligned}$$

we have

$$\begin{aligned} G_l(\alpha ) = \bigcup \limits _{n \ge 1} G_l^{n}(\alpha ) \;\;\;\; \text {and} \;\;\;\; G_r(\alpha ) = \bigcup \limits _{n \ge 1} G_r^{n}(\alpha ). \end{aligned}$$

Therefore, we conclude that $G(\alpha ) = G_l(\alpha ) \oplus G_r(\alpha )$ is the graph of a $C^{1+Lip}$ bimodal map $b_{\alpha }$ by using the same facts of Sect. 3.

The shift map $\sigma : \varSigma _3 \rightarrow \varSigma _3$ is defined as

$$\begin{aligned} \sigma (\alpha _1 \alpha _2 \alpha _3 \ldots ) = (\alpha _2 \alpha _3 \alpha _4 \ldots ). \end{aligned}$$

Proposition 5

The restricted maps $b_\alpha ^3 : [y_1,\;z_1] \rightarrow [y_1,\;z_1]$ and $b_\alpha ^3 : [y_1',\;z_1'] \rightarrow [y_1',\;z_1']$ are the unimodal maps for all $\alpha \in \varSigma _3.$ In particular, $b_\alpha$ is a renormalizable map and $R b_\alpha = b_{\sigma (\alpha )}.$

Proof

We know that $b_\alpha : [y_1,\;z_1] \rightarrow I_{2,l}^1$ is a unimodal and onto, $b_\alpha : I_{2,l}^1 \rightarrow I_{0,l}^1$ is onto and affine and also $b_\alpha : I_{0,l}^1 \rightarrow [y_1,\;z_1]$ is onto and affine. Therefore $b_\alpha ^3$ is a unimodal map on $[y_1,\;z_1]$. Analogously, $b_\alpha ^3$ is a unimodal map on $[y_1',\;z_1']$. The above construction implies

$$\begin{aligned} R b_\alpha = b_{\sigma (\alpha )}. \end{aligned}$$

$\square$

This gives us the following theorem.

Theorem 3

The renormalization operator R acting on the space of $C^{1+Lip}$ symmetric bimodal maps has unbounded topological entropy.

Proof

From the above construction, we conclude that $\alpha \longmapsto b_\alpha \in C^{1+Lip}$ is injective. The domain of R contains two copies, namely $\varLambda _1$ and $\varLambda _2,$ of the full 3-shift. As topological entropy $h_{top}$ is an invariant of topological conjugacy. Hence $h_{top}(\left. R\right| _{\varLambda _1 \cup \varLambda _2} ) > \ln 3.$ In fact, if we choose n different pairs of $C^{1+Lip}$ maps, say, $\phi _0,\;\phi _1,\;\phi _2,\ldots \phi _{n-1}$ and $\psi _0,\;\psi _1,\;\psi _2,\ldots \psi _{n-1},$ which extends $f_{s^*},$ then it will be embedded two copies of the full $n-\text {shift}$ in the domain of R. Hence, the topological entropy of R on $C^{1+Lip}$ symmetric bimodal maps is unbounded. $\square$

5 An $\epsilon$ Perturbation of the Scaling Data

In this section, we use an $\epsilon$ perturbation on the construction of the scaling data as presented in Sect. 2, to obtain the following theorem

Theorem 4

There exists a continuum of fixed points of the renormalization operator acting on $C^{1+Lip}$ symmetric bimodal maps.

Proof

Consider an $\epsilon$ variation on scaling data and we modify the construction which is described in Sect.  2.

Let us define the neighborhoods $N_{\epsilon }^l$ and $N_{\epsilon }^r$ about the respective points $(b_{c}^3(0),\; b_{c}^4(0))$ and $(b_{c}^3(1),\; b_{c}^4(1))$ as

$$\begin{aligned} N_{\epsilon }^l(b_{c}^3(0),\; b_{c}^4(0)) = \{ (b_{c}^3(0),\; \epsilon \cdot b_{c}^4(0)) \; : \; \epsilon> 0 \; \text { and} \; \epsilon \; \text {close to } 1 \} \\ N_{\epsilon }^r(b_{c}^3(1),\; b_{c}^4(1)) = \{ (b_{c}^3(1),\; \epsilon \cdot b_{c}^4(1)) \; : \; \epsilon > 0 \; \text { and} \; \epsilon \; \text {close to } 1 \} \end{aligned}$$

(i). The perturbed scaling data on $I_0^l,$ then the scaling ratios are defined as

$$\begin{aligned} s_{2,l}(c,\epsilon )&= \frac{b_{c}^3(0)}{b_{c}(0)} \\ s_{0,l}(c,\epsilon )&= \frac{b_{c}(0)-\epsilon b_{c}^4(0)}{b_{c}(0)} \\ s_{1,l}(c,\epsilon )&= \frac{b_{c}^2(0)-b_{c}(\epsilon b_{c}^4(0))}{b_{c}(0)}, \end{aligned}$$

where $c \in (0, \frac{3-\sqrt{3}}{6}).$ Also, we define

$$\begin{aligned} {\mathscr {R}}(c,\epsilon )&= \frac{b_{c}^2(0)-c}{s_{1,l}(c,\epsilon )}. \end{aligned}$$

From Sect. 2.1, we know that the map ${\mathscr {R}}$ which is defined in Eq. 5, has unique fixed point $c^*$. Consequently, for a given $\epsilon$ close to 1,  ${\mathscr {R}}(c,\epsilon )$ has only one unstable fixed point, namely $c_{\epsilon }^*.$ Therefore, we consider the perturbed scaling data ${s}_{l,\epsilon }^* : {\mathbb {N}} \rightarrow \varDelta ^3$ with

$$\begin{aligned} {s}_{l,\epsilon }^* = \left( \frac{b_{c_{\epsilon }^*}(0)-\epsilon b_{c_{\epsilon }^*}^4(0)}{b_{c_{\epsilon }^*}(0)}, \; \frac{b_{c_{\epsilon }^*}^2(0)-b_{c_{\epsilon }^*}(\epsilon b_{c_{\epsilon }^*}^4(0))}{b_{c_{\epsilon }^*}(0)}, \;\frac{b_{c_{\epsilon }^*}^3(0)}{b_{c_{\epsilon }^*}(0)} \right) . \end{aligned}$$

Then $\sigma (s_{l,\epsilon }^* ) = {s}_{l,\epsilon }^*$ and using Lemma 1, we have

$$\begin{aligned} R^lf_{{s}_{l,\epsilon }^*} = f_{{s}_{l,\epsilon }^*}. \end{aligned}$$

(ii). Considering the perturbed scaling data on $I_0^r,$ one has the scaling data ${s}_{r,\epsilon }^* : {\mathbb {N}} \rightarrow \varDelta ^3$ with

$$\begin{aligned} {s}_{r,\epsilon }^* = \left( \frac{\epsilon b_{c_{\epsilon }^*}^4(1)-b_{c_{\epsilon }^*}(1)}{1-b_{c_{\epsilon }^*}(1)}, \; \frac{b_{c_{\epsilon }^*}(\epsilon b_{c_{\epsilon }^*}^4(1))-b_{c_{\epsilon }^*}^2(1)}{1-b_{c_{\epsilon }^*}(1)}, \;\frac{1-b_{c_{\epsilon }^*}^3(1)}{1-b_{c_{\epsilon }^*}(1)} \right) . \end{aligned}$$

Then $\sigma (s_{r,\epsilon }^* ) = {s}_{r,\epsilon }^*$ and using Lemma 3, we have

$$\begin{aligned} R^rf_{{s}_{r,\epsilon }^*} = f_{{s}_{r,\epsilon }^*}. \end{aligned}$$

Moreover, $f_{{s}_{l,\epsilon }^*}$ and $f_{{s}_{r,\epsilon }^*}$ are the piece-wise affine maps which are infinitely renormalizable. For a given pair of proper scaling data ${s}_{\epsilon }^* = ({s}_{l,\epsilon }^*,\;{s}_{r,\epsilon }^*),$ we have

$$\begin{aligned} Rf_{{s}_{\epsilon }^*} = f_{{s}_{\epsilon }^*}. \end{aligned}$$

Now we use similar extension described in Sect. 3, then we get $g_{{s}_{\epsilon }^*}$ is the $C^{1+Lip}$ extension of $f_{{s}_{\epsilon }^*}$. This implies that $g_{{s}_{\epsilon }^*}$ is a renormalizable map. As $R g_{{s}_{\epsilon }^*}$ is an extension of $Rf_{{s}_{\epsilon }^*}.$ Therefore $R g_{{s}_{\epsilon }^*}$ is renormalizable. Hence, for each $\epsilon$ close to 1,  $g_{{s}_{\epsilon }^*}$ is a fixed point of the renormalization. This proves the existence of a continuum of fixed points of the renormalization. $\square$

Remark 3

In particular, for two different perturbed scaling data $s_{\epsilon _1^*}$ and $s_{\epsilon _2^*},$ one can construct two infinitely renormalizable maps $g_{s_{\epsilon _1^*}}$ and $g_{s_{\epsilon _2^*}}.$ Therefore, the respective Cantor attractors will have different scaling ratios. Consequently, it shows the non-rigidity for symmetric bimodal maps, whose smoothness is $C^{1+Lip}$.

6 Conclusions

In this paper, we have investigated the existence of fixed point of the renormalization operator which is defined on the space of symmetric bimodal maps with low smoothness. For a given pair of proper scaling data $s^* = (s_l^*,\; s_r^*),$ we have first constructed the piece-wise affine infinitely renormalizable map $f_{s^*}$ which is the only fixed point of the renormalization. We observe that the geometry of invariant Cantor set is more complex than the geometry of the Cantor set of piece-wise affine period doubling renormalizable map [11]. Further, we have extended this fixed point $f_{s^*}$ to a $C^{1+Lip}$ symmetric bimodal map. Moreover, we proved that the renormalization operator acting on the space of $C^{1+Lip}$ symmetric bimodal maps has infinite topological entropy. Finally, we proved the existence a continuum of fixed points of renormalization by considering a small perturbation on the scaling data. Consequently, it showed the non-rigidity of the Cantor attractors of infinitely renormalizable symmetric bimodal maps with low smoothness.