Abstract
We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C 1 conformal contractions on compact subsets of \({{\mathbb{R}}^d}\) . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.
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Colella D., Heil C.: The characterization of continuous, four-coefficient scaling functions and wavelets. IEEE Trans. Inform. Theory 38, 876–881 (1992)
Daubechies I., Lagarias J.C.: Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal. 23, 1031–1079 (1992)
Deng, Q.-R., He, X.-G., Lau, K.-S.: Self-affine meausres and vector-valued representations, Studia Math. (2008, in press)
Falconer K.J.: Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. 106, 543–554 (1989)
Falconer K.J.: Techniques in fractal geometry. Wiley, Chichester (1997)
Fan A.-H., Lau K.-S.: Iterated function system and Ruelle operator. J. Math. Anal. Appl. 231(2), 319–344 (1999)
Feng D.: The smoothness of L q-spectrum of self-similar measures with overlaps. J. Lond. Math. Soc. 68, 102–118 (2003)
Feng D.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195, 24–101 (2005)
Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Jin N., Yau S.S.T.: General finite type IFS and M-matrix. Comm. Anal. Geom. 13, 821–843 (2005)
Lau K.-S., Ngai S.-M.: Multifractal measures and a weak separation condition. Adv. Math. 141, 45–96 (1999)
Lau K.-S., Ngai S.-M.: A generalized finite type condition for iterated function systems. Adv. Math 208, 647–671 (2007)
Lau K.-S., Ngai S.-M., Rao H.: Iterated function systems with overlaps and self-similar measures. J. Lond. Math. Soc. (2) 63(1), 99–116 (2001)
Lau, K.-S., Rao, H., Ye, Y.: Corrigendum: “Iterated function system and Ruelle operator” [J. Math. Anal. Appl. 231 (1999), 319–344] by Lau, K.-S., Fan, A. H. J. Math. Anal. Appl. 262, 446–451 (2001)
Lau K.-S., Wang X.-Y.: Iterated function systems with a weak separation condition. Studia Math. 161, 249–268 (2004)
Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
Mauldin R.D., Urbański M.: Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73, 105–154 (1996)
Mauldin R.D., Urbański M.: Conformal iterated function systems with applications to the geometry of continued fractions. Trans. Amer. Math. Soc. 351, 4995–5025 (1999)
Mauldin, R.D., Urbański, M.: Graph directed Markov systems. Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003)
Ngai S.-M., Wang Y.: Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. (2) 63(3), 655–672 (2001)
Patzschke N.: Self-conformal multifractal measures. Adv. Appl. Math. 19(4), 486–513 (1997)
Peres Y., Rams M., Simon K., Solomyak B.: Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets. Proc. Amer. Math. Soc. 129(9), 2689–2699 (2001)
Peres, Y., Schlag, W., Solomyak, B.: Sixty years of Bernoulli convolutions. Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46. Birkhäuser, Basel, pp. 39–65 (2000)
Ye Y.-L.: Separation properties for self-conformal sets. Studia Math. 152(1), 33–44 (2002)
Ye Y.-L.: Multifractal of self-conformal measures. Nonlinearity. 18(5), 2111–2133 (2005)
Zerner M.P.W.: Weak separation properties for self-similar sets. Proc. Amer. Math. Soc. 124, 3529–3539 (1996)
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Communicated by K. Schmidt.
The authors are supported in part by an HKRGC grant.
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Lau, KS., Ngai, SM. & Wang, XY. Separation conditions for conformal iterated function systems. Monatsh Math 156, 325–355 (2009). https://doi.org/10.1007/s00605-008-0052-4
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DOI: https://doi.org/10.1007/s00605-008-0052-4
Keywords
- Conformal iterated function system
- Self-conformal measure
- Weak separation condition
- Finite type condition
- Singularity
- Absolute continuity
- Hausdorff measure