# Parabolic Cantor sets

Fundamenta Mathematicae (1996)

- Volume: 151, Issue: 3, page 241-277
- ISSN: 0016-2736

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topUrbański, Mariusz. "Parabolic Cantor sets." Fundamenta Mathematicae 151.3 (1996): 241-277. <http://eudml.org/doc/212194>.

@article{Urbański1996,

abstract = {The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.},

author = {Urbański, Mariusz},

journal = {Fundamenta Mathematicae},

keywords = {conformal measure; rigidity; Lipschitz classification; parabolic Cantor sets; Cantor repellors; invariant Cantor sets; fixed parabolic points},

language = {eng},

number = {3},

pages = {241-277},

title = {Parabolic Cantor sets},

url = {http://eudml.org/doc/212194},

volume = {151},

year = {1996},

}

TY - JOUR

AU - Urbański, Mariusz

TI - Parabolic Cantor sets

JO - Fundamenta Mathematicae

PY - 1996

VL - 151

IS - 3

SP - 241

EP - 277

AB - The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.

LA - eng

KW - conformal measure; rigidity; Lipschitz classification; parabolic Cantor sets; Cantor repellors; invariant Cantor sets; fixed parabolic points

UR - http://eudml.org/doc/212194

ER -

## References

top- [ADU] J. Aaronson, M. Denker, and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548. Zbl0789.28010
- [Be] T. Bedford, Applications of dynamical systems theory to fractals - a study of cookie-cutter Cantor sets, preprint, 1989.
- [Bo] R. Bowen, Equilibrium States and the Ergodic Theory for Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975. Zbl0308.28010
- [DU1] M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. 43 (1991), 107-118. Zbl0734.28007
- [DU2] M. Denker and M. Urbański, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579. Zbl0745.28008
- [DU3] M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66. Zbl0737.58030
- [DU4] M. Denker and M. Urbański, The capacity of parabolic Julia sets, Math. Z. 211 (1992), 73-86. Zbl0763.30009
- [DU5] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365-384. Zbl0722.58028
- [DU6] M. Denker and M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), 563-587. Zbl0745.58031
- [LS] R. de la Llave and R. P. Schafer, Rigidity properties of one dimensional expanding maps, preprint, 1994.
- [Ma] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. Zbl0819.28004
- [Po] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992. Zbl0762.30001
- [Pr1] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures, preprint, 1995.
- [Pr2] F. Przytycki, ${C}^{1+\epsilon}$ Cantor repellers in the line, scaling functions, an application to Feigenbaum’s universality, preprint, 1991.
- [Pr3] F. Przytycki, Sullivan's classification of conformal expanding repellers, preprint, 1991.
- [Pr4] F. Przytycki, On holomorphic perturbations of $z\mapsto {z}^{n}$, Bull. Polish Acad. Sci. Math. 34 (1986), 127-132.
- [PT] F. Przytycki and F. Tangerman, Cantor sets in the line: Scaling functions of the shift map, preprint, 1992.
- [PU] F. Przytycki and M. Urbański, Fractals in the complex plane - ergodic theory methods, to appear. Zbl1202.37001
- [PUZ] F. Przytycki, M. Urbański, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps II, Studia Math. 97 (1991), 189-225. Zbl0732.58022
- [Ru] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.
- [Sc] F. Schweiger, Number theoretical endomorphisms with σ-finite invariant measures, Israel J. Math. 21, (1975), 308-318.
- [SS] M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems, 5 (1985), 285-289. Zbl0583.58022
- [Su1] D. Sullivan, Quasiconformal homeomorphisms in dynamics, topology, and geometry, in: Proc. Internat. Congress of Math., Berkeley, Amer. Math. Soc., 1986, 1216-1228.
- [Su2] D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, in: The Mathematical Heritage of Herman Weyl, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., 1988, 15-23.
- [TT] S. J. Taylor and C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699. Zbl0537.28003
- [U1] M. Urbański, On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point, Studia Math. 97 (1991), 167-188. Zbl0727.58024
- [U2] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. Zbl0807.58025
- [Wa] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), 937-971. Zbl0318.28007

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