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On Hausdorff dimension of invariant sets for expanding maps of a circle

Published online by Cambridge University Press:  19 September 2008

Mariusz Urbański
Affiliation:
Institute of Mathematics, N. Copernicus University, Schopina 12/18, 87-100 Toruń, Poland
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Abstract

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Given an orientation preserving C2 expanding mapping g: S1Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

Type
Research Article
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[B1]Bowen, R., Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377397.Google Scholar
[B2]Bowen, R., Hausdorff dimension of quasi-circles. Publ. Math. IHES, 50 (1980), 1125.CrossRefGoogle Scholar
[K]Krzyzewski, K. & Szlenk, W.. On invariant measures for expanding differentiable mappings. Stud. Math. 33 (1968), 8392.CrossRefGoogle Scholar
[McC-M]McCluskey, H. & Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 251260.CrossRefGoogle Scholar
[P]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 168178.CrossRefGoogle Scholar
[R]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99107.CrossRefGoogle Scholar
[S1]Shub, M.. Endomorphisms of compact difierentiable manifolds. Amer. J. of Math. 91 (1969), 175199.CrossRefGoogle Scholar
[S2]Smale, S.. Differentiable dynamical systems. Bull. Amer. Soc. 73 (1967), 747817.CrossRefGoogle Scholar
[W1]Walters, P.. Equilibrium states for β-transformations and related transformations. Math. Z. 159 (1978), 6588.CrossRefGoogle Scholar
[W2]Williams, R. F.. The ‘DA’ maps of Smale and structural stability. In Global Analysis 14 (1970), 329334.CrossRefGoogle Scholar
[Y]Young, L. S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109124.CrossRefGoogle Scholar