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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
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: S1 → Sl 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
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 295 - 309
Copyright
Copyright © Cambridge University Press 1986

References

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