Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 6
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    CARMINATI, CARLO and TIOZZO, GIULIO 2016. The local Hölder exponent for the dimension of invariant subsets of the circle. Ergodic Theory and Dynamical Systems, p. 1.

    Gelfert, Katrin 2016. Horseshoes for diffeomorphisms preserving hyperbolic measures. Mathematische Zeitschrift, Vol. 283, Issue. 3-4, p. 685.

    Dettmann, Carl 2013. Open circle maps: small hole asymptotics. Nonlinearity, Vol. 26, Issue. 1, p. 307.

    Nilsson, Johan 2009. On numbers badly approximable by dyadic rationals. Israel Journal of Mathematics, Vol. 171, Issue. 1, p. 93.

    Labarca, Rafael and Moreira, Carlos Gustavo 2006. Essential Dynamics for Lorenz maps on the real line and the Lexicographical World☆☆Partially supported by Fondecyt-CHILE grant # 1000098 and PRONEX-BRAZIL on Dynamical Systems, Brazil.. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 23, Issue. 5, p. 683.

    Jenkinson, Oliver 2005. Maximum hitting frequency and fastest mean return time. Nonlinearity, Vol. 18, Issue. 5, p. 2305.

  • Ergodic Theory and Dynamical Systems, Volume 6, Issue 2
  • June 1986, pp. 295-309

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

  • Mariusz Urbański (a1)
  • DOI:
  • Published online: 01 September 2008

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).

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[B1]R. Bowen , Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377397.

[B2]R. Bowen , Hausdorff dimension of quasi-circles. Publ. Math. IHES, 50 (1980), 1125.

[P]W. Parry . Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 168178.

[S1]M. Shub . Endomorphisms of compact difierentiable manifolds. Amer. J. of Math. 91 (1969), 175199.

[S2]S. Smale . Differentiable dynamical systems. Bull. Amer. Soc. 73 (1967), 747817.

[W1]P. Walters . Equilibrium states for β-transformations and related transformations. Math. Z. 159 (1978), 6588.

[W2]R. F. Williams . The ‘DA’ maps of Smale and structural stability. In Global Analysis 14 (1970), 329334.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *