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Continuity of the hausdorff dimension for piecewise monotonic maps

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Abstract

In this paper a piecewise monotonic mapT:X→ℝ, whereX is a finite union of intervals, is considered. DefineR(T)=\(\mathop \cap \limits_{n = 0}^\infty \overline {T^{ - n} X} \). The influence of small perturbations ofT on the Hausdorff dimension HD(R(T)) ofR(T) is investigated. It is shown, that HD(R(T)) is lower semi-continuous, and an upper bound of the jumps up is given. Furthermore a similar result is shown for the topological pressure.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090, Wien, Austria

    Peter Raith

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  1. Peter Raith
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Additional information

Research was supported by Projekt Nr. P8193-PHY of the Austrian Fond zur Förderung der wissenschaftlichen Forschung

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Raith, P. Continuity of the hausdorff dimension for piecewise monotonic maps. Israel J. Math. 80, 97–133 (1992). https://doi.org/10.1007/BF02808156

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  • DOI: https://doi.org/10.1007/BF02808156

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