Invariant subsets of expanding mappings of the circle
- 1 December 1987
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 7 (4) , 627-645
- https://doi.org/10.1017/s0143385700004247
Abstract
The continuity of Hausdorff dimension of closed invariant subsets K of a C2-expanding mapping g of the circle is investigated. If g/K satisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that if f is a piecewise monotone continuous mapping of a compact interval and φ a continuous function with P(f,φ)> sup(φ), then the pressure P(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.Keywords
This publication has 13 references indexed in Scilit:
- On Hausdorff dimension of invariant sets for expanding maps of a circleErgodic Theory and Dynamical Systems, 1986
- Hausdorff dimension for horseshoesErgodic Theory and Dynamical Systems, 1983
- Ergodic properties of invariant measures for piecewise monotonic transformationsMathematische Zeitschrift, 1982
- On intrinsic ergodicity of piecewise monotonic transformations with positive entropy IIIsrael Journal of Mathematics, 1981
- Hausdorff dimension of quasi-circlesPublications mathématiques de l'IHÉS, 1979
- On intrinsic ergodicity of piecewise monotonic transformations with positive entropyIsrael Journal of Mathematics, 1979
- Theory of dynamical systems and general transformation groups with invariant measureJournal of Mathematical Sciences, 1977
- Equilibrium States and the Ergodic Theory of Anosov DiffeomorphismsPublished by Springer Nature ,1975
- Some systems with unique equilibrium statesTheory of Computing Systems, 1974
- On the entropy of a flowAmerican Mathematical Society Translations: Series 2, 1966