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Journal d’Analyse Mathématique

, Volume 78, Issue 1, pp 117–142 | Cite as

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

  • Peter Raith
Article

Abstract

Assume thatX is a finite union of closed intervals and consider aC 1-mapX→ℝ for which {c∈X: T′c=0} is finite. Set\(T\left( T \right) = \cap _{j = 1}^\infty \overline {T^{ - j} X} \). Fix ann ∈ ℕ. For ε>0, theC 1-map\(\tilde T:X \to \mathbb{R}\) is called an ε-perturbation ofT if\(\tilde T\) is a piecewise monotonic map with at mostn intervals of monotonicity and\(\tilde T\) is ε-close toT in theC 1-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation\(\tilde T\) ofT has a unique measure\(\tilde \mu \) of maximal entropy, and the map\(\tilde T \mapsto \tilde \mu \) is continuous atT in the weak star-topology.

Keywords

Topological Entropy Maximal Measure Finite Union Topological Pressure Finite Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Magnes Press, The Hebrew University 1999

Authors and Affiliations

  • Peter Raith
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

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