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


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.


Topological Entropy Maximal Measure Finite Union Topological Pressure Finite Partition 
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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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