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Journal of Statistical Physics

, Volume 89, Issue 3–4, pp 537–548 | Cite as

Existence of many ergodic absolutely continuous invariant measures for piecewise-expandingC 2 chaotic transformations in ℝ2 on a fixed number of partitions

  • Kourosh Adl-Zarabi
  • Harald Proppe
Article
  • 49 Downloads

Abstract

Let Ω be a region in ℝn and letp = Pi ) i 1m , be a partition ofΩ into a finite number of closed subsets having piecewise C2 boundaries of finite(n - 1 )dimensional measure. Let τ:Ω→Ω be piecewise C2 onP where, τi = τ¦pi is aC 2 diffeomorphism onto its image, and expanding in the sense that there exists α > 1 such that for anyi = 1, 2,...,m ‖Dτi -1 ‖ < α-1, where Dτi -1 is the derivative matrixτ i - 1 and ¦‖·‖ is the Euclidean matrix norm. By means of an example, we will show that the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we will construct a piecewise expanding C2 transformation on a fixed partition with a finite number of elements in ℝ2, but which has an arbitrarily large number of ergodic, absolutely continuous invariant measures

Key Words

Absolutely continuous invariant measures (acim) ergodic piecewise-C2 expanding transformation perturbation 

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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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