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Absolutely continuous invariant measures for piecewise expanding \(C^ 2\) transformations in \(\mathbb R^ n\) on domains with cusps on the boundaries. (English) Zbl 0856.58022
Let \(\Omega\) be a bounded region in \(\mathbb R^n\) and let \({\mathcal P}= \{P_i\}^m_{i= 1}\) be a partition of \(\Omega\) into a finite number of subsets having piecewise \(C^2\)-boundaries. The boundaries may contain cusps.
Let \(\tau: \Omega\to \Omega\) be piecewise \(C^2\) on \(\mathcal P\) and expanding in the sense that there exists \(\alpha >1\) such that for any \(i= 1,2,\dots, m\), \(|D\tau^{- 1}_i|< \alpha^{- 1}\), where \(D\tau^{- 1}_i\) is the derivative matrix of \(\tau^{- 1}_i\) and \(|\cdot|\) is the Euclidean matrix norm.
The main result provides a lower bound on \(\alpha\) which guarantees the existence of an absolutely continuous invariant measure for \(\tau\).

MSC:
37A05 Dynamical aspects of measure-preserving transformations
28D05 Measure-preserving transformations
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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