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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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##### References:
 [1] Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984) · Zbl 0545.49018 [2] Keller, C. R. Acad. Sri. 289 pp 625– (1979) [3] DOI: 10.2307/1996575 · Zbl 0298.28015 [4] DOI: 10.2307/1969514 · Zbl 0040.06502 [5] Yuri, Dynamical Systems and Nonlinear Oscillations 1 (1986) [6] Lasota, Probabilistic Properties of Deterministic Systems (1985) · Zbl 0606.58002 [7] DOI: 10.1007/BF02764946 · Zbl 0691.28004 [8] Kosyakin, Izv. VUZ Matematika 3 pp 32– (1972) [9] Krzyzewski, Studia Math. 33 pp 82– (1969) [10] DOI: 10.1007/BF01941798 · Zbl 0463.28011 [11] Candeloro, Atti Sem. Mat. Fis. Univ. Moena XXXV pp 33– (1987) [12] Ivanov, Absolutely continuous invariant measures for locally expanding transformations. Preprint No. 27 [13] Jablo?ski, Ann. Polon. Math. XLIII pp 185– (1983) [14] Schweiger, Invariant Measures and Ergodic Properties of Numbertheoretical Endomorphisms 23 pp 283– · Zbl 0689.10056
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