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Absolutely continuous invariant measures for piecewise expanding \(C^ 2\) transformations in \(\mathbb R^N\). (English) Zbl 0691.28004
The authors give sufficient conditions for the existence of absolutely continuous invariant measures for piecewise expanding, piecewise \(C^2\) transformations \(T\) of bounded regions in \(\mathbb R^n\). This generalizes work of the reviewer [C. R. Acad. Sci., Paris, Sér. A 289, 625–627 (1979; Zbl 0419.28007)] from dimension 2 to higher dimensions. The expansion condition relates \(\sigma:=\) the operator norm of the inverse of the Jacobian of \(T\) (i.e. the amount of expansion) to a cone condition for the domains of differentiability \(S_1,\dots,S_m\) of \(T\): If each boundary point of an \(S_i\) supports a cone pointing to the interior of \(S_i\) with angle \(2\alpha,\) then a sufficient condition is \(\sigma \cdot (1+| 1/\sin (\alpha)|)<1.\) For \(n=2\) a much weaker sufficient condition is stated in the reviewer’s thesis referred to in the above cited note.
Reviewer: G.Keller

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