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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

##### MSC:
 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
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