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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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[1] D. Candeloro,Misure invariante per transformazioni in piú dimensioni, Atti Sem. Mat. Fis. Univ. ModenaXXXV (1987), 33–42.
[2] H. Federer,Geometric Measure Theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801
[3] E. Giusti,Minimal Surfaces and Functions of Bounded Variation, Birkhauser, 1984. · Zbl 0545.49018
[4] F. Hofbauer and G. Keller,Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z.180 (1982), 119–140. · Zbl 0485.28016
[5] C. T. Ionescu Tulcea and G. Marinescu,Théorie ergodique pour des classes d’opérations non completement continues, Ann. of Math.52 (1950), 140–147. · Zbl 0040.06502
[6] V. V. Ivanov and A. G. Kachurowski,Absolutely continuous invariant measures for locally expanding transformations, Preprint No. 27, Institute of Mathematics AN USSR, Siberian Section (in Russian).
[7] M. Jabłoński,On invariant measures for piecewise C 2-transformations of the n-dimensional cube, Ann. Polon. Math.XLIII (1983), 185–195. · Zbl 0591.28014
[8] G. Keller,Ergodicité et mesures invariantes pour les transformations dilatantes par morcaux d’une région bornée du plan, C.R. Acad. Sci. Paris289, Série A (1979), 625–627. · Zbl 0419.28007
[9] G. Keller,Proprietés ergodiques des endomorphismes dilatants, C 2 par morceaux, des régions bornées du plan, Thesis, Université de Rennes, 1979.
[10] A. A. Kosyakin and E. A. Sandler,Ergodic properties of a class of piecewise-smooth transformations of an interval, Izv. VUZ Matematika (3)[118] (1972), 32–40. (English translation from the British Library, Translation Service.)
[11] A. A. Kosyakin and E. A. Sandler,Stochasticity of a certain class of discrete system, translated from Automatika and Telemekhanika No. 9 (1972), 87–94.
[12] K. Krzy\.zewski and W. Szlenk,On invariant measures for expanding differentiable mappings, Studia Math.33 (1969), 82–92. · Zbl 0176.00901
[13] A. Lasota and M. Mackey,Probabilistic Properties of Deterministic Systems, Cambridge University Press, 1985. · Zbl 0606.58002
[14] A. Lasota and J. A. Yorke,On the existence of invariant measures for piecewise monotonic transformations, Trans. Am. Math. Soc.186 (1973), 481–488. · Zbl 0298.28015
[15] M. Rychlik,Bounded variation and invariant measures, Studia Math.LXXVI (1983), 69–80. · Zbl 0575.28011
[16] F. Schweiger,Invariant measures and ergodic properties of numbertheoretical endomorphisms, Banach Center Publications, to appear. · Zbl 0689.10056
[17] E. Straube,On the existence of invariant absolutely continuous measures, Commun. Math. Phys.81 (1981), 27–30. · Zbl 0463.28011
[18] M. Yuri,On a Bernouilli property for multi-dimensional mappings with finite range structure, inDynamical Systems and Nonlinear Oscillations, Vol. 1 (Giko Ikegami, ed.), World Scientific, Singapore, 1986.
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