## SRB measures for non-hyperbolic systems with multidimensional expansion.(English)Zbl 0955.37012

The main goal of this work is to set up a framework for the study of statistical properties of systems with nonuniform expansion, which is motivated by the examples of multidimensional nonhyperbolic attractors constructed by M. Viana. The approach involves an inducing procedure, based on the notion of hyperbolic time that he introduces here, and contains a theorem of existence of absolutely continuous invariant measures for multidimensional piecewise expanding maps with countably many domains of smoothness.

### MSC:

 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A05 Dynamical aspects of measure-preserving transformations

### Keywords:

non-hyperbolic system; invariant measure; hyperbolic time
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### References:

 [1] K. ADL-ZARABI , Absolutely continuous invariant measures for piecewise expanding C2 transformations in \Bbb Rn with cusps on their boundaries , Ergodic Theory Dynamical Systems 16 ( 1996 ) 1-18. MR 96k:58129 | Zbl 0856.58022 · Zbl 0856.58022 [2] J.F. ALVES , CH. BONATTI and M. VIANA , SRB measures for partially hyperbolic systems whose central direction is mostly expanding , Preprint CMUP, University of Porto, 1999 . · Zbl 0962.37012 [3] M. BENEDICKS and L. CARLESON , On iterations of 1 - ax$$^{2}$$ on (-1, 1) , Ann. Math. 122 ( 1985 ) 1-25. MR 87c:58058 | Zbl 0597.58016 · Zbl 0597.58016 [4] M. BENEDICKS and L. CARLESON , The dynamics of the Hénon map , Ann. Math. 133 ( 1991 ) 73-169. MR 92d:58116 | Zbl 0724.58042 · Zbl 0724.58042 [5] M. BENEDICKS and L.-S. YOUNG , SRB-measures for certain Hénon maps , Invent. Math. 112 ( 1993 ) 541-576. MR 94e:58074 | Zbl 0796.58025 · Zbl 0796.58025 [6] CH. BONATTI , A. PUMARIÑ;O and M. VIANA , Lorenz-like attractors with arbitrary unstable dimension , C. R. Acad. Sci. Série I 325 ( 1997 ) 883-888. Zbl 0896.58043 · Zbl 0896.58043 [7] CH. BONATTI and M. VIANA , SRB measures for partially hyperbolic systems whose central direction is mostly contracting , Israel J. Math. (to appear). Zbl 0996.37033 · Zbl 0996.37033 [8] R. BOWEN and D. RUELLE , The ergodic theory of Axiom A flows , Invent. Math. 29 ( 1975 ) 181-202. MR 52 #1786 | Zbl 0311.58010 · Zbl 0311.58010 [9] J. BUZZI , A.c.i.m.’s for arbitrary expanding piecewise \Bbb R-analytic mappings of the plane , Preprint Luminy, 1998 . [10] L.C. EVANS and R.F. GARIEPY , Measure Theory and Fine Properties of Functions , Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992 . MR 93f:28001 | Zbl 0804.28001 · Zbl 0804.28001 [11] E. GIUSTI , Minimal Surfaces and Functions of Bounded Variation , Birkäuser, Basel, 1984 . MR 87a:58041 | Zbl 0545.49018 · Zbl 0545.49018 [12] P. GÓRA and A. BOYARSKY , Absolutely continuous invariant measures for piecewise expanding C2 transformations in \Bbb RN , Israel J. Math. 67 ( 1989 ) 272-286. MR 91c:58061 | Zbl 0691.28004 · Zbl 0691.28004 [13] P. GÓRA and A. BOYARSKY , On functions of bounded variation in higher dimensions , Amer. Math. Month. 99 (2) ( 1992 ) 159-160. MR 92k:26025 | Zbl 0757.26014 · Zbl 0757.26014 [14] M. JAKOBSON , Absolutely continuous invariant measures for one-parameter families of one-dimensional maps , Comm. Math. Phys. 81 ( 1981 ) 39-88. Article | MR 83j:58070 | Zbl 0497.58017 · Zbl 0497.58017 [15] G. KELLER , Ergodicité et mesures invariants pour les transformations dilatants par morceaux d’une région bornée du plan , C. R. Acad. Sci. Paris Série A 289 ( 1979 ) 625-627. MR 80k:28016 | Zbl 0419.28007 · Zbl 0419.28007 [16] A. LASOTA and J.A. YORKE , On the existence of invariant measures for piecewise monotonic maps , Trans. Amer. Math. Soc. 186 ( 1973 ) 481-488. MR 49 #538 | Zbl 0298.28015 · Zbl 0298.28015 [17] R. MAÑ;É , Ergodic Theory and Differentiable Dynamics , Springer, Berlin, 1987 . Zbl 0616.28007 · Zbl 0616.28007 [18] W. DE MELO and S. VAN STRIEN , One-Dimensional Dynamics , Springer, Berlin, 1993 . MR 95a:58035 | Zbl 0791.58003 · Zbl 0791.58003 [19] L. MORA and M. VIANA , Abundance of strange attractors , Acta Math. 171 ( 1993 ) 1-71. MR 94k:58089 | Zbl 0815.58016 · Zbl 0815.58016 [20] D. RUELLE , A measure associated with Axiom A attractors , Amer. J. Math. 98 ( 1976 ) 619-654. MR 54 #3763 | Zbl 0355.58010 · Zbl 0355.58010 [21] B. SAUSSOL , Absolutely continuous invariant measures for multi-dimensional expanding maps , Preprint Luminy, 1997 . [22] YA. SINAI , Gibbs measures in ergodic theory , Russ. Math. Surv. 27 (4) ( 1972 ) 21-69. MR 53 #3265 | Zbl 0255.28016 · Zbl 0255.28016 [23] D. SINGER , Stable orbits and bifurcations of maps of the interval , SIAM J. Appl. Math. 35 ( 1978 ) 260-267. MR 58 #13206 | Zbl 0391.58014 · Zbl 0391.58014 [24] M. TSUJII , Absolutely continuous invariant measures for piecewise real-analytic maps on the plane , Preprint Hokkaido Univ., 1998 . · Zbl 0989.37015 [25] M. VIANA , Strange attractors in higher dimensions , Bull. Braz. Math. Soc. 24 ( 1993 ) 13-62. MR 94k:58093 | Zbl 0784.58044 · Zbl 0784.58044 [26] M. VIANA , Multidimensional nonhyperbolic attractors , Publ. Math. IHES 85 ( 1997 ) 63-96. Numdam | MR 98j:58073 | Zbl 1037.37016 · Zbl 1037.37016
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