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Sinai-Bowen-Ruelle measures for certain Hénon maps. (English) Zbl 0796.58025
As it was proved by the first author and L. Carleson [Ann. Math. (2) 133, No. 1, 73–169 (1991; Zbl 0724.58042)], for a positive measure set of parameters \(0<a<2\), \(b>0\) that the maps \(T_{a,b} (x,y) = (1-ax^ 2 + y,bx)\) of the real plane to itself have a topologically transitive attractor \(\Lambda_{a,b}\). The authors study statistical properties of these attractors. The main result of the paper is that for a positive measure set of parameters there is a unique Sinai-Bowen-Ruelle invariant measure (i.e., such that it has a positive Lyapunov exponent, and its conditional measures on unstable manifolds are absolutely continuous). Moreover, its support coincides with \(\Lambda_{a,b}\), and the corresponding dynamical system is Bernoulli.

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37A05 Dynamical aspects of measure-preserving transformations
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References:
[1] [B] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. (Lect. Notes Math., vol. 470) Berlin Heidelberg New York Springer: 1975 · Zbl 0308.28010
[2] [BC1] Benedicks, M., Carleson, L.: On iterates ofx?1-ax 2 on (-1,1). Ann. Math.122, 1-25 (1985) · Zbl 0597.58016
[3] [BC2] Benedicks, M., Carleson, L.: The dynamics of the H?non map. Ann. Math.133, (1991), 73-169 · Zbl 0724.58042
[4] [BM] Benedicks, M. Moeckel, R.: An attractor for the H?non map. Z?rich: ETH (Preprint)
[5] [BY] Benedicks, M., Young, L.S.: Random perturbations and invariant measures for certain one-dimensional maps. Ergodic Theory Dyn. Syst. (to appear)
[6] [L] Ledrappier, F.: Propri?t?s ergodiques des mesures de Sinai. Publ. Math., Inst. Hautes Etud. Sci.59, 163-188 (1984) · Zbl 0561.58037
[7] [LS] Ledrappier, F., Strelcyn, J.-M.: A proof of the estimation from below in Pesin’s entropy formula. Ergodic Theory Dyn. Sys2 203-219 (1982)
[8] [LY] Ledrappier, F., Young, L.-S.: The metric entropy of diffemorphisms. Part I. Characterization of measures satisfying Pesin’s formula. Part II. Relations between entropy, exponents and dimension. Ann. Math.122, 509-539, 540-574 (1985) · Zbl 0605.58028
[9] [MV] Mora, L., Viana, M.: Abundance of strange attractors. IMPA reprint (1991) · Zbl 0815.58016
[10] [K] Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math., Inst. Hautes ?tud. Sci.51, 137-174 (1980) · Zbl 0445.58015
[11] [P1] Pesin, Ja.G.: Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR, Izv.10, 1261-1305 (1978) · Zbl 0383.58012
[12] [P2] Pesin, Ja.G.: Characteristic Lyaponov exponents and smooth ergodic theory. Russ. Math. Surv.32.4, 55-114 (1977) · Zbl 0383.58011
[13] [PS] Pugh, C., Shub, M.: Ergodic Attractors. Trans. Am. Math. Soc.312, 1-54 (1989)
[14] [Ro] Rohlin, V.A.: On the fundamental ideas of measure theory. Transl., Am. Math. Soc.10, 1-52 (1962)
[15] [Ru1] Ruelle, D.: A measure associated with Axiom A attractors. Am. J. Math.98, 619-654 (1976) · Zbl 0355.58010
[16] [Ru2] Ruelle, D.: Ergodic theory of differentialble dynamical systems. Publ. Math., Inst. Hautes ?tud. Sci.50, 27-58 (1979) · Zbl 0426.58014
[17] [S1] Sinai, Ya. G.: Markov partitions andC-diffeomorphisms. Func. Anal. Appl.2, 64-89 (1968) · Zbl 0182.55003
[18] [S2] Sinai, Ya. G.: Gibbs measures in ergodic theory. Russ. Math. Surv.27:4, 21-69 (1972) · Zbl 0246.28008
[19] [Y1] Young, L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory Dyn. Syst.2, 163-188 (1982) · Zbl 0523.58024
[20] [Y2] Young, L.-S.: A Bowen-Ruelle measure for certain piecewise hyperbolic maps. Trans. Am. Math. Soc.287, 41-48 (1985) · Zbl 0552.58022
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