SRB measures for some stretched Hénon-like maps
Abstract
UDC 517.9
We discuss the construction of SRB measures for some families of stretched Hénon-like maps.
References
R. Adler, $F$-expansions revisited, Lecture Notes in Math., 318, 1–5 (1973). DOI: https://doi.org/10.1007/BFb0061717
V. M. Alekseev, Quasi-random dynamical systems, {I}, Math. Sb., 5, № 1, 73–128 (1968). DOI: https://doi.org/10.1070/SM1968v005n01ABEH002587
D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967).
D. V. Anosov, Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22, 103–167 (1967). DOI: https://doi.org/10.1070/RM1967v022n05ABEH001228
M. Benedicks, L. Carleson, The dynamics of the Hénon map, Ann. Math., 133, 73–169 (1991). DOI: https://doi.org/10.2307/2944326
M. Benedicks, L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261, 13–56 (2000).
L. A. Bunimovich, A transformation of the circle, Mat. Zametki, 8, 205–216 (1970).
Yu-Ru Huang, Estimating measure of stochastic parameters nonadjacent to the Chebyshev value, PhD, University of MD (2012).
M. V. Jakobson, On smooth mappings of the circle into itself, Mat. Sb., 85, № 2, 163–188 (1971). DOI: https://doi.org/10.1070/SM1971v014n02ABEH002611
M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys., 81, 39–88 (1981). DOI: https://doi.org/10.1007/BF01941800
M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, Proc. Sympos. Pure Math., 69, 825–881 (2001). DOI: https://doi.org/10.1090/pspum/069/1858558
M. Jakobson, Method of parameter exclusion. Some recollections and some new results, Pure and Appl. Funct. Anal., 5, 1377–1394 (2020).
M. Jakobson, Topological and metric properties of one-dimensional endomorphisms, Soviet Math. Dokl., 19, № 6, 1452–1456 (1979).
M. V. Jakobson, S. E. Newhouse, A two-dimensional version of the Folklore theorem, Amer. Math. Soc. Transl. Ser. 2, 171, 89–105 (1996). DOI: https://doi.org/10.1090/trans2/171/09
M. V. Jakobson, S. E. Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle, Astérisque, 261, 103–160 (2000).
M. Jakobson, L. Simonelli, Countable Markov partitions suitable for thermodynamic formalism, J. Mod. Dyn., 13, 199–219 (2018). DOI: https://doi.org/10.3934/jmd.2018018
S. Luzzatto, H. Takahasi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19, 1657–1695 (2006). DOI: https://doi.org/10.1088/0951-7715/19/7/013
L. Mora, M. Viana, Abundance of strange attractors, Acta Math., 1–73 (1993). DOI: https://doi.org/10.1007/BF02392766
J. Palis, J.-C. Yoccoz, Implicit formalism for affine-like maps and parabolic composition, Global Anal. Dyn. Syst., 67–87 (2001).
J. Palis, J.-C. Yoccoz, Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles, Publ. Math. Inst. Hautes Études Sci., 110, 1–217 (2009). DOI: https://doi.org/10.1007/s10240-009-0023-x
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, V. V. Fedorenko, Dynamics of one-dimensional mappings, Naukova Dumka, Kiev (1989).
D. Ruelle, Repellers for real analytic maps, Ergodic Theory and Dynam. Systems, 2, 99–107 (1982). DOI: https://doi.org/10.1017/S0143385700009603
O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19, № 6, 1565–1593 (1999). DOI: https://doi.org/10.1017/S0143385799146820
O. Sarig, Thermodynamic formalism for countable Markov shifts, Proc. Sympos. Pure Math., 89, 81–117 (2015). DOI: https://doi.org/10.1090/pspum/089/01485
Ya. G. Sinai, Markov partitions and $C$-diffeomorphisms, Funct. Anal. Pril., 2, 64–89 (1968). DOI: https://doi.org/10.1007/BF01075361
S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marstone Morse), Princeton Univ. Press, 63–80 (1965). DOI: https://doi.org/10.1515/9781400874842-006
W. Tucker, D. Wilczak, A rigorous lower bound for the stability regions of the quadratic map, Physica D, 238 (18), 1923–1936 (2009). DOI: https://doi.org/10.1016/j.physd.2009.06.020
P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236, 121–153 (1978). DOI: https://doi.org/10.1090/S0002-9947-1978-0466493-1
Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147, № 3, 585–650 (1998). DOI: https://doi.org/10.2307/120960
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