We discuss the construction of SRB measures for some families of stretched Hénon-like maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Adler, “F-expansions revisited,” Lecture Notes in Math., 318, 1–5 (1973).
V. M. Alekseev, “Quasi-random dynamical systems, I,” Mat. Sb. (N.S.), 5, No. 1, 73–128 (1968).
D. V. Anosov, “Geodesic flows on closed Riemann manifolds with negative curvature,” Proc. Steklov Inst. Math., 90 (1967).
D. V. Anosov and Ya. G. Sinai, “Some smooth ergodic systems,” Russ. Math. Surveys, 22, 103–167 (1967).
M. Benedicks and L. Carleson, “The dynamics of the Hénon map,” Ann. Math. (2), 133, 73–169 (1991).
M. Benedicks and 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, No. 2, 163–188 (1971).
M. V. Jakobson, “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,” Comm. Math. Phys., 81, 39–88 (1981).
M. Jakobson, “Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions,” Proc. Sympos. Pure Math., 69, 825–881 (2001).
M. Jakobson, “Method of parameter exclusion. Some recollections and some new results,” Pure Appl. Funct. Anal., 5, 1377–1394 (2020).
M. Jakobson, “Topological and metric properties of one-dimensional endomorphisms,” Sov. Math. Dokl., 19, No. 6, 1452–1456 (1979).
M. V. Jakobson and S. E. Newhouse, “A two-dimensional version of the Folklore theorem,” Amer. Math. Soc. Transl. Ser. 2, 171, 89–105 (1996).
M. V. Jakobson and S. E. Newhouse, “Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle,” Astérisque, 261, 103–160 (2000).
M. Jakobson and L. Simonelli, “Countable Markov partitions suitable for thermodynamic formalism,” J. Mod. Dyn., 13, 199–219 (2018).
S. Luzzatto and H. Takahasi, “Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps,” Nonlinearity, 19, 1657–1695 (2006).
L. Mora and M. Viana, “Abundance of strange attractors,” Acta Math., 1–73 (1993).
J. Palis and J.-C. Yoccoz, “Implicit formalism for affine-like maps and parabolic composition,” in: Global Analysis of Dynamical Systems, Institute of Physics Publishing, Bristol (2001), pp. 67–87.
J. Palis and J.-C. Yoccoz, “Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles,” Publ. Math. Inst. Hautes Études Sci., 110, 1–217 (2009).
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of One-Dimensional Mappings, Naukova Dumka, Kiev (1989).
D. Ruelle, “Repellers for real analytic maps,” Ergodic Theory Dynam. Syst., 2, 99–107 (1982).
O. Sarig, “Thermodynamic formalism for countable Markov shifts,” Ergodic Theory Dynam. Systems, 19, No. 6, 1565–1593 (1999).
O. Sarig, “Thermodynamic formalism for countable Markov shifts,” Proc. Symp. Pure Math., 89, 81–117 (2015).
Ya. G. Sinai, “Markov partitions and C-diffeomorphisms,” Funct. Anal. Prilozh., 2, 64–89 (1968).
S. Smale, “Diffeomorphisms with many periodic points,” Differential and Combinatorial Topology (a Symposium in Honor of Marstone Morse), Princeton Univ. Press, Princeton, NJ, 63–80 (1965).
W. Tucker and D. Wilczak, “A rigorous lower bound for the stability regions of the quadratic map,” Phys. D, 238 (18), 1923–1936 (2009).
P. Walters, “Invariant measures and equilibrium states for some mappings which expand distances,” Trans. Amer. Math. Soc., 236, 121–153 (1978).
Lai-Sang Young, “Statistical properties of dynamical systems with some hyperbolicity,” Ann. Math. (2), 147, No. 3, 585–650 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 92–106, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.7725.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jakobson, M., Newhouse, S. SRB Measures for Some Stretched Hénon-Like Maps. Ukr Math J 76, 97–111 (2024). https://doi.org/10.1007/s11253-024-02310-1
Received:
Published:
Version of record:
Issue date:
DOI: https://doi.org/10.1007/s11253-024-02310-1