Viana maps driven by Benedicks-Carleson quadratic maps

Gao, Rui

doi:10.1090/tran/8249

Viana maps driven by Benedicks-Carleson quadratic maps
HTML articles powered by AMS MathViewer

by Rui Gao PDF

Trans. Amer. Math. Soc. 374 (2021), 1449-1495 Request permission

Abstract:

We study the measurable dynamics of a family of skew-product maps known as Viana maps. In our setting, those maps are constructed by coupling two quadratic maps, and we aim at showing the abundance of non-uniform hyperbolicity in this family. We prove that, for any polynomial coupling function of odd degree, when the parameter pair of the two factor quadratic maps is chosen from a two-dimensional positive measure set, the associated Viana map has two positive Lyapunov exponents and admit finitely many ergodic absolutely continuous invariant probability measures.

References

José Ferreira Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 1, 1–32 (English, with English and French summaries). MR 1743717, DOI 10.1016/S0012-9593(00)00101-4
Yuan-Ling Ye, Non-uniformly expanding dynamical systems: multi-dimension, Discrete Contin. Dyn. Syst. 39 (2019), no. 5, 2511–2553. MR 3927523, DOI 10.3934/dcds.2019106
José F. Alves, Christian Bonatti, and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), no. 2, 351–398. MR 1757000, DOI 10.1007/s002220000057
José F. Alves and Daniel Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3943–3955. MR 3091785, DOI 10.1090/S0002-9939-2013-11680-1
José F. Alves and Marcelo Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 1–32. MR 1889563, DOI 10.1017/S0143385702000019
Vitor Araujo and Javier Solano, Absolutely continuous invariant measures for random non-uniformly expanding maps, Math. Z. 277 (2014), no. 3-4, 1199–1235. MR 3229987, DOI 10.1007/s00209-014-1300-z
Artur Avila and Carlos Gustavo Moreira, Statistical properties of unimodal maps: the quadratic family, Ann. of Math. (2) 161 (2005), no. 2, 831–881. MR 2153401, DOI 10.4007/annals.2005.161.831
Viviane Baladi and Marcelo Viana, Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 4, 483–517. MR 1386223
Michael Benedicks and Lennart Carleson, On iterations of $1-ax^2$ on $(-1,1)$, Ann. of Math. (2) 122 (1985), no. 1, 1–25. MR 799250, DOI 10.2307/1971367
A. M. Blokh and M. Yu. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 545–573. MR 1132757
Henk Bruin, Stefano Luzzatto, and Sebastian Van Strien, Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 621–646 (English, with English and French summaries). MR 2013929, DOI 10.1016/S0012-9593(03)00025-9
Jérôme Buzzi, Olivier Sester, and Masato Tsujii, Weakly expanding skew-products of quadratic maps, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1401–1414. MR 2018605, DOI 10.1017/S0143385702001694
P. Collet and J.-P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983), no. 1, 13–46. MR 743027, DOI 10.1017/S0143385700001802
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
Bing Gao and Weixiao Shen, Summability implies Collet-Eckmann almost surely, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1184–1209. MR 3227153, DOI 10.1017/etds.2012.173
Rui Gao and Weixiao Shen, Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps, Discrete Contin. Dyn. Syst. 34 (2014), no. 5, 2013–2036. MR 3124724, DOI 10.3934/dcds.2014.34.2013
Sébastien Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France 134 (2006), no. 1, 1–31 (English, with English and French summaries). MR 2233699, DOI 10.24033/bsmf.2500
Jacek Graczyk and Grzegorz Światek, Generic hyperbolicity in the logistic family, Ann. of Math. (2) 146 (1997), no. 1, 1–52. MR 1469316, DOI 10.2307/2951831
Wen Huang and Weixiao Shen, Analytic skew products of quadratic polynomials over circle expanding maps, Nonlinearity 26 (2013), no. 2, 389–404. MR 3007895, DOI 10.1088/0951-7715/26/2/389
M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981), no. 1, 39–88. MR 630331
Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Tomasz Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems 5 (1985), no. 4, 611–616. MR 829861, DOI 10.1017/S0143385700003199
Tomasz Nowicki and Duncan Sands, Non-uniform hyperbolicity and universal bounds for $S$-unimodal maps, Invent. Math. 132 (1998), no. 3, 633–680. MR 1625708, DOI 10.1007/s002220050236
V. A. Pliss, On a conjecture of Smale, Differencial′nye Uravnenija 8 (1972), 268–282 (Russian). MR 0299909
Feliks Przytycki, Juan Rivera-Letelier, and Stanislav Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), no. 1, 29–63. MR 1943741, DOI 10.1007/s00222-002-0243-x
Juan Rivera-Letelier and Weixiao Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 6, 1027–1083 (English, with English and French summaries). MR 3297154, DOI 10.24033/asens.2233
Daniel Schnellmann, Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents, Ergodic Theory Dynam. Systems 28 (2008), no. 1, 245–266. MR 2380309, DOI 10.1017/S0143385707000429
Daniel Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map, Nonlinearity 22 (2009), no. 11, 2681–2695. MR 2550691, DOI 10.1088/0951-7715/22/11/006
Javier Solano, Non-uniform hyperbolicity and existence of absolutely continuous invariant measures, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 1, 67–103. MR 3077635, DOI 10.1007/s00574-013-0004-z
Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
Masato Tsujii, Physical measures for partially hyperbolic surface endomorphisms, Acta Math. 194 (2005), no. 1, 37–132. MR 2231338, DOI 10.1007/BF02392516
Paulo Varandas, Statistical properties of generalized Viana maps, Dyn. Syst. 29 (2014), no. 2, 167–189. MR 3195850, DOI 10.1080/14689367.2013.868868
Marcelo Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 63–96. MR 1471866