# American Institute of Mathematical Sciences

May  2014, 34(5): 2013-2036. doi: 10.3934/dcds.2014.34.2013

## Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps

 1 Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076, Singapore, Singapore

Received  September 2012 Revised  August 2013 Published  October 2013

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.
Citation: Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013
##### References:
 [1] J. F. Alves, A survey of recent results on some statistical features of non-uniformly expanding maps, Discrete Contin. Dyn. Syst., 15 (2006), 1-20. doi: 10.3934/dcds.2006.15.1. [2] J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4), 33 (2000), 1-32. doi: 10.1016/S0012-9593(00)00101-4. [3] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057. [4] J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics, Proc. Amer. Math. Soc., 141 (2013), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1. [5] J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32. doi: 10.1017/S0143385702000019. [6] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-product of quadratic maps, Ergodic Theory Dynam. Systems, 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694. [7] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. [8] W. Huang and W. Shen, Analytic skew products of quadratic polynomials over circle expanding maps, Nonlinearity, 26 (2013), 389-404. doi: 10.1088/0951-7715/26/2/389. [9] J. Milnor, Dynamics in One Complex Variable, Third Edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. [10] T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 611-616. doi: 10.1017/S0143385700003199. [11] D. Schnellmann, Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents, Ergodic Theory Dynam. Systems, 28 (2008), 245-266. doi: 10.1017/S0143385707000429. [12] D. Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map, Nonlinearity, 22 (2009), 2681-2695. doi: 10.1088/0951-7715/22/11/006. [13] M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96. doi: 10.1007/BF02699535.

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##### References:
 [1] J. F. Alves, A survey of recent results on some statistical features of non-uniformly expanding maps, Discrete Contin. Dyn. Syst., 15 (2006), 1-20. doi: 10.3934/dcds.2006.15.1. [2] J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4), 33 (2000), 1-32. doi: 10.1016/S0012-9593(00)00101-4. [3] J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057. [4] J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics, Proc. Amer. Math. Soc., 141 (2013), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1. [5] J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems, 22 (2002), 1-32. doi: 10.1017/S0143385702000019. [6] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-product of quadratic maps, Ergodic Theory Dynam. Systems, 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694. [7] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. [8] W. Huang and W. Shen, Analytic skew products of quadratic polynomials over circle expanding maps, Nonlinearity, 26 (2013), 389-404. doi: 10.1088/0951-7715/26/2/389. [9] J. Milnor, Dynamics in One Complex Variable, Third Edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. [10] T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 611-616. doi: 10.1017/S0143385700003199. [11] D. Schnellmann, Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents, Ergodic Theory Dynam. Systems, 28 (2008), 245-266. doi: 10.1017/S0143385707000429. [12] D. Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map, Nonlinearity, 22 (2009), 2681-2695. doi: 10.1088/0951-7715/22/11/006. [13] M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96. doi: 10.1007/BF02699535.
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