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

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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

Abstract
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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.

Keywords: Lyapunov exponent, Viana maps, non-uniform expansion, Misiurewicz-Thurston maps., absolutely continuous invariant measure.

Mathematics Subject Classification: Primary: 37D25; Secondary: 37C4.

Citation: Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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. doi: 10.1017/S0143385702001694.
[7]	L. Carleson and T. W. Gamelin, Complex Dynamics,, Universitext: Tracts in Mathematics. Springer-Verlag, (1993).
[8]	W. Huang and W. Shen, Analytic skew products of quadratic polynomials over circle expanding maps,, Nonlinearity, 26 (2013), 389. doi: 10.1088/0951-7715/26/2/389.
[9]	J. Milnor, Dynamics in One Complex Variable,, Third Edition, (2006).
[10]	T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents,, Ergodic Theory Dynam. Systems, 5 (1985), 611. 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. doi: 10.1017/S0143385707000429.
[12]	D. Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map,, Nonlinearity, 22 (2009), 2681. doi: 10.1088/0951-7715/22/11/006.
[13]	M. Viana, Multidimensional nonhyperbolic attractors,, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63. doi: 10.1007/BF02699535.

show all references

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. 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. 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. 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. 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. 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. doi: 10.1017/S0143385702001694.
[7]	L. Carleson and T. W. Gamelin, Complex Dynamics,, Universitext: Tracts in Mathematics. Springer-Verlag, (1993).
[8]	W. Huang and W. Shen, Analytic skew products of quadratic polynomials over circle expanding maps,, Nonlinearity, 26 (2013), 389. doi: 10.1088/0951-7715/26/2/389.
[9]	J. Milnor, Dynamics in One Complex Variable,, Third Edition, (2006).
[10]	T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents,, Ergodic Theory Dynam. Systems, 5 (1985), 611. 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. doi: 10.1017/S0143385707000429.
[12]	D. Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map,, Nonlinearity, 22 (2009), 2681. doi: 10.1088/0951-7715/22/11/006.
[13]	M. Viana, Multidimensional nonhyperbolic attractors,, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63. doi: 10.1007/BF02699535.

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