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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

Rui Gao 1, and  Weixiao Shen 1,

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.
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, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.  Google Scholar

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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.  Google Scholar

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J. Milnor, Dynamics in One Complex Variable, Third Edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[10]

T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 611-616. doi: 10.1017/S0143385700003199.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96. doi: 10.1007/BF02699535.  Google Scholar

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), 1-20. doi: 10.3934/dcds.2006.15.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[9]

J. Milnor, Dynamics in One Complex Variable, Third Edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[10]

T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 611-616. doi: 10.1017/S0143385700003199.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math., 85 (1997), 63-96. doi: 10.1007/BF02699535.  Google Scholar

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