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

Rui Gao; Weixiao Shen

doi:10.3934/dcds.2014.34.2013

Article Contents

2014, Volume 34, Issue 5: 2013-2036. Doi: 10.3934/dcds.2014.34.2013

This issue Previous Article Convergence analysis of the vortex blob method for the $b$-equation Next Article Dirichlet $(p,q)$-equations at resonance

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

Received: September 2012

Revised: August 2013

Published: May 2014

Abstract Related Papers Cited by

Abstract

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,
absolutely continuous invariant measure,
non-uniform expansion,
Viana maps,
Misiurewicz-Thurston maps.

Mathematics Subject Classification: Primary: 37D25; Secondary: 37C40.

Citation:

Related Papers

Cited by

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.