# American Institute of Mathematical Sciences

October  2013, 33(10): 4567-4578. doi: 10.3934/dcds.2013.33.4567

## An equivalent characterization of the summability condition for rational maps

 1 School of Mathematics and Information Science, Henan University, Kaifeng 475004, China

Received  October 2012 Revised  February 2013 Published  April 2013

We give an equivalent characterization of the summability condition in terms of the backward contracting property defined by Juan Rivera-Letelier, for rational maps of degree at least two which are expanding away from critical points.
Citation: Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567
##### References:
 [1] H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533. doi: 10.1007/s00222-007-0108-4. [2] H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306. [3] P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46. doi: 10.1017/S0143385700001802. [4] J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96. doi: 10.1007/s002220050239. [5] J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415. doi: 10.1007/s00222-008-0152-8. [6] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055. [7] H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176. doi: 10.4064/fm198-2-6. [8] H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677. doi: 10.1007/s11425-010-3134-4. [9] H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799. doi: 10.1093/imrn/rns105. [10] T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680. doi: 10.1007/s002220050236. [11] F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403. [12] F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178. doi: 10.1016/j.ansens.2006.11.002. [13] F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x. [14] J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636. doi: 10.1017/S0143385706000629. [15] J. Rivera-Letelier, Asymptotic expansion of smooth interval maps, preprint, arXiv:1204.3071. [16] J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions, preprint, arXiv:1004.0230.

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##### References:
 [1] H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533. doi: 10.1007/s00222-007-0108-4. [2] H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306. [3] P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46. doi: 10.1017/S0143385700001802. [4] J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96. doi: 10.1007/s002220050239. [5] J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415. doi: 10.1007/s00222-008-0152-8. [6] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055. [7] H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176. doi: 10.4064/fm198-2-6. [8] H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677. doi: 10.1007/s11425-010-3134-4. [9] H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799. doi: 10.1093/imrn/rns105. [10] T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680. doi: 10.1007/s002220050236. [11] F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403. [12] F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178. doi: 10.1016/j.ansens.2006.11.002. [13] F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x. [14] J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636. doi: 10.1017/S0143385706000629. [15] J. Rivera-Letelier, Asymptotic expansion of smooth interval maps, preprint, arXiv:1204.3071. [16] J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions, preprint, arXiv:1004.0230.
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