ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 01 Oct 2023 01:44:36 GMT2023-10-01T01:44:36Z50101- Decay of geometry for unimodal maps: The C2 casehttps://scholarbank.nus.edu.sg/handle/10635/103103Title: Decay of geometry for unimodal maps: The C2 case
Authors: Shen, W.
Abstract: We prove that a C2 unimodal interval map with critical order not greater than 2 has the decay of geometry property, by showing that all the cross-ratio estimates needed in the previous proof for the C3 case remain true. © 2009 Science in China Press and Springer-Verlag GmbH.
Mon, 01 Jun 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031032009-06-01T00:00:00Z
- Combinatorial rigidity for unicritical polynomialshttps://scholarbank.nus.edu.sg/handle/10635/103001Title: Combinatorial rigidity for unicritical polynomials
Authors: Avila, A.; Kahn, J.; Lyubich, M.; Shen, W.
Abstract: We prove that any unicritical polynomial fc : z → zd + C which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the "Multibrot set") is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030012009-01-01T00:00:00Z
- Parapuzzle of the multibrot set and typical dynamics of unimodal mapshttps://scholarbank.nus.edu.sg/handle/10635/103915Title: Parapuzzle of the multibrot set and typical dynamics of unimodal maps
Authors: Avila, A.; Lyubich, M.; Shen, W.
Abstract: We study the parameter space of unicritical polynomials fc: z zd + c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the "principal nest" of parapuzzle pieces. © European Mathematical Society 2011.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039152011-01-01T00:00:00Z
- On stochastic stability of expanding circle maps with neutral fixed pointshttps://scholarbank.nus.edu.sg/handle/10635/103759Title: On stochastic stability of expanding circle maps with neutral fixed points
Authors: Shen, W.; Van Strien, S.
Abstract: It is well known that the Manneville-Pomeau map with a parabolic fixed point of the form is stochastically stable for ≥ 1 and the limiting measure is the Dirac measure at the fixed point. In this paper, we show that if ∈ (0, 1), then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero. © 2013 Copyright Taylor and Francis Group, LLC.
Sun, 01 Sep 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037592013-09-01T00:00:00Z
- An improved real Ck Koebe principlehttps://scholarbank.nus.edu.sg/handle/10635/102837Title: An improved real Ck Koebe principle
Authors: Li, S.; Shen, W.
Abstract: We prove an improved Ck Koebe principle for a Ck interval map with non-flat critical points, where k ≥ 3, that requires no disjointness of the intervals involved. © 2009 Cambridge University Press.
Fri, 01 Oct 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028372010-10-01T00:00:00Z
- Analytic skew products of quadratic polynomials over circle expanding mapshttps://scholarbank.nus.edu.sg/handle/10635/102862Title: Analytic skew products of quadratic polynomials over circle expanding maps
Authors: Huang, W.; Shen, W.
Abstract: We prove that a Viana map with an arbitrarily non-constant real-analytic coupling function admits two positive Lyapunov exponents almost everywhere. © 2013 IOP Publishing Ltd & London Mathematical Society.
Fri, 01 Feb 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028622013-02-01T00:00:00Z
- Analytic skew-products of quadratic polynomials over misiurewicz-thurston mapshttps://scholarbank.nus.edu.sg/handle/10635/102863Title: Analytic skew-products of quadratic polynomials over misiurewicz-thurston maps
Authors: Gao, R.; Shen, W.
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.
Thu, 01 May 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028632014-05-01T00:00:00Z
- Topological invariance of a strong summability condition in one-dimensional dynamicshttps://scholarbank.nus.edu.sg/handle/10635/104387Title: Topological invariance of a strong summability condition in one-dimensional dynamics
Authors: Li, H.; Shen, W.
Abstract: We say that a rational map satisfies a strong summability condition if, for each critical value v of f belonging to the Julia set, we have for any β>0. We give an equivalent formulation of this property in terms of backward contracting properties of f. We prove that the strong summability condition is a topological invariant for rational maps with one critical point in the Julia set and without parabolic cycles. For unimodal interval maps, we obtain that the strong summability condition is invariant under quasisymmetric conjugacy. © 2012 The Author(s).
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043872013-01-01T00:00:00Z
- On stochastic stability of non-uniformly expanding interval mapshttps://scholarbank.nus.edu.sg/handle/10635/103760Title: On stochastic stability of non-uniformly expanding interval maps
Authors: Shen, W.
Abstract: We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic stability. ©2013 London Mathematical Society.
Fri, 01 Nov 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037602013-11-01T00:00:00Z
- On non-uniform hyperbolicity assumptions in one-dimensional dynamicshttps://scholarbank.nus.edu.sg/handle/10635/103730Title: On non-uniform hyperbolicity assumptions in one-dimensional dynamics
Authors: Li, H.B.; Shen, W.X.
Abstract: We give an essentially equivalent formulation of the backward contracting property, defined by Juan Rivera-Letelier, in terms of expansion along the orbits of critical values, for complex polynomials of degree at least 2 which are at most finitely renormalizable and have only hyperbolic periodic points, as well as all C3 interval maps with non-flat critical points. © 2010 Science China Press and Springer-Verlag Berlin Heidelberg.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037302010-01-01T00:00:00Z