Workshop on Non-uniformly Hyperbolic and Neutral One-dimensional Dynamics
(23 - 27 Apr 2012)


~ Abstracts ~

 

On the Whitney-holder differentiability of the SRB measure in the quadratic family
Viviane Baladi, Ecole Normale Supérieure, France


For a smooth one-parameter family of smooth hyperbolic discrete-time dynamics (i.e. Anosov systems, which are structurally stable), the SRB measure depends differentiably on the parameter, say t, and the derivative is given by an explicit "linear response" formula (Ruelle, 1997). When structural stability does not hold, the linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where the regularity can be only t log (t) for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case) and Baladi-Smania (slow recurrence case) recently obtained linear response for fully tangential families (remaining in a topological class). We now study the transversal case (e.g. the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).
Joint work with M. Benedicks and D. Schnellmann.

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Uniform hyperbolicity and absolutely continuous invariant measures for double standard maps
Michael Benedicks, Royal Institute of Technology, Sweden


The double standard family is a family of maps of the unit circle which are double covers of the circle with an inflexion point, which may be critical. The work presented is a study of the parameter space of this family. For different parameters the maps have be uniformly expanding, have attractive periodic orbits or have absolutely continuous invariant measures. The parameter space can to a large extent be understood.

This is joint work with Michal Misiurewicz and Ana Rodrigues.

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The attractors of complex quadratic polynomials with an irrationally indifferent fixed point
Davoud Cheraghi, University of Warwick, UK


It is well-known that the local (and semi-local) dynamics of a holomorphic germ near an irrationally indifferent fixed point may be very complicated. The near parabolic renormalization of Inou-Shishikura is an scheme introduced to study very large iterates near such fixed points of complex quadratic polynomials. In this talk we explain how one can use this scheme to describe the geometry of the attractors of complex quadratic polynomials with an irrationally indifferent fixed point.

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A near parabolic renormalization invariant class for unisingular holomorphic maps, a la Inou-Shishikura
Arnaud Cheritat, Mathematics Institute of Toulouse, France


The work of Inou and Shishikura has strong applications to the study of quadratic polynomials' dynamics. We will explain how we adapt it to unicritical polynomials, and how we plan to do the same for the exponential family.

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Shishikura trees associated with disconnected Julia sets
Guizhen Cui, Chinese Academy of Sciences, China


We discuss the topological structures of disconnected Julia sets of sub-hyperbolic rational maps. At first, we define the canonical stable multicurve for a sub-hyperbolic rational map, and a continuous map on its dual tree - we call it Shishikura tree. Then we give a sufficient and necessary condition under which a tree map can be realized as a Shishikura tree map. In the last, we will count the number of cycles of complex type Julia component by its Shishikura tree map.

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On deterministic perturbations of non-uniformly expanding interval maps
Bing Gao, National University of Singapore


We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map f satisfying a summability condition. For a one-parameter family f_t of maps with f_0=f, we prove that t=0 is a Lebesgue density point of the set of parameters for which f_t satisfies both the Collect_Eckmann condition and a strong polynomial recurrence condition.

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Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps
Rui Gao, National University of Singapore


We study skew-products of quadratic maps over certain Misiurewicz-Thurston maps coupled by non-constant polynomial functions. We prove that these systems admit two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probablity measure.

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On domains and ranges of near-parabolic renormalizations
Hiroyuki Inou, Kyoto University, Japan


Shishikura and the speaker defined a class of holomorphic maps with a fixed point at the origin invariant under parabolic and near-parabolic renormalizations. We prove that the domain is contained in the range for the (near-)parabolic renormalization of a given map in this class. As an application, we discuss the non-existence of invariant line fields on the Julia sets for infinitely renormalizable quadratic polynomials in the sense of near-parabolic renormalization.

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Topological invariance of a strong summability condition in one-dimensional dynamics
Huaibin Li, Pontifical Catholic University of Chile, Chile


We say that a rational (or interval) map satisfies a strong summability condition if it satisfies summability condition with any positive exponent. First we give an equivalent formulation of this property in terms of backward contracting properties. Then 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.
Joint work with Weixiao Shen

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Reciprocal relations between geometrical and statistical properties for nonuniformly expanding maps
Stefano Luzzatto, The Abdus Salam International Centre for Theoretical Physics, Italy


Statistical properties of dynamical systems are often deduced from particular geometric properties (e.g. Markov structures) which are assumed or can be proved to exist in certain situations. In this talk I will discuss the extent to which such geometric properties are necessary as well as sufficient conditions, i.e. whether they themselves can be deduced follow from the statistical properties.

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Low temperature phase transitions in the quadratic family
Juan Rivera-Letelier, Pontifical Catholic University of Chile, Chile


We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.
This is a joint work with Daniel Coronel.

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On topological complexity of Cantor attractors for unimodal maps
Weixiao Shen, National University of Singapore


We study the topological complexity of a regular unimodal map restricting to a Cantor attractor. We show that for any open cover U, the complexity function p( U, n) is of order n\log n. This is a joint work with Li Simin.

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Straight brush model for irrationally indifferent fixed points of holomorphic functions
Mitsuhiro Shishikura, Kyoto University, Japan


We propose a topological model for the local invariant sets (hedgehogs) for irrationally indifferent fixed points of holomorphic functions. Main tools are the near-parabolic renormalization (joint work with H. Inou) and dynamical charts.

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On a conjecture of Milnor on monotonicity of entropy and Tresser's version
Sebastian van Strien, Imperial College London and Warwick University, UK


A few years ago, Henk Bruin and I solved a 20 years old conjecture of Milnor asserting that isentropes are connected within the space of real polynomials with only real critical points. (An isentrope is the set of maps of a given topological entropy.) More recently Tresser postulated a related conjecture, dropping the assumption that all critical points are real. In this talk I will discuss recent progress on Tresser's question.

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Stability, bifurcation and classification of minimal sets in random complex dynamics
Hiroki Sumi, Osaka University, Japan


We consider random complex dynamics on the Riemann sphere. There are many new phenomena in random complex dynamics which cannot hold in the usual iteration dynamics of a single rational map. In this talk, we show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears due to the automatic cooperation of many kinds of maps in the system (cooperation principle). Moreover, we investigate the bifurcation of limit states of one-parameter family of random complex dynamics. To show these results, we introduce classification of minimal sets of systems.

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Large deviation principle for Benedicks-Carleson quadratic maps
Hiroki Takahasi, University of Tokyo, Japan


Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue (acips). For a positive measure set of these quadratic maps we take the acip as a reference measure, and establish the large deviation principle, i.e. give a rate function which estimates the exponential probability that empirical distributions stay away from the acip. Joint work with Yong Moo Chung (Hiroshima University)

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Spectrum of transfer operators for geodesic flows on negatively curved manifolds
Masato Tsujii, Kyushu University, Japan


We consider one parameter families of transfer operators associated to geodesic flows on negatively curved manifolds (or any contact Anosov flows). It is known that the generators of those families of transfer operator have discrete spectrum if we take an appropriate function spaces for them to act. The question is how the discrete spectra are distributed on the complex plane. In the case of surface of constant negative curvature, we know a complete description of the spectrum through a classical results of Selberg on the zeta function. We try to see how that picture is true in the case of variable curvature.

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Dynamics of McMullen maps
Yongcheng Yin, Zhejiang University, China


We study the dynamics for a special family of rational maps with the form $z^n+\lamda/z^n$, $n>2$. It is proved that the unbounded Fatou component is always a Jordan domain except the Julia set is totally disconnected and each hyperbolic component in the parameter space is a Jordan domian. These results were conjectures raised by R.Devaney.
These are joint works with W.Qiu, P.Roesch and X.Wang.

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