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.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... 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
« Back... 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.
« Back... 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.
These are joint works with W.Qiu, P.Roesch and X.Wang.
« Back...
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