Viviane Baladi. Linear response for intermittent maps (joint work with M. Todd)

Transcription

1 Viviane Baladi Linear response for intermittent maps (joint work with M. Todd) We consider the one-parameter family T_t of Pomeau-Manneville type interval maps, with the associated absolutely continuous invariant probability measure m_t. For t in (0,1), Sarig and Gouëzel proved that the system mixes only polynomially (in particular, there is no spectral gap). We show that for any L^q observable g, the integral of g with respect to m_t is differentiable (as a function of t) for 0<=t <1-1/q, and we give a (linear response) formula for the value of the derivative. For t above 0.5, we need the faster decorrelation obtained by Gouëzel for some zero-average observables. We will also mention related work of A. Korepanov. Hiroyuki Inou On wiggly features and self-similarity of multicorns Oleg Kozlovski Cyclicity in families of one dimensional maps Is the number of attracting periodic trajectories of smooth maps of an interval or a circle is niversally bounded for a "typical" family of maps? The answer on this question is not straightforward and depends on what we define as a "typical" family and on the smoothness as well. Stefano Luzzatto SRB measures for nonuniformly hyperbolic surface We show that any surface diffeomorphism with non-zero Lyapunov exponents and admitting a natural recurrence conditions admits a Young Tower with first return inducing time, and consequently an SRB measure, thus proving Viana s conjecture in two-dimensions. As a Corollary of our argument we get that every hyperbolic measure is liftable to a Young tower. (joint work with V. Climenhaga and Y. Pesin). Marco Martens Renormalization and Symmetry The renormalizations of an infinitely renormalizable smooth unimodal map with critical exponent $\alpha>1$, say of period doubling type, converges to a fixed point of renormalization. These renormalization fixed points have a stable manifold of co-dimension at most 1. The rate of convergence can be calculated in terms of the scaling structure of the fixed point.

2 Unimodal maps with critical exponent larger than 1 do not have a structure which can be explored to study renormalization. Unlike the holomorphic context when the exponent is even. However, repeatedly renormalizing a unimodal map, creates a so-called internal structure of the map. There is a Banach space of a priori possible internal structures. This space carries a representation of the dyadic rationals, subgroup of the circle. The internal structures of the limits of renormalization are characterized by symmetry. They correspond to the trivial part of the representation. The core of the argument is the study of this representation. Similar results hold for general combinatorics. Welington de Melo Rigidity in Dynamics I will discuss several rigidity results in dynamics including my joint work with Guarino and Guarino-Martens on the rigidity of smooth critical cicle maps. Michal Misiurewicz Entropy locking Consider discontinuous piecewise linear interval maps with two pieces, where the map is increasing on one piece and decreasing on the other piece. Often the topological entropy depends only on the slopes, not on the size of the jump at the discontinuity point. We present a simple explanation of this phenomenon. This is joint work with David Cosper. Tomasz Nowicki If you are a hammer everything looks like a nail: Dynamics in Algorithms Livana Palmisano Ergodic properties of bimodal circle endomorphisms We give sufficient and necessary conditions that characterize the existence of an absolutely continuous invariant measure for a degree one C^2 endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Moreover, we prove that those conditions hold for Lebesgue almost every rotation interval. The measure obtained is a global physical measure, and it is hyperbolic. This is a joint work with S. Crovisier and P. Guarino. Juan Rivera-Letelier The large deviation principle in one-dimensional dynamics

3 For uniformly hyperbolic diffeomorphisms, the large deviation principle was established in the late 1980s by Takahashi, Orey and Pelikan, Kifer, and Young. For quadratic interval maps, the critical point is a serious obstruction to uniform hyperbolicity. Nevertheless, we show that the large deviation principle holds under a mild topological hypothesis, namely, for every nonrenormalizable quadratic map the full (level-2) large deviation principle holds for empirical means. This includes the quadratic maps without physical measure found by Hofbauer and Keller, and leads to a somewhat paradoxical conclusion: Averaged statistics hold, even for some systems without average asymptotics. This is a joint work with Yong Moo Chung and Hiroki Takahasi. Mitsuhiro Shishikura Tropical Complex Dynamics For a rational map with non-empty Fatou set, one can associate a piecewise linear map on a tree. From this ``tree map ', on ``toropicalized complex dynamics, we can derive some information on whether certain type of dynamics can be realized, or at which degree such dynamics can be realized. This tree map is supposed to describe the degeneration of rational maps under the limit of quasiconformal deformation. In this talk, we will discuss various problems related to the tropical complex dynamics. Daniel Smania Solenoidal attractors with bounded combinatorics are shy We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lesbesgue measure. Sebastian van Strien On the interface between real and complex dynamics: some questions Charles Tresser Bounding the errors for convex dynamics on one or more polytopes I will discuss the error diffusion, a greedy algorithm for approximating a sequence of inputs in a family of polytopes lying in affine spaces by an output sequence made of vertices of the respective polytopes. More precisely, as in other works now lead by Tomasz Nowicki, I consider here the case when the greed of the algorithm is dictated by the Euclidean norms of the successive cumulative errors. This algorithm can be interpreted as a time-dependent dynamical system in the vector space, where the errors live, or as a time-dependent dynamical system in an affine space containing copies of all the original polytopes (as noticed by Kitchen and Nogueira in the case of one polytope). The strategy that was used in joint work with Adler, Kitchens, Martens, Pugh, Shub, where the work was done in the affine space, cannot work here. One can already see that for two distinct intervals in a line ( a first paper on the many polytopes

4 study, that I will briefly present: THE one dimensional aspect of all that I will tell), yet the main result from the 6 authors paper, once re-interpreted, saves the day: that main result was obtained for several polytopes at once, but considered for sequences of inputs in a single one of them, a generality that turned out crucial for the general problem. A series of "new images" (= new math?) illustrates what happens if one starts with {0} as the error space and the error space grows under iteration (considering all possible next steps at once). Would I have known these complicated images, I would not have been pulling so much for obtaining a proof as we eventually put together. I will try to replace this story both in what can be loosely called "convex dynamics", and in a collection of important industrial questions: after all the birthday boy spent many years in a technical institute! The algorithm described above is a generalization of a classical technology for digital printing that I improved in joint work with Chai Wah Wu, and could serve for further practical scheduling problems. A toy problem consists in a carpool assignment problem. Said carpool is our problem where the set of polytopes is made of one standard simplex and all the simplices that happen as parts of the boundary, of the boundary of the boundary pieces. The algorithm to be discussed is a gready algorithm for a type of questions that has been abundantly studied in the Netherlands. I will present briefly a new possible path for the 6 authors theorem as we have discussed, so far informally, with Tomasz, and from there use the fact that a proof has been completed and published. Edson Vargas Invariant measure for critical covering maps of the circle We introduce a generalization of the Fibonacci combinatorics for critical covering maps of the circle and, under the additional hypothesis of negative Schwarzian derivative and critical point of order between 1 and 2, we prove that they have an absolutely continuous invariant measure. Björn Winckler Renormalization without universality One tenet of renormalization has up to now been that "topology determines geometry". That is, that two topologically conjugate infinitely renormalizable maps asymptotically have the same geometry. This "sameness" of geometry can be quantified via the concept of metric universality as well as the stronger notion of rigidity. In this talk I will show how this tenet fails in a spectacular fashion in the space of 1d Lorenz maps. I will focus on a topological class of infinitely renormalizable Lorenz maps of stationary combinatorial type. This class contains a renormalization fixed point but there is no (metric) universality, let alone rigidity. Some maps in the class have a priori bounds but others degenerate under renormalization. Contrast this with unimodal renormalization where stationary

5 combinatorics implies rigidity. The topological classes of Lorenz maps are much more intricate than their unimodal counterparts. The above example is not contrived. There are countably many such classes and we conjecture that there are finitely many (but nonzero) topological classes where we do see rigidity.