# THEMATIC PROGRAMS

May 22, 2022

### Audio of talks available here.

 May 31, 1:30 p.m. ** (Note special time) Hui Guo (Fields Institute) Integrable Teichmuller spaces We introduce a new kind of subspaces of the universal Teichmuller space. Some characterizations of them are given in terms of univalent functions, Beltrami coefficients and quasisymmetric homeomorphisms of the boundary of the unit disc. May 1 Mitsuhiro Shishikura (Kyoto University and Fields Institute) Teichmuller contraction and renormalization According to Royden-Gardiner theorem, any holomorphic mapping between Teichmuller spaces does not expand the Teichmuller distance. (Just like Schwarz-Pick theorem says that any holomorphic mapping between hyperbolic Riemann surfaces does not expand the Poincare metric.) In the theory of renormalizations, one often wants to show that the renormalization map (which is defined in a transcendental way) on the space of certain dynamical systems is hyperbolic or contracting. Therefore Royden-Gardiner theorem is an obvious candidate of tools to obtain the contraction. This idea was first used by Sullivan in his work on generalized Feigenbaum type renormalization. In this talk, we will discuss two applications of Teichmuller theory to renormalizations: 1. Parabolic renormalization for parabolic fixed points and their perturbations. 2. Rigidity of real quadratic polynomials only using Yoccoz's combinatorial a priori bounds. For these cases, we deal with the Teichmuller spaces of a disk or a punctured disk and show that a certain inclusion map induces a contraction in Teichmuller distance. Apr. 17 Jeremy Kahn (SUNY Stony Brook / Fields Institute) The complex a priori bounds, continued We describe how to reduce the a priori bounds for bounded-primitive type renormalization to a simple combinatorial statement regarding the combinatorial dynamics of arcs. We show how the combinatorial statement follows from the expanding dynamics on the Hubbard tree. Apr. 10 Mary Rees (Liverpool) A fundamental domain for V_{3}. Abstract: We consider the space V_{n} of quadratic rational maps with one named critical point of period n, quotiented by M\"obius conjugacy preserving named critical points. Thus, the space V_{1} is the space of all quadratic polynomials up to affine conjugacy. The parameter space V_{1} must be one of the most studied in all dynamics. But V_{1} differs from most other parameter spaces in an important respect: there is a natural base map'' and, within the Mandelbrot set, natural paths, up to a natural homotopy, to any other map in the Mandelbrot set. The simplest parameter space for which this fails to be true is V_{3}. One can say (truthfully) that there is no canonical choice of fundamental domain for V_{3,m}, which is obtained from V_{3} by removing a natural dynamically defined puncture set. I shall exhibit a fundamental domain, using the dynamical planes of three maps within V_{3} and a theory known as the Resident's View''. This enables one to at least formulate an analogue of MLC for this family. Three parts of the fundamental domain are straightforward (although the proof in one of these cases is probably new). The structure of the fourth part is much more interesting, involving a spiral in the dynamical plane of the so-called aeroplane'' quadratic polynomial z\mapsto z^{2}+c for the (unique) real parameter c for which the critical point 0 has period 3. Apr. 3 Matilde Martinez (Fields Institute) Measures on hyperbolic surface laminations We will consider laminations by hyperbolic Riemann surfaces, and different measures that can be associated with these objects (holonomy-invariant measures, harmonic measures, measures invariant under the geodesic and horocycle flows). We will state some results that show how these measures are related. We will mainly focus on two families of examples: Riccati foliations and Hilbert modular foliations. Mar. 20 Peter Makienko Remarks on Ruelle Operator, Invariant Differentials and invariant line fields problem. Abstract: TBA Mar. 13th **12PM (Note special time**) Special Brown Bag Lunch Dynamical Systems Seminar Dierk Schleicher (IU Bremen), Johannes Rueckert (IU Bremen), Magnus Aspenberg (Fields), and Vladim Timorin (Stony Brook) + anyone who wants to can give input. Combinatorics of Rational maps Mar. 6 Roland Roeder (Fields) Newton's Method in two complex variables: Indeterminant points and the topology of basins of attraction The equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, but the Newton Map N: C^2 \rightarrow C^2 for finding the four roots has very complicated dynamics: N is four-to-one and N has points of indeterminacy. Furthermore, high iterates of N have many points of indeterminacy. By restricting to parameters |1-B|>1, all of these points of indeterminacy are in the set X_l = Re(x) < 1/2, which is invariant under N. If one wants to consider the homotopy type of a basin of attraction, W(r_i), for one of the roots r_i \in X_l under N, one encounters a kind of topological indeterminacy.'' When studying the homotopy type of a loop \gamma in W(r_i), should one consider the homotopies that hit the points of indeterminacy of N^k or should one avoid them? Both seem reasonable. To avoid such questions, one can perform blow-ups at the points of indeterminacy of all iterates of N, obtaining a new space X_l^\infty from X_l on which all iterates of N are defined. We show how to make precise the notion of linking numbers in X_l^\infty, overcoming a different kind of indeterminacy that is a result of the fact that H_2(X_l^\infty) is infinitely generated. Having developed this technology, I will explain how we can use it to study the homotopy type of the basins of attraction W(r_i) within X_l^\infty. Feb. 27 Robert Devaney (Boston University) Rings around the McMullen Domain Feb. 13 Andrzej Bis (University of Illinois at Chicago) Dynamics of foliated spaces in codimension greater than one Abstract. Dynamics of foliated spaces can be characterized by the exceptional minimal sets and topological entropy. Dynamical theory of codimension one foliations was developed by Hector, Cantwell and Conlon, and others. We present some results on exceptional minimal sets, which are analogous to the attractors in a standard dynamical system, of foliated spaces in codimension greater than one. Feb. 6 Yoel Feler (Fields Institute) Holomorphic endomorphisms of configuration spaces Abstract: The most traditional configuration space C(X,n) of a complex space X consists of all n point subsets ("configurations") Q in X. If X carries an additional geometric structure, it may be taken into account; say if X=CP^m or C^m, the space C(X,n;gp) of geometrically generic configurations consists of all n point configurations Q such that no hyperplane in X contains more than m points of Q. An automorphism T of X (preserving an additional geometric structure, whenever it is relevant) produces a holomorphic endomorphism f of the configuration space via f(Q)=TQ. If the automorphism group Aut X is a complex Lie group, one may take T=T(Q) depending analytically on a configuration Q and define the corresponding holomorphic endomorphism f by f(Q)=T(Q)Q. Such a map f is called tame. In the talk, we shall see that for every non-hyperbolic Riemann surface X all "non-degenerate" holomorphic endomorphisms of configuration spaces C(X,n) are tame. To some extent, this is true also for spaces of geometrically generic configurations. Jan. 30 Tomoki Kawahira (Fields Institute) Tessellation and Lyubich-Minsky laminations associated with quadratic maps Abstract: In 1990s, M.Lyubich and Y.Minsky introduced the hyperbolic 3-laminations associated with rational maps as an analogue of the hyperbolic 3-manifolds associated with Kleinian groups. In this talk I will present a new method to describe topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degeneration of quadratic maps. The method is based on tessellation of filled Julia sets, which gives a nice organization of the dynamics inside the filled Julia set like external rays outside. Jan. 23 Dierk Schleicher (International University Bremen) Dynamics of transcendental entire functions from the point of view of polynomials Abstract: In this talk, we will discuss some fundamentals of iterated entire functions and indicate why and how they differ from polynomial dynamics, with a special focus on the simplest representatives in both cases, and a view towards generalization. Jan. 16 Dmitrii V. Anosov (Steklov Mathematical Institute) A lemma about families of epsilon pseudo-trajectories revisited In hyperbolic dynamics there are results related to the existence of a true trajectory near an epsilon pseudo-trajectory for sufficiently small epsilon (although, formally, some of these results are expressed rather differently.) Many years ago I found a lemma which covers most of these questions. The proof is rather involved and some famous mathematicians expressed complaints about the difficulty. In this talk, I will present a simplified version of the proof. Dec. 5 Viviane Baladi (Institut Mathematiques de Jussieu) Anisotropic spaces of distributions and dynamical zeta functions Abstract: (Joint work with M. Tsujii) The Ruelle transfer operator is a powerful tool in ergodic theory, which involves composition with the dynamics. Many relevant dynamical systems are hyperbolic, i.e. they involve contracting and expanding directions. Composition with a contraction improves regularity - but composing with an expanding map "worsens" regularity: It has been an open problem for many years to find a space of distributions on which composition by a hyperbolic diffeomorphism (of finite smoothness) can be well understood. Last year we constructed such a space and estimated the essential spectral radius of the transfer operator on this space. After recalling this result, we shall describe more recent progress including spectral interpretation of zeroes of dynamical determinants. Nov. 28 Kristian Bjerklov (University of Toronto) The dynamics of the quasi-periodic Schroedinger cocycle at the lowest energy of the spectrum Abstract: We will study properties of the quasi-periodic Schroedinger equation at the lowest energy of the spectrum. This will lead us into the study of phase transitions. Moreover, we will answer a question by M. Herman concerning the geometry of a certain minimal set - a non-chaotic strange atractor - of the projective Schroedinger cocycle. We study the case of large coupling constant and Diophantine frequency. Nov. 25, 1:10 p.m. **Note: special day and time Mitchell Feigenbaum (The Rockefeller University) Exponents in period doubling Nov. 24, 1:10 p.m. **Note: special day and time Yulij Ilyashenko (Cornell University) Topological properties of polynomial and analytic foliations Abstract: Geometrical study of holomorphic foliations of the complex plane, both projective and affine, lies on the boundary of differential equations, topology and complex analysis. Foliations of \Bbb CP^2 have an algebraic origin: they are defined by polynomial vector fields, but their behavior is highly transcendental. Their properties are drastically different from those of real polynomial vector fields. Properties of density of leaves, absolute rigidity and existence of a countable number of limit cycles were discovered by different authors in 60s and 70s. The talk will present these results together with a survey of the further development and open problems. Foliations of \Bbb C^2 have an analytic origin: they are defined by analytic vector fields. Generic properties of these fields were studied very recently. Yet genericity of density of leaves and existence of the infinite number of complex limit cycles is recently proved. Moreover, generic leaves of such foliations are either disks, or cylinders. These results are obtained by graduate students Firsova, Kutuzova and Volk. Nov. 21 Konstantin Khanin (University of Toronto) Minimizers for random Lagrangian systems Abstract: We shall discuss random Aubry-Mather theory and prove that for time-dependent random Lagrangian systems on compact manifolds there exists a unique global minimizer. In the one-dimensional case we show that the global minimizer corresponds to a hyperbolic invariant measure for the random Lagrangian flow. We also discuss dynamical properties of shocks and show that their global structure is quite rigid and reflects the topology of the configuration manifold. Nov. 14 ** talk at 4:10 p.m. Israel Sigal (University of Toronto). Spectral Renormalization Group and Theory of Radiation Abstract: Non-relativistic quantum electrodynamics describes interaction of charged particles (electrons and nuclei) with quantized electro-magnetic field (photons). The key problem here is to describe emission and absorption of radiation by systems of matter such as atoms and molecules. In this talk I will present some recent rigorous results on the problem of radiation and describe a novel renormalization group technique used in proving these results. I will not assume any prior knowledge of quantum field theory or quantum mechanics. Nov. 7 Marco Martens (University of Groeningen) Henon renormalization (I) Abstract: This mini-course will introduce a renormalization operator for dissipative Henon-like maps. The fixed point of the one-dimensional renormalization operator will also be a hyperbolic fixed point of the Henon-renormalization operator. This corresponds to universal geometrical properties of the Cantor attractor of infinitely renormalizable Henon-like maps. However, the two-dimensional theory is richer than the unimodal case. In particular, the Cantor attractor is not rigid, does not lie on a smooth curve and generically does not have bounded geometry. The quantitative aspects of these phenomena are controlled by the average Jacobian. The global topological properties of finitely renormalizable Henon-like maps in phase and parameter space are also controlled by the average Jacobian. In particular, density of hyperbolicity will be discussed in a neighborhood of the infinitely renormalizable maps. Oct. 31 Hans Koch (University of Texas, Austin) Renormalization of vector fields (I) Abstract: This mini-course covers some of the recent developments in the renormalization of flows - mainly Hamiltonian flows and skew flows. After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and invariant tori, including shearless tori, and non-differentiable tori for critical Hamiltonians. Then we discuss properties related to the spectrum of the linearized renormalization transformation, such as the accumulation rates for sequences of closed orbits. The last part describes extensions from "simple" to Diophantine rotation vectors. This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions. Whenever appropriate, the discussion of details will be restricted to special cases where inessential technical complications can be avoided. Oct. 24 Michael Shub (U Toronto) Lower bounds for the entropy in several families of dynamical systems Abstract: Using soft techniques we prove lower bounds for the maximum entropy of a system in a family in terms of the entropy of a random product of the systems in the family. We accomplish this for two families of immersions of the circle. For one family this is joint work with Leonel Robert and Enrique Pujals and the other Rafael de la Llave and Carles Simo. Time permitting we recall a two dimensional analog whose entropy properties are still unknown, but for which partial results have been achieved in joint work with Francois Ledrappier, Carles Simo and Amie Wilkinson. Oct. 17 Robert MacKay (University of Warwick, UK) Some robustly mixing fluid flows Abstract: I suggest an example of a C^3 divergence-free vector field in a domain of R^3 with smooth boundary, vanishing on the boundary, which is mixing and remains so for all small perturbations in this class. Two other candidates are also presented. Oct. 10 Thankgiving holiday Oct. 3 Charles Pugh (UC Berkeley) Smoothing Topological Manifolds Abstract: The Cairns-Whitehead Smoothing Theorem is proved by dynamical systems methods, namely the Invariant Section Theorem. Sept. 26 Roland Roeder (Fields Institute) Super-stable manifolds for Newton's methods in two complex variables Abstract: While the equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, the dynamics of the Newton map N(x,y) for finding the four roots is quite complicated. In particular, N is many-to-one and N has points of indeterminacy. The two vertical lines x=0 and x=1 are invariant under N and super-attracting. Within these lines the ''circles'' Re(y) = 1/2 and Re(y) = (1-B)/2, respectively, are hyperbolically repelling with multiplier 2. In this talk we will prove that these circles have superstable manifolds of real dimension 3 using the technique of holomorphic motions. These manifolds extend to all points with Re(x) < 1/2 and Re(x) > 1/2 respectively and provide insight into the topology of the basins of attraction for the four roots. This work follows the ideas of John H. Hubbard and Sebastien Krief. Sep. 19 M. Shishikura, Kyoto University Renormalization for irrationally indifferent fixed points of holomorphic maps Abstract: Indifferent fixed points of holomorphic maps give rise to delicate problems such as linearization problems, discontinuous Julia sets etc. In this talk, we review the study of those phenomena from the renormalization point of view. We define a certain class of holomorphic maps with "non-degenerate" parabolic fixed points. The parabolic renormalizaion is defined for this class and shown to leave the class invariant. Then the class is still invariant under a small perturbation, therefore we can handle irrationally indifferent fixed points with large continued fraction coefficients. This is a joint work with Hiroyuki Inou. We will compare this approach with Yoccoz's and McMullen's renormalizations. We will also mention possible applications such as Buff-Cheritat's work toward positive measure Julia sets.