UofT Dynamics Seminar - Fall 2021

The seminar meets online at 3.10 PM on Mondays.

Zoom link: https://utoronto.zoom.us/j/88134686264
Passcode: 452271

September 27, 2021
Amadeu Delshams, Universitat Politècnica de Catalunya
Celestial Mechanics tools for studying the hydrogen atom [video]

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian depending on one parameter K>0, which is a perturbation of the standard Kepler problem. The associated Hamiltonian system has two equilibria: L1 (center-saddle for all K) and L2 (center-center for small K and complex-saddle otherwise). Associated to L1 there is a family of Lyapunov periodic orbits that form a normally hyperbolic invariant manifold (NHIM). In this talk, we compute the primary transversal homoclinic orbits to the NHIM (and therefore the associated scattering maps) combining Poincaré-Melnikov methods with numerical methods. It should be noted that the transversality of these homoclinic orbits is exponentially small in K (in analogy with the libration point L3 of the R3BP). This is a joint work with Mercè Ollé and Juan R. Pacha (Universitat Politècnica de Catalunya).

October 18, 2021
Kirill Lazebnik, University of Toronto
Transcendental Julia Sets of Minimal Hausdorff Dimension [video]

We discuss an approach to the construction of entire functions with Julia sets having minimal Hausdorff dimension. This talk will not assume a background in complex dynamics. This talk is based on joint work with Jack Burkart.

November 1, 2021
Paulo Varandas, Universidade Federal da Bahia and Universidade de Porto
Irregular behavior for semigroup actions [video] [slides]

The foundations of the classical ergodic theory rely on ergodic theorems, which guarantee an almost everywhere convergence of time averages (with respect to invariant probability measures). Notwithstanding, it is often the case that the set of points with irregular behavior, sometimes referred to as historic behavior, is quite relevant (e.g. has full entropy, full Hausdorff dimension, full metric mean dimension, etc). In this talk I will present some of the recent advances and challenges to understand the set of points with irregular behavior in the context of (semi)group actions, depending on the type of averaging. For instance, one can define spherical averages or Birkhoff averages determined by each infinite path in the Cayley graph of the (semi)group, whose irregular behavior is often unrelated. In the context of finitely generated semigroup actions I will provide sufficient conditions under which the Birkhoff averages associated to the 'majority' of infinite paths in the Cayley graph of the semigroup exhibit irregular behavior. We will exemplify these results in the context of certain locally constant SL(2,R) cocycles, discussing set of matrix products and projective directions that present Lyapunov irregular behavior.

November 8, 2021
Petr Kosenko, University of Toronto
The fundamental inequality for random walks on cocompact Fucshian groups [video]

Given a finitely presented group (G, S), a left-invariant distance d on G and a nice enough random walk (X_n) on G, we can associate three invariants: the Avez entropy h, the drift l, and the logarithmic volume v. The fundamental inequality states that h <= l*v whenever all three invariants are well-defined. However, we don't always end up having an equality h = l*v, and classifying all triples (G, d, (X_n)) for which we have the equality seems extremely difficult. Moreover, even if we restrict ourselves to the case when G is a Fuchsian group generated by the side-pairing transformations relative to its fundamental polygon and set d to be the word metric or the hyperbolic metric induced by the action on H^2, the problem still remians quite tricky. In this talk we will discuss the recent progress made in the case of the hyperbolic metric on cocompact Fuchisan groups. We will give all necessary definitions, explain the relation between the fundamental inequality and Patterson-Sullivan theory, and, if time permits, we will briefly talk about the ideas used in the proofs themselves. Joint work with G. Tiozzo.

November 22, 2021
Dan Thompson, Ohio State University
Stronger ergodic properties for equilibrium states in non-positive curvature [video]

Equilibrium states for geodesic flows over compact rank 1 non-positive curvature manifolds and sufficiently regular potential functions were studied by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe some recent results on the dynamical properties of these unique equilibrium states. We show that these equilibrium states have the Kolmogorov property (joint with Ben Call), and that approximations of the equilibrium states by regular closed geodesics asymptotically satisfy a type of Central Limit Theorem (joint with Tianyu Wang). Time permitting, I will explain some of the main ideas behind the proofs, focusing on the MME case to ease the exposition.

November 29, 2021 [Note: special time 10 AM]
Inhyeok Choi, KAIST
Random walks on mapping class groups favor pseudo-Anosovs [video]

It has been believed that pseudo-Anosovs are generic in the mapping class group in certain aspects. One way to pick a random isometry is to consider a random walk on the mapping class group. In this setting, Maher proved that the probability for non-pseudo-Anosovs at step n decreases to 0. Later, Baik, Kim and I built upon Maher-Tiozzo's theory and proved that almost every sample path becomes pseudo-Anosov eventually. Meanwhile, one can instead pick a random isometry inside a radius R ball of the mapping class group and see the asymptotic proportion. This counting problem and the random walk estimates are widely different since one cannot count group elements exactly once from the latter perspective. In this talk, I will explain how to establish the exponential genericity of pseudo-Anosovs in the counting problem via the random walk method. The essential ingredient is that if we generate the random walk with an 'almost Schottky' generating set, the probability for non-pseudo-Anosovs at step n decays exponentially. If time permits, I will explain how this strategy can be implemented to random walks on automatic structures of groups, following Gekhtman-Taylor-Tiozzo.

December 6, 2021
Nguyen-Bac Dang, Université Paris Saclay
Spectral interpretation of dynamical degrees and degree growth [video]

In this talk based on a joint work with Charles Favre, I will explain how, using techniques from functional analysis, one can understand some problems in algebraic dynamics: namely study the growth of the degrees of the iterates of a given rational self-map on the projective n-space. Our method relies on endowing certain space of cohomology classes of an "infinite blow-up space" with a Banach norm, on which we uncover some spectral gap phenomena.

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