Jairo Bochi

Title: Angles between Oseledets spaces

Abstract: This talk is based on joint work with Pablo Lessa. We provide an example of a probability distribution on the group GL(2,R) with finite first moment such that the corresponding random product of i.i.d. matrices has two distinct Lyapunov exponents, but the angle between the Oseledets directions is not log-integrable. We prove that, on the other hand, if the second moment is finite, then this angle, if defined, is log-integrable. Next, we turn our attention to general GL(2,R)-cocycles over ergodic automorphisms, and ask ourselves if there is any criterion for log-integrability of the angle between the Oseledets directions in terms of a suitable integrability condition. The answer is negative. In fact, we show the following flexibility result: given any ergodic automorphism T of a non-atomic Lebesgue probability space, we can find a GL(2,R)-cocycle over T whose Lyapunov exponents and joint distribution of Oseledets spaces are prescribed a priori, and meeting any prescribed integrability condition. 

Aaron Brown

Title: Regularity of stationary measures 

Abstract: I will discuss some recent results about stationary measures for random dynamical systems.  In particular, I'll discuss questions about classification and absolute continuity of stationary measures.  I'll mostly focus on random dynamics on surfaces with some remarks on higher-dimensions.  

Emilio Corso

Title: Closed geodesics, periodic torus orbits and the distribution of shapes of unit lattices in number fields.

 Abstract: The quest to understand the asymptotic behaviour of periodic orbits of subgroup actions on homogeneous spaces bears substantial geometric and number-theoretic significance. The talk is concerned with one of the main strands of research in this direction, which focuses on compact orbits of diagonalizable subgroups on finite-volume quotients of Lie or algebraic groups. We will survey aspects of the interplay between geometry, dynamics and number theory, starting with the classical rank-one results on equidistribution of closed geodesics on hyperbolic manifolds; subsequently, we will turn our focus to the higher-rank case, mentioning the relevant generalizations in the works of Einsiedler-Lindenstrauss-

Michel-Venkatesh and Dang-Li, and discussing the arithmetic interpretation of periodic torus orbits in terms of orders, and their unit groups, in number fields. Finally, we address long-standing questions of Margulis and Gromov on the asymptotic distribution of shapes of periodic orbits, presenting recent joint work with F. Rodriguez Hertz, improved upon by Dang, Li and Gargava, on the topology of the set of shapes in the case of cubic fields.

David Fisher

Title: Finiteness of totally geodesic submanifolds.

Abstract:  I will give a survey of recent results concerning finiteness of totally geodesic manifolds in negativelycurved manifolds. 

Basak Gürel

Title: Invariant Sets and Hyperbolic Periodic Orbits

Abstract: The presence of hyperbolic periodic orbits or invariant sets often has an effect on the global behavior of a symplectic dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards a higher-dimensional Franks’ theorem or forced existence of periodic orbits for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Ginzburg and asserting that no periodic orbit of a Hamiltonian pseudo-rotation of a complex projective space is locally maximal. The talk is based on a joint work with Erman Cineli, Viktor Ginzburg and Marco Mazzucchelli.

Misha Guysinsky 

Title: Distribution of the areas of the countries.

Abstract: At the end of the 1990s, V. Arnold suggested the following model that could explain certain properties of the areas of countries. Let the areas of the countries be denoted as x_1, x_2, ... , x_N. We fix a k x k column stochastic matrix M. In each iteration, we randomly select k countries x_{i_1}, x_{i_2},..., x_{i_k}, form a vector \mathbf{x}, and calculate a new vector M \mathbf{x}. The coordinates of this new vector are then used as the new areas for the selected countries. For example, during an iteration, two countries may merge, or one country may split into two countries of equal area. Specifically, if we select three countries with areas x, y, and z, we replace them with areas x + y, z/2, and z/2. It was observed in computer experiments that, for large N, the distribution of the areas of the countries converges to the stable distribution. We will prove it rigorously. This problem is equivalent to analyzing random products of N x N stochastic matrices and proving that, as N goes to infinity, the corresponding stationary measure on (N-1)-dimensional simplices concentrates around points whose coordinates form the same distribution. This is a joint work with J. Paik.

Bryna Kra 

Title : The density finite sums theorem

Abstract : Since Szemeredi's Theorem and Furstenberg's proof thereof using ergodic  theory, dynamical methods have been used to show the existence of numerous configurations in sets of positive upper density. Until recently, a common feature of all of these configurations is that they were all finite. Resolving questions and conjectures of Erdos, we use dynamical methods to prove a density version of the finite sums theorem of Hindman. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.

Egor Shelukhin

Title: A symplectic Hilbert-Smith conjecture

Abstract: A well-known open question in group actions and topological dynamics is the Hilbert-Smith conjecture. It extends Hilbert's fifth problem and states that a locally compact group acting faithfully on a manifold must be a Lie group. We discuss a recent result regarding p-adic actions by homeomorphisms of symplectic nature, which provides new evidence to support this conjecture. 

Pablo Shmerkin

Title: Higher rank Furstenberg slicing

Abstract: I will discuss upper bounds for the dimensions of the affine and smooth slices of Cartesian products of Cantor sets invariant under multiplication by p_i on the circle. These upper bounds generalize the case of linear slices of products of two invariant Cantor sets, which is the original Furstenberg slicing problem. The higher rank version requires several new tools and ideas. Joint work with Emilio Corso.

Masato Tsuji

Title: Regularity of Stable and Unstable Foliations in Anosov Flows and Partially Hyperbolic Dynamical Systems

Abstract: This talk addresses the regularity of the stable and unstable foliations in Anosov flows and partially hyperbolic dynamical systems. Even when assuming the systems to be smooth, it is known that the holonomy maps of the stable and unstable foliations are generally not smooth, but only Hölder continuous. This irregularity presents significant challenges in the analysis of smooth dynamical systems. We introduce a new approach to this problem using "non-stationary normal trivializations" of the normal bundle of the stable and unstable manifolds. We then explore a few implications of this method for the ergodic properties of Anosov flows and partially hyperbolic dynamical systems.

Hao Wu

Title: A temporal central limit theorem for irrational rotations

Abstract: Dynamical systems are deterministic systems. However, in chaotic systems where the entropy is positive, the ergodic sums often behave similarly to the sums of independent random variables and satisfy the spatial central limit theorems (CLT) according. In zero-entropy systems, the spatial CLT often fails due to the lack of (fast) mixing properties. But in some cases, such as irrational rotations of bounded type, one can retrieve the central limit theorem if we study single orbit statistics, where we fix the starting point and randomise time, hence the word “temporal”. In this talk, I will present an ongoing joint work with Bromberg and Ulcigrai, where we generalise their previous method of coding and Markov chains to obtain a temporal central limit theorem for a broader class of observables over bounded irrational rotations.

SHORT TALKS:

Jason Day

Title: Topologically mixing suspension flows over shift spaces

Abstract: We show the necessary and sufficient conditions for suspension flows over certain families of shift spaces to be topologically mixing.  We also show that the set of roof functions that produce non-mixing suspension flows is dense in the set of all continuous roof functions.  This is joint work with the late Todd Fisher.

Gregory Hemenway

Title: Conjugacies of Expanding Skew Products

Thomas O’Hare 

Title: Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms

Abstract: Let f,g be C^2 area-preserving Anosov diffeomorphisms on T^2 which are topologically conjugated by a homeomorphism h. It was proved by de la Llave in 1992 that the conjugacy h is automatically C^{1+} if and only if the Jacobian periodic data of f and g are matched by h for all periodic orbits. We prove that if the Jacobian periodic data of f and g are matched by h for all points of some large period N, then f and g are ``approximately smoothly conjugate." That is, there exists a a C^{1+\alpha} diffeomorphism h_N that is exponentially close to h in the C^0 norm, and such that f and f_N:=h_N^{-1}\circ g\circ h_N is exponentially close to f in the C^1 norm.

Abror Pirnapasov

Title: C^{0} Stability of Topological Entropy for Reeb and Geodesic Flows

Abstract: In this talk, we establish the lower semi-continuity of topological entropy for closed contact 3-manifolds in the C^0topology for C^\infty-generic contact forms. By applying these techniques to geodesic flows on surfaces, we show that non-degenerate metrics are among the points of lower semi-continuity. In particular, for a geodesic flow with positive topological entropy, this property persists under sufficiently small C^0-perturbations of the metric. This work is joint with Marcelo Alves, Lucas Dahinden, and Matthias Meiwes

Chengyang Shao

Title: Paradifferential calculus and dynamical conjugacy problems

Abstract: In this talk, I will introduce how paradifferential calculus can be applied to dynamical conjugacy problems involving "loss of regularity". A powerful tool in modern harmonic analysis, paradifferential operators share the algebraic structure of usual differential and Fourier integral operators, while on the other hand provide a scheme in which a nonlinearity can be manipulated as if it were linear, with delicate control of regularity. Therefore, paradifferential calculus not only serves as a promising alternative to the usual KAM/Nash-Moser method, but also applies to problems where the "loss of regularity" disables the latter. The talk is based on joint works with Thomas Alazard, together with a recent preprint of Giovanni Forni. 

.