Tentative plan — Autumn 2024

Effective equidistribution of orbits of semisimple groups and an effective closing lemma
In recent years, quantitative equidistribution results have been a major theme in homogeneous dynamics. In the first half of this talk, we discuss a polynomially effective equidistribution result for orbits of semisimple groups on congruence quotients. This extends work of Einsiedler, Margulis, and Venkatesh by removing the assumed triviality of the centralizer in the ambient group. An essential ingredient of the proof is an effective closing lemma for unipotent flows from work with Lindenstrauss, Margulis, Mohammadi, and Shah. We discuss this result in the second half of the talk.

AI using singular measures
We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $\mµ_n$ on $\Bbb R$. Assume that this sequence of operators constitutes an $L^p$-norm approximate identity for some $1\le p<\infty$. We ask, under what additional conditions do we have almost everywhere convergence for all $f\in L^p(\Bbb R)$. We focus on the particular case of a sequence of contractions $C_{t_n}\mu$ of a single Borel probability measure $\mu$, with $t_n\to0$, so that that the sequence of operators is an $L^p$-norm approximate identity. If $\mu$ is discrete, then no sequence of such contractions can give a.e. convergence for all of $L^p(\Bbb R)$. If $\mu$ is absolutely continuous with respect to Lebesgue measure, then there is a sequence $(t_n)$ such that a.e. convergence holds on all of $L^1(\Bbb R)$. But when the measure µ is continuous and singular to Lebesgue measure, obtaining a.e. results for some sequence $(t_n)$, is more challenging. Such results can always be obtained on $L^2(\Bbb R)$ when $\mu$ is a Rajchman measure. For non-Rajchman measures obtaining a.e. results on $L^2(\Bbb R)$ is sometimes possible, but not easy. In fact, it may be the case that there is a continuous, singular probability measure $\mu$ for which there is no sequence $(t_n)$ tending to zero with $C_{t_n}\mu\ast f\to f$ a.e., even just for all $f\in L^\infty(\Bbb R)$.

Fluctuation of ergodic averages and other stochastic processes
For an ergodic map T and a non-constant, real-valued L^1 function f, the ergodic averages converge for almost every x, but the convergence is never monotone. Depending on particular properties of the function f , the averages may or may not actually fluctuate around the mean value infinitely often almost everywhere. One of the main results that we will discuss in this talk is that almost everywhere fluctuation around the mean is the generic behavior. That is, for a fixed ergodic T , the generic L^1 function f has the property that the averages fluctuating around the mean infinitely often for almost every x. We will also talk about fluctuation for other stochastic processes, namely the ergodic averages along a subsequence, convolution operators and martingales.
This is a joint work with Joseph Rosentblatt and Máté Wierdl.

Marked Length Spectrum Rigidity
Given a closed Riemannian manifold with everywhere negative sectional curvature, there exists a unique geodesic inside of every non-trivial free homotopy class. The marked length spectrum is defined to be the function which takes a free homotopy class and returns the length of this geodesic; this can be thought of as the function that records the length and locations of closed geodesics. It was conjectured by Burns and Katok that the marked length spectrum determines a Riemannian metric up to isometry. In this talk, I’ll discuss the history of this conjecture, as well as discuss a variation of this conjecture in the setting of so-called “magnetic surfaces.” In particular, I will discuss how this new setting leads to deeper insights on when the lengths of closed curves determine the geometry of the underlying space.

Sub-actions and marked length spectrum rigidity for some locally CAT(-1) spaces
Let $\phi_t$ be a flow on $X$ and $A:X\to\mathbb{R}$ a Holder function. If $\int_0^T A(\phi_t x)dt \geq V(\phi_T x)-V(x)$ for all $x$ and $T$, $V$ is a sub-action for $A$. A clear necessary condition for the existence of a sub-action is that the integral of $A$ around any closed orbit is nonnegative. Lopes and Thieullen prove that this condition is sufficient for Anosov flows, providing what can be thought of as a positive Livsic theorem. In this talk I’ll explain how we generalize their work to geodesic flow on locally CAT(-1) spaces. We then apply this result to certain locally CAT(-1) spaces where marked length spectrum rigidity is known (surface amalgams and Fuchsian building quotients provide some examples). We prove that if there is a marked length spectrum inequality between two spaces, yet their volume is the same, they are in fact isometric, using a proof strategy due to Croke and Dairbekov. This is joint work with Elvin Shrestha and Yandi Wu.

Invariant Families of Measures in Non-stationary Dynamics
Historically, the Ruelle-Perron-Frobenius (RPF) theorem provided a fundamental framework for understanding the long-term behavior of dynamical systems and their associated equilibrium states. In the 1990s and early 2000s, mathematicians like Kifer and Simmons-Urbanski studied equilibrium states for random systems, where the dynamics are chosen probabilitisically from a class of maps at each iteration. In particular, they used fiberwise transfer operators to construct families of measures that are invariant under the dynamics almost everywhere.

We will discuss new results on the construction of families of measures for some non-stationary systems (a generalization of random systems). Specifically, we will demonstrate how the Hilbert metric, a powerful tool for analyzing convex cones in Banach spaces, can be employed to establish a non-stationary RPF theorem.

L^q-spectra of dynamically driven self-similar measures: the multi-dimensional case
A great deal of interest in fractal geometry centres on determining the dimensional properties of self-similar sets and measures, as well as of their projections and convolutions. In a seminal contribution dating from nearly a decade ago, Hochman achieved substantial progress towards the celebrated exact overlaps conjecture, establishing that the Hausdorff dimension of self-similar sets and measures on the real line matches the similarity dimension whenever the generating iterated function system satisfies exponential separation. The result was subsequently refined by Shmerkin, who established the analogue for the full L^q-spectrum of self-similar measures and successfully applied it to settle long-standing conjectures in dynamics and fractal geometry, most notably Furstenberg’s intersection conjecture for the action of multiplicatively independent integers on the torus. In joint work with Shmerkin, we extend the dimensional result to any ambient dimension under an additional unsaturation assumption; as in the one-dimensional case, our framework consists of the class of dynamically driven self-similar measures, which allows for a unified treatment of self-similar and stochastically self-similar measures, their projections and convolutions. The argument relies crucially on an inverse theorem for the L^q-norm of convolutions of discrete measures in Euclidean spaces, recently established by Shmerkin, akin in spirit to the asymmetric version of the Balog-Szemerédi-Gowers theorem due to Tao and Vu.

Local Product Structure for Equilibrium States of Geodesic Flows
Local product structure of a measure has been used as a key ingredient for showing various mixing properties of dynamical systems. One prominent example of this is Ornstein and Weiss’s proof of the Bernoulli property. I will discuss recent work establishing local product structure using a nonuniform Gibbs property for a class of equilibrium states for the geodesic flow on rank one nonpositively curved manifolds, as well as flat surfaces with cone singularities. Time permitting, I will also discuss how this can be used to show the Bernoulli property for these measures. This is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.

Invariant measures of minimal homeomorphisms on manifolds
It is well known that a topological dynamical system’s set of invariant measures is a nonempty metrizable Choquet simplex. In 1991, Downarowicz showed that all such simplices arise as the set of invariant measures of a class of minimal subshifts. Hence, we can ask: Which nonempty Choquet simplices can be realized as sets of invariant measures for minimal homeomorphisms on manifolds? It is known that for one-dimensional manifolds, the geometry of manifolds restricts the available dynamics. In particular, minimal homeomorphisms of the circle are uniquely ergodic. During the talk, we will discuss a class of minimal homeomorphisms of the Cantor set whose dynamics can be quite faithfully realized as minimal homeomorphisms on manifolds. Consequently, we will prove that every Choquet simplex is realized as a set of invariant measures of a minimal homeomorphism on every manifold of dimension two and higher carrying a minimal uniquely ergodic homeomorphism. The talk is based on results obtained in cooperation with Jernej Činč, Till Hauser, Dominik Kwietniak and Piotr Oprocha.

Quasi-fuchsian flows and the coupled vortex equations
In 1992, Ghys introduced a remarkable class of Anosov flows on closed 3-manifolds
having smooth weak foliations. The class is parametrized by a pair of points in Teichm\”uller space.
I will discuss an alternative description of this class using the so-called coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of a Blaschke metric uniquely determined by a conformal class and a holomorphic quadratic differential.
This is joint work with Mihajlo Cekić.