# Dynamics, Geometry and Combinatorics

#### EPFL, CIB

November 18, 2019

#### Counting in random lattices and higher rank actions

We count the number of points in the intersection of a random lattice in R^d with level sets of a product of linear forms.

The discrepancy functions of this counting are shown to satisfy a non-degenerate central limit theorem if d is large enough.

Our techniques are dynamical in nature, and exploit effective exponential mixing of all orders for the group of diagonal matrices acting on the space of lattices in R^d, combined with a novel technique to estimate higher order cumulants.

Joint work with Alexander Gorodnik (Zürich).

#### Determinantal Point Processes and Extrapolation

Consider a Gaussian Analytic Function on the disk, that is, a random series whose coefficients are independent complex Gaussians. In joint work arXiv:1612.06751 with Yanqi Qiu and Alexander Shamov, we show that the zero set of a Gaussian Analytic Function is a uniqueness set for the Bergman space on the disk: in other words, almost surely, there does not exist a nonzero square-integrable holomorphic function having these zeros. The key rôle in our argument is played by the determinantal structure of the zeros due to Peres and Viràg. In general, we prove that the family of reproducing kernels along a realization of a determinantal point process generates the whole ambient Hilbert space, thus settling a conjecture of Lyons and Peres.

The key lemma in our argument is that the determinantal property is preserved under conditioning on a fixed restriction of our configuration in a part of the phase space. In full generality, the explicit description of the kernels of these conditional measures remains an open problem.

It is nevertheless possible to write the analogue of the Gibbs property for one-dimensional determinantal point processes with integrable kernels: for these processes, conditional measures with respect to fixing the configiuration in the complement of an interval is an orthogonal polynomial ensemble with explicitly found weight (arXiv:1605.01400). Note here that a reproducing kernel must have integrable form as soon as the corresponding Hilbert space satisfies a weak form of division property (joint work with Roman Romanov); in this case, our space must be a Hilbert space of holomorphic functions.

The proof of the Lyons-Peres conjecture raises the problem of extrapolating a function from its restriction to a realization of a determinantal point process. In joint work with Yanqi Qiu (arXiv:1806.02306), extrapolation for Bergman functions from zero sets of Gaussian Analytic Functions is obtained using the Patterson-Sullivan construction.

#### Ergodic properties of low complexity symbolic systems

The topological entropy of a subshift is the exponential growth rate of the number of words of different lengths in its language. For subshifts of entropy zero, finer growth invariants constrain their dynamical properties. In this talk we will survey how the complexity of a subshift affects properties of the ergodic measures it carries. In particular, we will see some recent results (joint with B. Kra) relating the word complexity of a subshift to its set of ergodic measures as well as some applications.

#### On mixing properties of infinite measure preserving systems

We present several new results concerning mixing properties of hyperbolic systems preserving an infinite measure making a particular emphasis on mixing for extended systems. This talk is based on a joint work with Peter Nandori.

#### Correlations of the Liouville function along deterministic and independent sequences.

The Liouville function is equal to 1 on integers with an even number of prime factors and -1 otherwise. Its values are expected to be distributed randomly, and based on this, Chowla conjectured that its self correlations of all orders vanish. The conjecture is still widely open, but significant partial results were recently obtained. One by Matomaki, Radziwill and Tao, who established the conjecture "on average", and another by Tao and Terravainen who established a logarithmically averaged variant of the conjecture for all odd order correlations. We use tools from ergodic theory in order to obtain significant strengthenings of these results. Our main strategy is to study measure preserving systems naturally associated with the Liouville function and then transfer the results obtained back to number theory.

This is in part based on joint work with Bernard Host.

#### Extreme problems in Diophantine Approximation and Dynamics

We will explore the situation when the Dirichlet Theorem and the Minkowski Theorem can be improved and study the distribution of such improvements. We shall show that the improvements exhibit random behavior and relate this question to distribution of extremes for iterations of dynamical systems.

#### Ruelle resonances for linear pseudo-Anosov maps

The Ruelle resonances of a dynamical system are spectral characteristics of the system, describing the precise asymptotics of correlations. While their existence can often be shown by abstract spectral analysis arguments, it is in general not possible to compute them exactly. After explaining the general background, illustrated by classical examples, I will focus on a specific example: in the case of linear pseudo-Anosov maps, Ruelle resonances can be completely described in terms of the action of the map on cohomology.

Joint work with Faure and Lanneau

#### A short proof of a conjecture of Erdös proved by Moreira, Richter and Robertson

We give a short proof of a sumset conjecture of Erdös, recently proved by Moreira, Richter and Robertson: every subset of the integers of positive density contains the sum of two infinite sets.

The proof uses only classical ergodic theory.

#### Bernoulli and K properties in smooth dynamics

A measure preserving system $(T,X,\mu)$ is called ${Bernoulli}$ if it is isomorphic to a Bernoulli shift. $(T,X,\mu)$ is a ${K-system}$, if every (non-trivial) factor of $T$ has positive entropy. One of the central class for which $K$ and Bernoulli properties was studied is that of smooth systems on manifolds preserving a smooth measure. I will review classical results and discuss some recent developments. I will focus on partially hyperbolic (homogeneous) actions and skew products over hyperbolic systems.

#### Uniform approximation in ergodic theory with shrinking targets

Let $(X,\mu,T)$ be a finite measure-preserving system, and let $\{B_n\}$ be a sequence of nested subsets of $X$. Following a recent paper by Dubi Kelmer, let us say that $x\in X$ eventually always hits $\{B_n\}$ if for every sufficiently large $N$ there exists $n\in \{1,\dots,N\}$ such that $T^{n} x\in B_N$. This is a uniform analogue of the hitting property usually considered in ergodic theory with shrinking targets via dynamical Borel-Cantelli lemmas. For circle rotations this corresponds to uniform inhomogeneous Diophantine approximation.

In general not much can be said about the magnitude of the sets of eventually always hitting points. In this work, joint with Florian Richter and Ioannis Konstantoulas, we focus our attention on systems where translates of targets exhibit near perfect mutual independence. For such systems we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is null or co-null.

#### Rigidity in negative curvature

I will survey some results and problems related to recognizing compact manifolds with negative curvature that are locally symmetric spaces.

#### Geodesic planes in hyperbolic 3-manifolds

Let M be a hyperbolic 3-manifold, a geodesic plane in M is a totally geodesic immersion of the hyperbolic plane into M. In this talk we will give an overview of results which highlight how geometric, topological, and arithmetic properties of M are related to the behavior of geodesic planes in M.

The talk is based on joint projects with McMullen and Oh, and in another direction with Margulis.

#### Multicorrelation sequences and primes

Multicorrelation sequences are fundamental objects of study in ergodic Ramsey theory. The structure of multicorrelation sequences for a single transformation was described by Bergelson, Host and Kra, up to a negligible error. A similar description was later obtained by Frantzikinakis for several commuting transformations, with an arbitrarily small (but no longer negligible) error. Recently Le proved an analogue of the Bergelson-Host-Kra theorem for averages over the set of prime numbers. In this talk I will explain how to obtain a similar analogue of Frantzikinakis theorem for averages over the primes.
This talk is based on joint work with Anh Le and Florian Richter.

#### Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay

We show that the Fourier transform of the Patterson-Sullivan measure of a Kleinian Schottky subgroup of PSL2(C) enjoys polynomial decay. This generalizes a result of Bourgain-Dyatlov for convex co-compact Fuchsian groups, and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups. This is a joint work with Jialun Li and Fréderic Naud.

#### A nonlinear version of Bourgain's projection theorem, with applications

Bourgain's projection theorem is an important extension of his celebrated discretized sum-product estimate. I will discuss a generalization of the projection theorem from the family of linear projections to parametrized families of smooth maps satisfying a technical but mild condition. I will present some applications to distance sets and radial projections, as well as a relaxation of the hypotheses of the theorem that is new even in the linear case. The proofs are based on applying the original version of the theorem to measures in a suitable multiscale decomposition, that I will describe if time allows.

#### Thermodynamical formalism and Martin boundary

Let $(X,\theta)$ be a topologically mixing subshift of finite type equipped with a Hölder potential, $G$ a discrete group and $\gamma$ a locally constant map from $X$ to $G$. The skew product $T: (x,g) \mapsto (\theta(x),g\gamma(x))$ defines a further, locally compact shift space whose time evolution in the second coordinate generalizes the classical notion of a random walk on $G$. As in the classical case, one may define the Martin boundary as the set of equivalence classes of limits
of
$\frac{\sum_{k=0}^\infty r^k \mathcal{L}^k(1_{X \times \{g\}} )\circ T^n(x,h) }{\sum_{k=0}^\infty r^k \mathcal{L}^k(1_{X \times \{id\}})\circ T^n (x,h)},$
where $\mathcal{L}$ is Ruelle's operator and $r$ is less or equal than the radius of convergence $R$. However, as observed recently by Shwartz, the boundary in this situation is related to $\sigma$-finite conformal measures instead of harmonic functions.

If $G$ is a hyperbolic group, ideias of Ancona, Gouzel and Lalley then allow to identify the Martin boundary with the visual boundary and the set of minimal conformal measures. Moreover, this identification holds for all $r\leq R$ and it turns out that
a suitably defined Green metric is bi-Hölder equivalent to the visual metric on $\partial G$.

Joint work with Sara Ruth Pires Bispo.

#### Mixing time-changes of nilflows

Nilflows (flows on nilmanifolds, quotients of nilpotent Lie groups by a lattice) are one of the fundamental examples of parabolic dynamical systems. The simplest example is given by Heisenberg nilflows on quotients of the Heisenberg group.The talk will concern the ergodic properties, in particular mixing, of parabolic flows which are obtained by time-changes (or time-reparametrizations) of niflows. It is well known that nilflows are never mixing (for the presence of a toral factor). On the contrary, we will show that mixing is typical among (a class of) time-changes and we describe the mechanism which produces it. Our main result in particular generalizes to any nilflows (of step at least 2) previous results by Avila-Forni and myself for Heisenberg nilflows and Ravotti for filiform nilflows.

Joint work with Artur Avila, Giovanni Forni and Davide Ravotti.

#### The dimension of Bernoulli convolutions

The Bernoulli convolution with parameter $\lambda\in(0, 1)$ is the measure $\nu_\lambda$ on the real line that is the distribution of the random power series $\sum\pm\lambda^n$, where $\pm$ are independent fair coin tosses. These measures are natural objects from several points of view including fractal geometry, dynamics and number theory. I will discuss recent results about their dimension.

#### Diophantine Approximation on manifolds: lower bounds for Haudsorff dimension

There has recently been much activity in the theory of Diophatine approximation in which the points of interest are restricted to a sub-manifold M. I plan to provide a review of some of this activity. The goal is to describe a new approach based on the Mass Transference Principle which yields sharp lower bounds for the dimension of well approximable sets restricted to M.