ICTS

Lagrangian flow describes the motion of a passive tracer particle (e.g., a mote of dust) in a fluid, giving rise to a flow of diffeomorphisms on the fluid domain which is volume preserving when the fluid is incompressible. When the fluid is subjected to forcing, it is anticipated that this flow is chaotic in terms of sensitivity with respect to initial conditions (i.e., a positive Lyapunov exponent) and exponentially fast decay of correlations (equivalently, fast mixing of passively adverted scalars, such as chemical concentration). Unfortunately, rigorously proving these chaotic properties for most types of deterministic forcing seems hopelessly out of reach.

On the other hand, the presence of stochastic driving often renders tractable the problem of rigorously verifying chaotic regimes. In a joint work with Jacob Bedrossian (UMD) and Sam Punshon-Smith (U Brown), we prove these chaotic properties for the Lagrangian flow for an incompressible fluid evolving according to the Navier-Stokes equations on the torus subjected to Sobolev-regular, white-in-time forcing. As a consequence, for the models we consider, we resolve an open question in the study of passive scalar turbulence in the Batchelor regime (i.e., fixed viscosity with molecular diffusivity going to zero).

Nilflows on nilmanifolds are classical examples of parabolic homogeneous flows. Their ergodic properties are well-understood: although almost every nilflow is uniquely ergodic, they are never weak mixing. The absence of mixing is the result of an algebraic obstruction, namely a factor isomorphic to a linear flow on a torus. On the contrary, we show that for generic smooth time-changes of any uniquely ergodic nilflow, mixing holds unless the time-change function is measurably trivial. This result confirms some heuristic principle on the ergodic properties of smooth parabolic flows.

This is a joint work with Artur Avila, Giovanni Forni, and Corinna Ulcigrai.

Optimal rates of mixing (of arbitrarily high order) for the periodic Z^d billiard map with finite horizon have been obtained recently by F. Pene. First order mixing for the infinite horizon case is also known. Rates of mixing (higher order mixing) in the infinite horizon case poses new challenges. In joint work with F. Pene, we obtain second order mixing.

We consider skew tent maps Tα,β(x) with Tα,β(x) = βαx for 0 ≤ x ≤ αand Tα,β(x) = β1−α(1 − x) for α < x ≤ 1. With this choice of parametersTα,β maps [0, 1] into [0, 1] and (α, β) is the vertex of Tα,β. The dynamics ofTα,β for (α, β) ∈ [0, 1]2
is interesting when (α, β) ∈ U = {(α, β) : 0.5 < β ≤1, 1 − β < α < β}. Denote by h(α, β) the topological entropy of Tα,β. It is well-known that h(α, β) is strictly monotone increasing along vertical line segments in U. It is natural to ask what happens if we move in the horizontal direction, that is for fixed β we consider h(α) = h(α, β), for (α, β) ∈ U. Turned out that h(α) is strictly monotone increasing on (1 − β, β). To deal with this question one needs to consider equi-topological entropy curves in the square. We denote by M = K(α, β) the kneading sequence of Tα,β and by Λ = Λα,β its Lyapunov exponent. For a given kneading squence M we consider isentropes (or equi-topological entropy, or equi-kneading curves), (α, ΨM(α)) such that K(α, ΨM(α)) = M. On these curves the topological
entropy h(α, ΨM(α)) is constant. We show that Ψ0 M(α) exists and the Lya-punov exponent Λα,β can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function ΘM a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.

The ergodic theory of Lyapunov exponents, initiated by the work of Furstenberg and Kesten at the dawn of the 1960s, has been a remarkably active field of mathematics over all these decades and, through the works of Oseledets, Pesin and many, many others, found extremely important applications in the realm of smooth dynamics.  The speaker will survey some of the classical results and more recently developments, with a view to the future of the field.

A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.

Generically in Diff1m(M3), the existence of a minimal expanding invariant foliation implies stable Bernoulliness. Also, generically in Diff1m (M3), if an expanding f-invariant foliation of dimension u is minimal and there is a periodic point of unstable index u,then the foliation is stably minimal.For higher dimensions, we prove that generically in Diff1m(M), if an expanding f-invariant foliation of dimension u is minimal, and there is a periodic point of unstable index u, then the foliation is stably minimal. Also there is a hyperbolic ergodic component that is essentially dense.

This is joint work with Gabriel N´u˜nez.

The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this mini-course we will be interested by the non-uniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.

We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, Buzzi-Crovisier-Sarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.

Reference: arXiv:1811.02240.

Partially hyperbolic surface endomorphisms are a family of not necessarily invertible surface maps which are associated with interesting dynamics. The dynamical behaviour of these maps is less understood than their invertible counterparts, and existing results show that they can exhibit properties not possible in the invertible setting. In this talk, I will discuss recent results regarding the classification of partially hyperbolic surface endomorphisms. We shall see that either the dynamics of such a map is in some sense similar to a linear map, or that the map falls into a special class of interesting examples. This is joint work with Layne Hall.

Consider a family of probability measures $\{\mu_{x}\}_{x\in X}$ on a topological space $M$ parametrised by a given set $X$. Representing $\{\mu_{x}\}_{x\in X}$ consists in finding a map $F: X \times \Omega \to M$ such that, for each $x \in X$, we have, $\mu_{x} = F(x,\cdot)_{*}\mathbb{P}$, where $(\Omega, \mathbb{P})$ is an auxiliary probability space. Such questions arise in Probability, Geometry and several other areas. In particular, in Dynamical Systems, it appears in the context of disintegration of measures, and in random perturbation of dynamics, where one is interested in finding a probability on the spaces of maps which mimics a given Markov chain. In this talk, we provide sufficient conditions such that family of probabilities on manifolds can be represented by regular maps. The talk is based on a joint work with Jost, Matveev, and Portegies.

Decay of correlations for Axiom A flows remains an open and notoriously difficult problem. A reasonable conjecture is that an open and dense set of Axiom A flows have exponential decay of correlations for all nontrivial hyperbolic basic sets, but despite some isolated positive results this conjecture seems beyond modern technology.

A more tractable problem is to prove rapid mixing (mixing at all polynomial orders). This property is known to be open and dense, but only for observables that are sufficiently smooth. In this talk, we explain how to remove this restriction, proving that for an open and dense set of Axiom A flows the nontrivial hyperbolic basic sets are rapid mixing for Holder observables. Consequences and extensions will be mentioned. (Joint work with Caroline Wormell.)
 

A homeomorphism of the annulus without any periodic point is called a pseudo-rotation. Each pseudo-rotation has a unique rotation number in \mathbb R / \mathbb Z. We show that for a Baire generic rotation number \alpha \in \mathbb{R} / \mathbb{Z}, the set of area preserving C^\infty pseudo-rotation of the annulus \mathbb{A} with rotation number \alpha equals to the closure of the set of area preserving C^\infty pseudo-rotations which are smoothly conjugate to the rotation R_{\alpha} . As a corollary, a C^\infty generic area preserving pseudo-rotation of the annulus with a Baire generic rotation number \alpha is weakly mixing. This is a joint work with Barney Bramham.
 

We consider the component-wise action of the multiplicative semigroup of nonsingular $n \times n$ matrices with integer entries, on the Cartesian product of $p \leq n-1$ copies of the Euclidean space $\mathbb R^n$, and discuss the effectiveness of the approximation of any target point by the orbit of a given initial point under the action. The exponent of approximation is proved to be $(n-p)/p$ for all initial points outside a null set that can be described by a certain Diophantine condition.
 

In a joint work with Jeremy Kahn and Francois Labourie we prove that any uniform lattice in a simple complex Lie group G contains a surface group. (I.e. the the fundamental group of an orientable surface of genus at least 2).
This theorem is a generalization of the celebrated Kahn–Markovic Theorem which deals with the case of G=PSL(2, C).

In the spirit of H. Maass and W. Schmidt we consider two-dimensional rational subspaces of the four-dimensional Euclidean space. To any such subspace one can attach two CM points originating from the subspace and
its orthogonal complement. The accidental (local) isomorphism between SO(4) and SO(3) x SO(3) allows us to define two further CM points naturally attached to the subspace. We show simultaneous
equidistribution of subspaces of a fixed discriminant together with all their CM points under two splitting conditions. A crucial input is the classification of joinings of higher-rank diagonalizable actions on
homogeneous spaces by Einsiedler and Lindenstrauss. This is joint work with Menny Aka and Manfred Einsiedler.


 

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strombergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

In this talk I will outline a proof of Ratner's theorem on unipotent flows on quotients of semisimple groups by co-compact lattices.

The proof makes use of results from representation theory, in particular Casselman's theorem of imbedding unitary modules in principal series.

A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.

An important goal of classical Diophantine approximation is quantitative analysis of the denseness of the set of rational vectors in their ambient Euclidean space. A challenging and substantial extension of the classical theory is to establish quantitative aspects of the denseness of rational points (and rational points with constrained denominators) in general homogeneous algebraic varieties. Here the denseness in question refers to the rational points on the variety itself, and so it is natural to call this subject "intrinsic Diophantine approximation". This problem was raised by Serge Lang more than half a century ago, but until recently material progress towards it was achieved only in the case of Abelian algebraic groups.

A systematic approach to this problem for homogeneous algebraic varieties associated with non-amenable algebraic groups was initiated and developed in joint work with A. Ghosh and A. Gorodnik over the last decade. This approach is based on dynamical arguments and effective ergodic theory, and some of its results apply even more generally, to actions of arbitrary lattice subgroups on homogeneous spaces. It leads to the derivation of uniform and almost sure Diophantine exponents for rational points and lattice orbits, as well as to an analog of Khinchin's dichotomy, and an analog of Schmidt's solution counting theorem. Furthermore, some of the results established using this approach are in fact best possible. We will explain some of the main results obtained and some of the ingredients in their proof, and present a variety of examples to demonstrate them, as time permits.

Under certain congruence conditions, the elliptic curves defined over the complex numbers with complex multiplication (CM) by a given order can be reduced to supersingular curves (SSC) defined over a finite field of prime characteristic. The (finite) set of isomorphism classes of SSC curves carries a natural probability measure. It was shown by Philippe Michel via progress on the subconvexity problem that the reductions of CM curves equidistribute among the SSC curves when the discriminant of the order diverges along the congruence conditions. We will describe a proof of equidistribution in the product of the simultaneous reductions with respect to several distinct primes of CM curves of a given order using a recent classification of joinings for certain diagonalizable actions by Einsiedler and Lindenstrauss. This is joint work with Menny Aka, Philippe Michel, and Andreas Wieser.
 

 Let G be a linear algebraic group over Q, H be an observable Q-subgroup and \Gamma be an arithmetic lattice in G. Given a sequence of elements g_n in G(R), we study the limiting behavior of g_nH(R)\Gamma in G/\Gamma. For a bounded open set O in H(R), we give a criterion on when g_nO\Gamma diverges topologically to infinity in G/\Gamma using the work of Kleinbock--Margulis. And when the full orbit does not diverge, we classify all possible limiting measures. This analysis is based on the early work of Eskin--Mozes--Shah and Shapira--Zheng which ultimately relies on the Ratner's theorem on unipotent rigidity and the linearization technique developed by Dani--Margulis.

Recently there has been a surge of activity in quantifying the density of values of generic quadratic forms and other polynomials at integer points. I will survey what is known (the work of Ghosh-Gorodnik-Nevo, Athreya-Margulis, Kelmer-Yu and others) and present some new and quite general results in this direction obtained via the so-called Siegel-Rogers method, which has recently been utilized by Athreya and Margulis, and later by Kelmer and Yu. This work is joint with Mishel Skenderi. If time allows I'll comment on other related problems.
 

In this talk I will discuss some simple consequence of various rates of mixing and propose some problems related to it.

Recently there has been a surge of activity in quantifying the density of values of generic quadratic forms and other polynomials at integer points. I will survey what is known (the work of Ghosh-Gorodnik-Nevo, Athreya-Margulis, Kelmer-Yu and others) and present some new and quite general results in this direction obtained via the so-called Siegel-Rogers method, which has recently been utilized by Athreya and Margulis, and later by Kelmer and Yu. This work is joint with Mishel Skenderi. If time allows I'll comment on other related problems.
 

A cut-and-project set is constructed by restricting a lattice L in (d+m)-space to a domain bounded in the last m coordinates, and projecting it to the the space spanned by its d-dimensional orthogonal
complement. These point sets constitute an important example of so-called quasi-crystals.
 
During the talk, we shall present and give some classification results of the moduli spaces of cut-and-project sets, which were introduced by Marklof-Strömbergsson. These are obtained by considering the orbit
closure of the special linear group in d-space acting on the lattice L inside the space of unimodular lattices of rank d+m. Theorems of Ratner imply that these are meaningful objects.

This is a joint endeavour with Yotam Smilansky and Barak Weiss.