Hebrew University Dynamics Seminar
The lunch seminar meets in the coffee lounge of the math building and lasts around 40 minutes. The room is equipped with a large blackboard and sandwiches.
The afternoon seminar meets at 14:00 in Math 209 and typically runs for 1 hour, though longer talks are possible with a break in the middle. The room has a large blackboard and digital projector with standard connectors (but if you have a mac, you may need to bring an adaptor). If you require us to provide a computer, an overhead projector, or other special equipment, let the organizers know an advance.
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Schedule 2014-15 (previous years: 2013-14, 2012-13)
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Victor Kleptsyn
Towards the understanding of group actions on the circle
My talk, based on joint works with B. Deroin, D. Filimonov and A. Navas, will be devoted to the recent progress in the understanding of (pseudo)-group actions on the circle, as well as foliations of real codimension one.
One large class of such actions is those that are sufficiently rich: there are local flows in local closure. Roughly speaking, restricting the dynamics on some subinterval $J$ and closing it in $C^1(J)$, one finds a one-parameter subgroup generated by some vector field (to be more precise, a neighborhood of identity in such subgroup: the flow is no longer defined once the points leave $J$). In this case, it is easy to obtain the Lebesgue-ergodicity of the action as a corollary of the one of such local flow (and there are some other interesting conclusions) -- as do Loray, Nakai, Rebelo, Scherbakov. As Ghys' commutator technique shows, an analytic action is guaranteed to fall in this class provided that there is a free subgroup, generated by the elements sufficiently close to the identity.
Another large class consists of the actions admitting a Markov partition. The presence of such a partition is quite restrictive, giving us a good control on the action. An example of such action is the standard action of PSL(2,Z), or (in the non-minimal case) the Schottky group.
Recent results, obtained in a joint project with B. Deroin, D. Filimonov, A. Navas suggest (though do not establish in its full generality) that there is nothing else but these two classes. In other words, the following alternative seems to hold: an action either admits a Markov partition, or has local flows in its local closure.
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Ron Peled
Separating signal from noise
Suppose that a sequence of numbers $(x_n)$ (a signal) is transmitted through a noisy channel. The receiver observes a noisy version of the signal, $(x_n + \xi_n)$, where the $(\xi_n)$ are independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space $X$ of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted?
We study the case in which signals are infinite sequences and the recovery or detection are required to hold with probability one. We provide conditions on the space $X$ for checking whether detection or recovery are possible and illustrate their applicability on examples related to statistics, harmonic analysis, ergodic theory and probability. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible. Joint work with Nir Lev and Yuval Peres.
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Rongang Shi
Pointwise equidistribution for one-parameter diagonal flow on homogeneous space
Let $F$ be the positive ray of a one-parameter diagonal subgroup of a Lie group $L$ and let $U$ be a connected unipotent subgroup of the unstable horospherical subgroup of $F$. Let $x$ be an element of the finite volume homogeneous space $X=L/\Gamma$ where $\Gamma$ is a discrete subgroup of $L$. For certain group $U$ we show that the trajectory Fux is equidistributed with respect to the probability Haar measure on $X$. The proof of this result uses a new height function of Benoist-Quint on homogeneous space and the method of quantitative equidistribution of Chaika-Eskin.
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Omri Sarig
Title: Ergodic properties of the measure of maximal of entropy for smooth three dimensional flows
Geodesic flows on compact connected surfaces of non-positive, non-identically zero curvature are Bernoulli with respect to the Liouville measure (Ornstein-Weiss, Ratner, Pesin). We show that this is the case for the measure of maximal entropy. Some of our results hold for all smooth three dimensional flows. Joint ongoing work with Ledrappier and Lima. -
Michael Björklund
A few remarks on the roles of left and right in ergodic theory
The classical (strong) mean ergodic theorem is usually stated as follows:
Let $G$ be a countable amenable group and let $(F_n)$ be a RIGHT Folner sequence in $G$. Then, for every unitary representation $H$ of $G$, and for every $v$ in $H$, the averages of $\pi(g)v$ as $g$ ranges over $F_n$ converge in the STRONG (norm) topology to the projection of $v$ onto the $G$-invariant vectors in $H$.
It is not hard to show that if the words RIGHT and STRONG above are changed to LEFT and WEAK, then the same conclusion holds.
The aim of this talk is to discuss to which extent the words LEFT and STRONG can both remain in the statement. We shall see that this heavily depends on the structure of the conjugacy classes of $G$.
Furthermore, the question about the co-existence of the words LEFT and STRONG admits a reformulation which does not require amenability. If time permits, I will try to explain why these words always can co-exist in Property (T) groups, while every non-property (T) group admits an embedding into a countable group so that a relative version of this question has a very negative answer.
Joint work with A. Fish (Sydney).
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Menny Akka
Integer points on spheres and their orthogonal lattices
Linnik proved in the late 1950's the equidistribution of integer points on large spheres of the 3-dimensional space, under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec). We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. Using unipotent dynamics we obtain stronger equidistribution results of the higher dimensional analogs.
I will present these, and related, arithmetic problems in detail, their translation into dynamics on homogeneous spaces and then discuss the tools which enable to prove the corresponding dynamical result. Prerequisites will be kept to a minimum; the dynamical result is on S-arithmetic homogeneous spaces so in the second part of the talk p-adic numbers will be assumed.
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Felix Pogorzelski
On a Banach space-valued ergodic theorem for amenable groups
This talk is devoted to an abstract ergodic theorem for a class of Banach space-valued functions defined on finite subsets of a countable, amenable group. The key assumptions for the functions under consideration are given by some invariance condition with respect to a finite colouring of the group, as well as by an (almost-)additivity property with respect to pairwise disjoint families of sets. Under mild ergodicity criteria, the normalized versions of those functions converge along Folner sequences. We explain the theorem and sketch the basic idea of its proof, the latter being a refinement of the celebrated quasi tiling technique developed by Ornstein and Weiss in the late eighties. We also draw some links to classical ergodic theorems for amenable groups. If time permits, we demonstrate how the convergence result can be applied to obtain uniform convergence of the spectral distribution function (integrated density of states) for finite hopping range, random operators on amenable Cayley graphs. This is joint work with Fabian Schwarzenberger.
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Uri Shapira
Around Minkowski's conjecture.
I will discuss the history and current state of an old conjecture about products of linear forms which is attributed to Minkowski. The geometric way to state this conjecture is that if $L$ is a lattice of co-volume $1$ in euclidean $d$-space, and if $S$ denotes the "star shape" consisting of all vectors whose product of coordinates is less than or equal to $2^{-d}$, then the translates of $S$ by the lattice points cover the whole space.
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Georgie Knight
Follow the fugitive: an application of the method of images to open systems
Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open dynamical systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behaviour, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. This is joint work with G. Cristadoro and M. Degli Esposti
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Liviana Palmisano
On circle endomorphisms with a flat interval and Cherry flows
We study $C^2$ weakly order preserving circle maps with a flat interval. We prove that, if the rotation number is of bounded type, then there is a sharp transition from the degenerate geometry to the bounded geometry depending on the degree of the singularities at the boundary of the flat interval. The general case of functions with rotation number of unbounded type is also studied. The situation becomes more complicated due to the presence of underlying parabolic phenomena. Moreover, the results obtained for circle maps allow us to study the dynamics of particular flows on the two-dimensional torus called Cherry flows. We analyse their metric, ergodic and topological properties.
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Tom Meyerovitch
Automorphisms for shifts of finite type over countable groups
Let $G$ be a countable group. A $G$-subshift is a compact, $G$-invariant subset $X$ of $\mathbb{N}^G$ (equipped with the induced $G$-action). A $G$-subshift is of finite-type if it not a strictly decreasing intersection of subshifts. There are several classical and some newer results expressing the idea that ``dynamically rich'' shifts of finite type have automorphism groups which are ``algebraically rich'' (Boyle-Lind-Rudolph for irreducible $\mathbb{Z}$-shifts of finite, Ward for strongly irreducible $\mathbb{Z}^d$-shifts of finite type, Hochman for positive entropy $\mathbb{Z}^d$-shifts of finite type). A key ingredient in many of these results is the existence of ``Markers'', introduced in this context by Hedlund and his collaborators. I will explain some of these old results and some new results. The relevant notions will be explained during the talk.
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Younghwan Son
Uniform distribution of functions with polynomial growth along primes and applications
A Hardy field is a set of germs of real functions at infinity which is closed under differentiation and which forms a field under the usual addition and multiplication. A classical example of a Hardy field is the field L of logarithmico-exponential functions, that is, the collection of all functions that can be constructed using the real constants, exponential and logarithmic functions, and the operations of addition, multiplication, division and composition of functions.
For a polynomial growth function f(x) belonging to a Hardy field, Boshernitzan obtained a necessary and sufficient condition that f(n), n=1,2,3,..., is uniformly distributed mod 1. In our talk, we will present the following recent result: Under Boshernitzan's condition, f(p), p = 2,3,5,..., is uniformly distributed mod 1. We will also discuss some applications of this result to ergodic theory and combinatorial number theory. This is a joint work with V. Bergelson and G. Kolesnik.
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Boris Solomyak
Holder property for the spectrum of translation flows in genus two
We study suspension flows (with a piecewise-constant roof function) over interval exchange transformations. Equivalently, these are translation flows on a specific stratum of the moduli space of Abelian differentials. Such flows are typically uniquely ergodic, weakly mixing (under natural assumptions on combinatorics in genus $\ge 2$), but not strongly mixing. Sinai asked (personal communication) if one can find local asymptotics for spectral measures of such flows. As a first step in this direction, we establish Hoelder continuity of spectral measures for almost all translation flows corresponding to Abelian differentials with one zero of order two on surfaces of genus two. This corresponds to interval exchanges on four intervals with permutation (4321). This is a joint work with A. I. Bufetov.
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Alexander I. Bufetov
Blaschke products and Palm distributions of determinantal point processesinduced by Hilbert spaces of holomorphic functions
The talk will be devoted to quasi-symmetries of determinantal point processes corresponding to Hilbert spaces of holomorphic functions. The main examples are the Ginibre ensemble and the point process corresponding to the Bergman kernel, as well as their weighted analogues. In the Bergman case, we shall see that Radon-Nikodym derivatives are given by Blaschke products. The talk is based on this preprint. Joint work with Yanqi Qiu.
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Tuomas Orponen
Quantifying rectifiability via projections
The classical Besicovitch projection theorem in the plane states that a compact 1-set K has positively many projections of positive length, if and only if a positive fraction of K can be covered by a Lipschitz curve of finite length. In the talk, I will discuss ways to quantify Besicovitch’s result.
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Mikhail Sodin
Spectral measures of $\{0, 1\}$-stationary sequences
If the spectral measure of a stationary sequence of random variables that attain two values has a lacuna in its support, then it is supported by the roots of unity of fixed order (and, hence, the sequence is periodic).
I plan to sketch two proofs of this observation. The first proof was given in a recent joint work with Borichev and Nishry. The second proof was suggested by Fedor Nazarov.
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Noam Berger
Random walk with cookies
I will describe a model of self-interacting random walk which is called "cookie random walk". I will survey several results and will then prove a recent 0-1 law which was established together with Gideon Amir and Tal Orenshtein.
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Oliver Sargent
Stationary measures on the space of rank 2 discrete subgroups of R^3.Stationary measures on the space of rank 2 discrete subgroups of R^3.
The space of rank 2 discrete subgroups of $\mathbb{R}^3$ can be realised as a homogeneous space $G/H$. However $H$ is not a lattice or even a discrete subgroup of $G$. Using recent developments of Y. Benoist and J.F. Quint we are able to analyse stationary measures for Zariski dense subgroups on this space. This work is joint with Uri Shapira.
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Tuomas Sahlsten
Itō calculus and the Fourier dimension of Brownian graphs
Salem sets are sets on the Euclidean space such that their Fourier dimension agree with the Hausdorff dimension. Specific Salem sets are difficult to find and most well-known arise in Diophantine approximation (after Kaufman). However, many random constructions such as images of compact sets with respect to Brownian motion, are Salem (Kahane). In the light of this, Kahane (1993) presented a problem on checking whether the graph or level sets of random processes such as the standard Brownian motion are Salem sets or not. Level sets for Brownian motion were considered by Kahane and Fouché-Mukeru (2013) but the graph case remained wide open until 2014 when together with T. Orponen (Helsinki) and J. Fraser (Manchester) we established with geometric measure theoretic means that the standard Brownian graph cannot be a Salem set. However, this result did not tackle the actual value of the Fourier dimension of the Brownian graph and now we present a new approach based on stochastic analysis (Itō calculus) to find its value. If time permits, I will also discuss some consequences of this in dynamics. Based on a recent joint work with J. Fraser (Manchester).