Dynamics Archives for Fall 2014 to Spring 2015
Flat surface models of ergodic systems
When: Thu, September 12, 2013 - 2:00pmWhere: Math 1311
Speaker: Rodrigo Trevino- Tel Aviv University
Abstract: A Bratteli-Vershik transformation, also known as an adic transformation, is a measure preserving transformation of a zero dimensional set. It is defined by an infinite directed graph called a Bratteli diagram with some lexicographic order on the set of infinite paths through this graph. Using Rokhlin towers one can construct a flat surface (generically of infinite topological type) and a flow on it which is measurably isomorphic to the dynamics of the Bratteli-Vershik (adic) transformation. In fact, using Rolkhlin's lemma, one can do this construction for any aperiodic automorphism of a standard measure space.
Using recently-developed tools for the study of translation flows on flat surfaces of infinite type, we can study the ergodic properties through the flat surface model in very general terms. I will state a criterion for unique ergodicity for adic transformations, explain where it comes from, and talk a bit about the space of all Bratteli diagrams, which in some sense serves as a moduli space for all flat surfaces of finite area, and the systems which they model. I will end with some open questions about the structure of this space and relevant questions about dimension groups.
This is joint work with Kathryn Lindsey from Cornell University.
Rigid and weak mixing constructions preserving an infinite measure
When: Thu, September 19, 2013 - 2:00pmWhere: Math 1311
Speaker: Kelly Yancey- University of Maryland- kyancey@umd.edu
Abstract: In the setting of infinite ergodic theory, measure-preserving transformations that are rigid and spectrally weakly mixing are generic in the sense of Barie category. During this talk we will discuss rigid verses various types of weakly mixing in infinite ergodic theory. We will also construct examples of transformations that have these desired properties. Our examples will be via the method of cutting and stacking.
This is joint work with Rachel Bayless.
SOME PROBABILISTIC RESULTS USING DYNAMICS
When: Thu, September 26, 2013 - 2:00pmWhere: Math 1311
Speaker: Yuri Lima- UMD-
Abstract: We will discuss two problems that are probabilistic in nature and are solved using dynamics.
The first one considers annihilation and coalescence on finite binary trees. Given an infection configuration in the leaves, and given a set of spreading rules, the infection spreads along the nodes of the tree. What is the limiting distribution at the root node, as the height of the tree grows? We will present results for some instances of this problem. The proofs use quadratic equations. Joint work with I. Benjamini.
The second is a Polya's urn with graph based interactions. Given a finite connected graph, place a bin at each vertex, and call two bins a pair if they share an edge. At discrete times, a ball is added to each pair of bins as follows: one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power alpha > 0. What is the limiting behavior of the proportion of balls in the bins? We will present results for alpha ≤ 1. The proofs use stochastic approximation algorithms. Joint work with M. Benaım, I. Benjamini, and J. Chen.
Connections between Complex Dynamics and Ergodic Theory
When: Thu, October 3, 2013 - 2:00pmWhere: Math 1311
Speaker: Jane Hawkins- NSF
Abstract:While the Julia sets of rational maps of the sphere usually conjure up images of interesting topological features, they also possess many measure theoretic properties worth studying. Every rational map has several distinguished invariant measures: one is the unique invariant measure of maximal entropy and the other is a more geometric measure. Only in rare instances do they coincide. The geometric measure called a conformal measure. There is often a nonatomic invariant measure equivalent to conformal measure, sometimes infinite and sometimes finite. We give families of examples of these. We also discuss one-sided Bernoulli properties and which maps rule out one-sided Bernoulli behavior. When the Julia set is the entire sphere (or a smooth submanifold of the sphere), the ergodic properties of Lebesgue measure can be studied.
Norm Approximation in Ergodic Theory
When: Thu, October 10, 2013 - 2:00pmWhere: Matrh 1311
Speaker: Joe Rosenblatt- University of Illinois at Urbana-Champaign- http://www.math.uiuc.edu/~jrsnbltt
Abstract: Classical ergodic averages give good norm approximations, but these averages are not necessarily
giving the best norm approximation among all possible averages. We consider
1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function,
2) when these optimal Cesaro norm approximations are not comparable to the norm of the usual ergodic average,
3) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average,
and
4) the oscillatory nature of optimal norm approximations as well as the norm of the usual ergodic average.
Complexity for polygonal billiards (note special day)
When: Wed, October 16, 2013 - 12:30pmWhere: Math 0102
Speaker: Dmitry Scheglov-dvs117@gmail.com
Abstract: Polygonal billiard is a simple dynamical system, whose definition is immediately clear even for non-mathematicians. While for rational angles the subject is well developed, for irrational angles it is full of famous open questions such as the existence of periodic orbits, ergodicity, orbit growth and others. In this talk we will discuss recent advances in the orbit growth for irrational polygonal billiards, some being established and some being work in progress.
NO SEMINAR (PSU WORKSHOP)
When: Thu, October 17, 2013 - 2:00pmWhere:
Tower multiplexing and slow weak mixing
When: Thu, October 24, 2013 - 2:00pmWhere: Math 1311
Speaker: Terry Adams- Department of Defense
Abstract: Fix a Lebesgue probability space. Endow the set of invertible measure preserving transformations with the weak topology. It is well known that both the properties of weak mixing and rigidity are generic properties in this topological space. This is interesting since the key characteristics of these two properties contrast greatly. Weak mixing occurs when a system equitably spreads mass throughout the probability space for most times. Rigidity occurs when a system evolves to resemble the identity map infinitely often. Since both of these behaviors exist simultaneously in a large class of transformations, it is natural to ask what types of rigidity sequences are realizable by weak mixing transformations. In this talk, we show all rigidity sequences generated by ergodic measure preserving transformations are also generated by the class of weak mixing transformations. To solve this problem, we introduce a technique for combining two ergodic measure preserving transformations using an infinite chain of isomorphisms defined on sequences of Rokhlin towers.
Aubry-Mather type results for a system of elliptic PDE's
When: Thu, October 31, 2013 - 2:00pmWhere: Math 1311
Speaker: Paul Rabinowitz- University of Wisconsin Madison
Abstract: Moser initiated the development of an Aubry-Mather Theory for PDE's. He and subsequently others obtained results in this direction for a nonlinear elliptic PDE model. We discuss further extensions to an system of PDE's.
Cutting and resewing pillow cases
When: Thu, November 7, 2013 - 2:00pmWhere: Math 1311
Speaker: Pat Hooper-Mathematics at City College of New York
Abstract: I will discuss the dynamics of a fairly simple piecewise isometry of a square pillowcase. We cut the pillowcase along two horizontal edges we obtain a cylinder, which we can rotate and then sew back together. We can then do the same in the vertical direction. The composition of these two cutting and resewing operations yields a piecewise isometry of the pillowcase with interesting dynamics. We will describe how in
some cases the collection of aperiodic points forms a fractal curve, and the dynamics on this curve is topologically conjugate to a rotation (modulo concerns related to discontinuities). Properties of this map such as the existence of this curve depend
on the interaction of the even continued fraction expansions of the two rotation parameters.
The distal and equicontinuous structure relations
When: Thu, November 14, 2013 - 2:00pmWhere: Math 1311
Speaker: Joe Auslander-UMD
Abstract: We present new proofs of some "trivial" theorems.
Shilnikov bifurcations in analytic unfoldings of the Hopf-zero singularity
When: Thu, November 21, 2013 - 2:00pmWhere: Math 1311
Speaker:Oriol Castejon- UMD
Abstract: The Hopf-zero singularity consists in a vector field X in $R^3$ having the origin as a critical point, and the differential DX at this point having a pure imaginary pair and a simple zero eigenvalue. The dynamics of the analytic unfoldings of this singularity has yet not been completely understood.
More precisely, if one studies the truncation of the normal form (at any finite order) of some of these unfoldings, one can see that it has two saddle-focus critical points connected by a one- and a two-dimensional heteroclinic manifolds. However, if one considers the whole vector field, one expects these heteroclinic connections to be destroyed. This can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation.
Recently, it has been seen that the last step to prove the existence of such bifurcations requires a complete understanding of the splittings of the heteroclinic connections. We will show how asymptotic formulas for these splittings can be found.
This is a joint work with I. Baldoma' and T.M. Seara.
Mixing phenomena in infinite measure and examples in rank-one transformations
When: Thu, December 5, 2013 - 2:00pmWhere: Math 1311
Speaker: Cesar Silva- Williams College
Abstract: We will discuss notions such as weak mixing, power weak mixing, rational ergodicity and rational weak mixing for infinite measure-preserving transformations. We will consider examples in the class of rank-one transformations. These notions will also be considered in the context of measurable sensitivity.
Cocycles of point-distal flows
When: Thu, December 12, 2013 - 2:00pmWhere: Math 1311
Speaker: Gernot Greschonig- Vienna University of Technology
Abstract: We will discuss the generalisation of results on cocycles of distal minimal compact metric flows for a class of point distal flows, the so-called strict AI systems. These can be represented by a simplified Veech tower. In this simplified Veech tower the (possibly) transfinite induction process of isometric and almost 1-1 extensions takes place within factors the original flow, while in the general case of the Veech structure theorem the induction process yields an almost 1-1 extension of the original point distal flow.
Rigidity of higher rank abelian actions
When: Wed, January 8, 2014 - 2:00pmWhere: Math 3206 (note special day and room)
Speaker: Zhiren Wang-Yale University
Abstract: A $Z$-action generated by a single toral automorphism is not rigid, in the sense that invariant measures and subsets can behave quite arbitrarily. However, this is not the case for generic $Z^r$-actions by toral automorphisms. In this talk, we will describe rigidity properties of algebraic $Z^r$-actions on nilmanifolds in the categories of measures, subsets and smooth structures. We show that, up to smooth conjugacy, all Anosov $Z^r$-actions without rank-one actions on tori and nilmanifolds act by automorphisms. We will also explain how one can establish similar rigidity statements for actions by higher rank lattices by studying their abelian subgroups. This talk is partially based on joint works with Aaron Brown and Federico Rodriguez Hertz.
Parry's topological transitvity
When: Thu, January 30, 2014 - 2:00pmWhere: Math 1311
Speaker: Robbie Robinson-George Washington University
Abstract: In his 1964 paper on f-expansions, Parry studied piecewise-continuous, piecewise-monotonic interval maps F, and introduced a notion of topological transitivity different from any of the modern definitions. This notion, which we call "Parry topological transitivity", is that the backward orbit O^-(x)={F^{-n}(x)} of some x is dense. We show that topological transitivity (i.e., a dense forward orbit) implies Parry topological transitivity, but that the converse is generally false. We discuss Parry's application of these ideas to the theory of f-expansions.
Local Limit Theorems for dynamical systems
When: Thu, February 6, 2014 - 2:00pmWhere: Math 1311
Speaker: Dima Dolgopyat-University of Maryland
Abstract: We describe some recent progress and open problems in the
field of local limit theorems for dynamical systems. We obtain several applications of these results including non-equilibrium density profile for Lorentz gas and a several limit theorems for discrepancy of Kronecker sequences.
Teichmuller dynamics and algebraic geometry (CANCELED)
When: Thu, February 13, 2014 - 2:00pmWhere: Math 1311
Speaker: Simion Filip-University of Chicago
Abstract: The space of pairs (Riemann surface, holomorphic 1-form) admits an action of the group SL(2,R). This action is intimately related to more classical dynamical systems, such as billiards in polygons and interval exchanges. It can be interpreted as a renormalization dynamics and questions about these systems can often be answered using the SL(2,R) action. Eskin and Mirzakhani recently established a strong rigidity of SL(2,R)-invariant measures. In particular, these are of Lebesgue-class supported on manifolds. In this talk I will explain a proof that these manifolds are in fact algebraic varieties, with interesting arithmetic properties. The techniques also give restrictions on dynamics, in particular on the Kontsevich-Zorich cocycle. I will begin by introducing some background and motivating examples from Teichmuller dynamics. I will then introduce the necessary concepts from algebraic geometry, in particular Hodge theory. No background in either subject will be assumed.
Teichmueller dynamics and algebraic geometry
When: Wed, February 26, 2014 - 1:00pmWhere: Math 0302 (NOTE SPECIAL TIME AND ROOM)
Speaker: Simion Filip-University of Chicago
Abstract: The space of pairs (Riemann surface, holomorphic 1-form) admits an action of the group SL(2,R). This action is intimately related to more classical dynamical systems, such as billiards in polygons and interval exchanges. It can be interpreted as a renormalization dynamics and questions about these systems can often be answered using the SL(2,R) action. Eskin and Mirzakhani recently established a strong rigidity of SL(2,R)-invariant measures. In particular, these are of Lebesgue-class supported on manifolds. In this talk I will explain a proof that these manifolds are in fact algebraic varieties, with interesting arithmetic properties. The techniques also give restrictions on dynamics, in particular on the Kontsevich-Zorich cocycle. I will begin by introducing some background and motivating examples from Teichmuller dynamics. I will then introduce the necessary concepts from algebraic geometry, in particular Hodge theory. No background in either subject will be assumed.
Dynamical Borel-Cantelli Lemmas
When: Thu, February 27, 2014 - 2:00pmWhere: Math 1311
Speaker: Matthew Nicol- University of Houston
Abstract: Borel-Cantelli Lemmas are useful for establishing the almost sure behavior of stochastic processes. Suppose $T$ is a measure preserving transformation of a probability space $(X,\mu)$ and $\{B_i\}$ is a sequence of sets in $X$ such that $\sum_i \mu (B_i)$ diverges. We may ask: does $T^{i} (x) \in B_i$ for infinitely many $i$ for $\mu$ a.e. $x\in X$? This is an analog of the Borel-Cantelli lemma from probability theory. If $B_i$ is a sequence of nested balls in a metric space this question is sometimes called the shrinking target problem. We will discuss results progress in establishing Borel-Cantelli Lemmas for dynamical systems with some degree of hyperbolicity.
Symbolic dynamics for three dimensional flows with positive topological entropy
When: Thu, March 6, 2014 - 2:00pmWhere: Math 1311
Speaker: Yuri Lima-UMD
Abstract: Since Hadamard, the construction of symbolic models for dynamical systems has been successfully implemented in many scenarios. E.g. geodesic flows of surfaces with negative curvature are coded by suspensions over finite Markov shifts.
In a joint work with Sarig, we deal with non-singular flows on three dimensional manifolds. These include geodesic flows of surfaces, and Reeb vector fields. Provided the flow has positive topological entropy, we code it by a suspension over a countable Markov shift.
Here is an application: for almost every metric on the two-sphere, there are C,h>0 such that there are at least Ce^{Th}/T closed geodesics of size at most T.
The automorphism group of a shift of subquadratic growth
When: Thu, March 13, 2014 - 2:00pmWhere: Math 1311
Speaker: Van Cyr-Bucknell University
Abstract: The automorphism group of a symbolic dynamical system $(X,\sigma)$ is the group of homeomorphisms of $X$ that commute with $\sigma$. For many natural systems, this group is extremely large and complicated (e.g. a theorem of Boyle, Lind, and Rudolph shows that if $X$ is a topologically mixing SFT, then $\Aut(X)$ contains isomorphic copies of all finite groups, the free group on two generators, and the direct sum of countably many copies of $\mathbb{Z}$). This can be interpreted as a manifestation of the ``high complexity'' of these shifts.
In this talk I will discuss recent joint work with B. Kra which places restrictions on the automorphism group of any topologically transitive subshift (not necessarily an SFT) of ``low complexity.'' This class contains the Sturmian shifts, the Rauzy-Arnoux shifts, any other transitive subshift whose factor complexity function grows subquadratically. One of our main results is that, for these shifts, if $H$ is the subgroup of $\Aut(X)$ generated by $\sigma$ then $\Aut(X)/H$ is a periodic group.
NO SEMINAR (SPRING BREAK)
When: Thu, March 20, 2014 - 2:00pmWhere:
Nonlinear patterns in large sets of countable fields
When: Thu, March 27, 2014 - 2:00pmWhere: Math 1311
Speaker: Joel Moreira- Ohio State University
Abstract: An old lemma of Schur states that, given any finite partition of the positive integer numbers, there exists a triple of the form {x,y,x+y} in the same cell of the partition. There also exists a triple of the form {x,y,xy} in the same cell, but it has been an open problem ever since whether one can find a quadruple {x,y,x+y,xy} in the same cell. I will describe a dynamical approach to this type of problems and present recent joint work with V. Bergelson on analogues of this and similar questions with the set of positive integers replaced with a field.
Cellular automata over sofic groups (NOTE SPECIAL DAY)
When: Tue, April 1, 2014 - 2:00pmWhere: Math B0427
Speaker: Benjy Weiss-Einstein Institute of Mathematics (Hebrew Univ of Jerusalem)
Abtract: If A is a finite alphabet then a cellular automaton over A^Z is just a continuous map from A^Z to itself which commutes with the shift. It is a classical theorem of Hedlund that such a map is surjective if and only if it takes the uniform measure \mu on A^Z to itself. I shall discuss to what extent this theorem extends to sofic groups - which are a class of groups that contain all amenable groups and all residually finite groups.
NO SEMINAR
When: Thu, April 3, 2014 - 2:00pmWhere: Math 1311
The tau-topology, the Derived Group, and the Regionally Proximal Relation.
When: Thu, April 10, 2014 - 2:00pmWhere: Math 1311
Speaker: Dave Ellis- Beloit College
Abstract: The universal minimal flow (M, T) for minimal right actions of a group T, and its group G of automorphisms have a rich structure which can be exploited to obtain interesting results on minimal flows. In this talk I will show how an understanding of the structure of the flow (G,M), the tau-topology on G and the derived group G' leads to a better understanding of the regionally proximal relation Q(X) for a minimal flow (X, T). I will outline a proof that if (X,T) satisfies a certain group condition, then an apriori smaller relation Q^c(X) is equal to Q(X) (and both are equivalence relations). One interesting consequence: if in addition X is metrizable, then the Veech relation is also an equivalence relation.
Quantum ergodicity for ray-splitting (branching) billiards
When: Thu, April 17, 2014 - 2:00pmWhere: Math 1311
Speaker: Dmitry Jakobson- McGill University
Abstract: After giving an overview of Quantum Ergodicity results on compact Riemannian manifolds with ergodic geodesic flow (due to Shnirelman, Zelditch, Colin de Verdiere and others), we discuss several related examples, including QE for the Hodge Laplacian and Dirac operators, and the relationship to frame flows. We next discuss recent joint work with Yury Safarov and Alex Strohmaier, which concerns the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. We prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics. If time permits, we outline the first example (provided by Y. Colin de Verdiere) of a system where the ergodicity assumption holds for the discontinuous system. We end with a list of open problems.
Transitivite points in hyperspaces
When: Thu, April 24, 2014 - 2:00pmWhere: Math 1311
Speaker: Anima Nagar-Indian Institute of Technology and UMD
Abstract: Given a system (X,T), where X is a compact metric space and T is
a self map on X, we look into the dynamics of the induced system (2^X,T)
where 2^X is the space of all non-empty closed subsets of X. We look into the structure of transitive points in this induced system. We investigate the structure of such transitive points and discuss their implications on the dynamical properties of (X,T).
Anomalous modulus of continuity for the theta process and logarithm laws for geodesics.
When: Thu, May 1, 2014 - 2:00pmWhere: Math 1311
Speaker: Francesco Cellarosi-University of Illinois at Urbana Champaign
Abstract: Theta sums are particular exponential sums with deep number theoretical and physical connections. The planar curves obtained by linearly interpolating their partial sums are sometimes called 'curlicues' because of their rich geometric structure of spirals arranged in an approximate multi-fractal structure. Analogously to the construction of the Brownian Motion starting from simple symmetric random walks, I will briefly explain how to obtain a random process (the Theta Process) using equidistribution of horocycles under the action of the geodesic flow on a suitable hyperbolic manifold. Among the properties of this process, I will discuss the anomalous modulus of continuity of typical realizations of this process (different from that of a typical Brownian path), and derive this property using a logarithm law for geodesics due to Kleinbock and Margulis. This implies, in particular H\"older continuity of typical realizations for any exponent less than 1/2. Joint work with Jens Marklof (Bristol)
The trajectories asymptotic to KAM torus
When: Thu, May 8, 2014 - 2:00pmWhere: Math 1311
Speaker: Jianlu Zhang-Nanjing University
Abstract: In nearly integrable Hamiltonian systems with three degrees of freedom, there are asymptotic orbits to KAM torus, that is, the ω-limit set or α-limit set of these orbits is certain KAM torus.