Dynamics Archives for Academic Year 2014

Organizational meeting

When: Thu, September 4, 2014 - 2:10pm
Where: MATH 1311
Speaker: Organizational meeting

Dynamical systems modulo null sets

When: Thu, September 11, 2014 - 2:10pm
Where: MATH 1311
Speaker: Mike Boyle (UMD) - http://www.math.umd.edu/~mboyle/
Abstract: This is joint work with Jerome Buzzi.

Say two automorphisms of a standard Borel space are almost-Borel conjugate if they are conjugate on the complement of universal null sets (sets with measure zero for all nonatomic invariant Borel probabilities). We give a complete almost Borel conjugacy classification for countable state Markov shifts and C^{1+} surface diffeomorphisms (apart from also neglecting for some of the surface
diffeomorphisms some sets supporting only zero entropy measures). This relies on work of Hochman and Sarig.

In related work with Ricardo Gomez, we show that SPR mixing Markov shifts of equal finite entropy (e.g. mixing shifts of finite type) are Borel conjugate on the complement of the periodic points.

In related work with Kevin McGoff, we prove up to finite index a conjecture of Rufus Bowen: two mixing shifts of finite type of equal entropy are "entropy conjugate". Bowen's conjecture in general remains open.

On rank-one transformations that are rationally ergodic and rigid

When: Thu, September 18, 2014 - 2:10pm
Where: MATH 1311
Speaker: Kelly Yancey (UMD) - http://www2.math.umd.edu/~kyancey/
Abstract: The generic transformation in the set of all invertible transformations that preserve a finite measure is both rigid and weakly mixing. There have been many recent developments toward characterizing the possible rigidity sequences for weakly mixing maps.

In joint work with Rachel Bayless, we are interested in what happens when you move to preserving an infinite measure. I will discuss some new results from our paper. In particular, I will give a construction to show compatibility of rigidity with rational ergodicity and also analyze rigidity sequences for rationally ergodic, rigid transformations.

Transition chains of invariant tori around L_1 in the Sun-Earth system

When: Thu, September 25, 2014 - 2:10pm
Where: MATH 1311
Speaker: Pablo Roldan (UMD) -
Abstract: We consider the spatial Restricted Three-Body Problem modelling the Sun-Earth system. We focus on the center manifold W^c(L_1) of the collinear equilibrium point L_1, with linear type center x center x saddle. We present a systematic numerical exploration of heteroclinic connections between different invariant tori in the center manifold W^c(L_1).

The results show that, as energy increases, there exist longer transition chains of tori. For high enough energy, there exist transition chains linking almost all tori in the energy manifold.

Effective stability for generic elliptic equilibrium points

When: Thu, October 2, 2014 - 2:10pm
Where: MATH 1311
Speaker: Abed Bounemoura (CNRS - CEREMADE) - http://sites.google.com/site/abedbou/
Abstract: We will explain why solutions of a Hamiltonian system starting sufficiently close to an ellitpic equilibrium are, generically, stable for an interval of time which is super-exponential large with respect to the inverse of the distance to the equilibrium. This is joint work with Bassam Fayad and Laurent Niederman.

On the work of Fields Medalists Artur Avila and Maryam Mirzakhani

When: Fri, October 10, 2014 - 3:15pm
Where: MATH 3206
Speaker: Giovanni Forni (UMD) -
Abstract: This is a colloquium. Note different date, time and room.

The cited work of both Avila and Mirzakhani includes contributions to the study of area-preserving flows on surfaces, (and related systems, such as Interval Exchange Transformations and Billiards in Rational Polygons) and/or to the study of the corresponding ''renormalization dynamics'', that is, the so-called Teichmueller geodesic flow on the moduli space of Riemann surfaces. In this talk we will survey their main results on these topics and discuss their significance mainly from the point of view of dynamical systems.

Semi-annual Workshop in Dynamical Systems and Related Topics

When: Thu, October 16, 2014 - 12:00pm
Where: Penn State University
Speaker: http://www.math.psu.edu/dynsys/dw_2014/

Nearly integrable systems with orbits accumulating to KAM tori

When: Thu, October 23, 2014 - 2:10pm
Where: MATH 1311
Speaker: Marcel Guardia (Jussieu) - http://www.imj-prg.fr/~marcel.guardia/#research
Abstract: The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit.

This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.

Full groups: word problem and decidability of the subshift

When: Thu, October 30, 2014 - 2:10pm
Where: MATH 1311
Speaker: Konstantin Medynets (USNA) - http://www.usna.edu/Users/math/medynets/index.php
Abstract: With any minimal subshift (X,T), one can associate a countable group [[T]], termed the full group, that completely determines the class of flip conjugacy of T. The fact that the algebraic properties of [[T]] are completely determined by the dynamical properties of the subshift (X,T) has been recently used to establish the existence of simple finitely generated amenable groups. In this talk, we will give a brief review of known algebraic properties of [[T]] and their relationship to the underlying dynamics. Amongst the new results, we will prove that word problem is solvable if and only if the minimal subshift has recursive language. All necessary facts from geometric group theory will be presented.

Dynamical properties of conformally symplectic systems

When: Thu, November 6, 2014 - 2:10pm
Where: MATH 1311
Speaker: Rafael de la Llave (Georgia Institute of Technology) - http://www.math.gatech.edu/users/rll6
Abstract: When one considers the dynamics of mechanical systems with a friction proportional to the velocity one obtains a system with the remarkable property that a symplectic form is transformed into a multiple of itself. The same phenomenon happens when one minimizes the action after discounting it by an exponential factor (these models are very common in economics when one minimizes the present cost and includes inflation). We will present several results for this systems:

1) A KAM theory for these systems that leads to efficient algorithms.
2) Absence of Birkhoff invariants near Lagrangian tori.
3) Numerical experiments at the breakdown of tori
4) Analyticity domains of expansions for KAM tori.

All these works are in collaboration with R. Calleja and A. Celletti (the numerical work reported is by R. Calleja, A. Celletti, J.L - Figueras)

WAP systems and the space of labels

When: Thu, November 13, 2014 - 2:10pm
Where: MATH 1311
Speaker: Ethan Akin (CCNY) - http://math.sci.ccny.cuny.edu/people?name=ethan_akin
Abstract: I will introduce the Ellis semigroup of a dynamical system and use it to define WAP (= weakly almost periodic systems). We will be concentrating on certain shift spaces defined using rapidly expanding functions and the space of labels. Skimming over the technical details, I will describe the sorts of examples we are able to obtain.

Subdiagrams of Bratteli diagrams and invariant measures

When: Thu, November 20, 2014 - 2:10pm
Where: MATH 1311
Speaker: Sergey Bezuglyi (University of Iowa and Institute for Low Temperature Physics, Ukraine) -
Abstract: It is known that any homeomorphism T of a Cantor set can be realized as a transformation acting on the path space of a Bratteli diagram. This remarkable result gives a convenient method for the study of T-invariant measures by considering ergodic measures on the path space of a corresponding Bratteli diagram which are invariant with respect to the tail equivalence relation. In my talk, I'm going first to survey some recent results about such measures defined on special classes of Bratteli diagrams. I will also discuss new theorems about measures supported by subdiagrams of a Bratteli diagram in the case of general Bratteli diagrams. These results are obtained in a joint work with M. Adamska, O. Karpel, and J. Kwiatkowski.

On a conjecture of Charles Tresser

When: Mon, December 8, 2014 - 3:00pm
Where: MATH 1308
Speaker: Enrique Pujals (IMPA) - http://w3.impa.br/~enrique/
Abstract: It was conjectured by C. Tresser that in the space of C^k orientation preserving embedding of the two disk which are area contracting, maps which belongs to the boundary of positive topological entropy exhibit a period doubling cascade.

In a joint work with S. Crovisier and C. Tresser we prove such conjecture assuming also that the embedding is in the boundary of Morse-Smale.

Tameness of Margulis Spacetimes

When: Thu, December 11, 2014 - 2:10pm
Where: MATH 1311
Speaker: William Goldman (UMD) - http://www.math.umd.edu/~wmg/
Abstract: A complete affine manifold is a quotient of Euclidean space by a discrete group of affine transformations acting properly. A Margulis spacetime is a 3-dimensional complete affine 3-manifold with free fundamental group of finite rank at least 2. Such a flat manifold corresponds to a deformation of a noncompact hyperbolic surface S. It is conjectured that such 3-manifolds are homeomorphic to solid handlebodies. We apply dynamical properties of the geodesic flow on S to prove this conjecture when S has no cusps. Not all quotient manifolds of Euclidean 3-space by free groups are solid handlebodies, however.

This is joint work with Suhyoung Choi. I will also discuss an independent alternative approach to this question developed by Jeff Danciger, François Guéritaud and Fanny Kassel, which uses Lorentzian geometry and applies to Lorentzian manifolds of negative curvature as well.

A microlocal toolbox for hyperbolic dynamics

When: Thu, January 29, 2015 - 2:10pm
Where: MATH 1311
Speaker: Semyon Dyatlov (MIT) - http://math.mit.edu/~dyatlov/
Abstract: I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation (X+\lambda)u=f, where X is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space.

I will show meromorphic continuation of the resolvent of X; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani-Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou.

Automorphisms of nonorientable surfaces

When: Thu, February 5, 2015 - 2:10pm
Where: MATH 1311
Speaker: Bruce Kitchens (IUPUI-NSF) - http://www.math.iupui.edu/~kitchens/
Abstract: The (well-studied) family of birational maps f_{a,b}(x,y) = (y,(y-b)/(x-a)) can be thought of as defining maps from the real projective plane to itself. Some of these maps give rise to automorphisms of nonorientable surfaces. The dynamics of these maps are interesting. Some of them are zero entropy and can be understood as piecewise rotations and translations. Others can be seen to have positive entropy because of their action on the homology but the dynamics is not easy to understand. It appears that some have contain strange atttractors and others "standard" horseshoes. This is joint work with Roland Roeder.

An averaging theory for nonlinear PDEs

When: Thu, February 12, 2015 - 2:10pm
Where: MATH 1311
Speaker: Guan Huang (UMD) - http://www-math.umd.edu/people/item/1186-guan.html
Abstract: Consider a small perturbation of some nonlinear integrable or linear PDEs. Assume that the unperturbed system admits a full set of action-angle variables. We show that if the size of the perturbatio a>0 is small enough, for typical initial data, the curves formed by the action variables, calculating for the solutions of the perturbed system on time interval of order a^{-1}, can be well approximated by solutions of a certain averaged equation. Our model equations are perturbed KdV equations and weakly nonlinear CGL equations.

Geodesic flows on surfaces with nonpositive curvature that are Bernoulli

When: Thu, February 19, 2015 - 2:10pm
Where: MATH 1311
Speaker: Yuri Lima (UMD) - http://www2.math.umd.edu/~yurilima/
Abstract: We prove that the measure of maximal entropy of the geodesic flow on a nonflat smooth surface with nonpositive curvature is Bernoulli. This is consequence of a more general result for smooth flows with positive speed on three dimensional manifolds: if an equilibrium measure of a Holder potential has positive metric entropy then the flow is a Bernoulli flow, or it is isomorphic to a Bernoulli flow times a rotational flow. Joint work with Francois Ledrappier and Omri Sarig.

Two special measures for random beta-transformation

When: Thu, February 26, 2015 - 2:10pm
Where: MATH 1311
Speaker: Karma Dajani (University of Utrecht) - http://www.staff.science.uu.nl/~kraai101/
Abstract: It is well known that if beta is a non-integer greater than 1, then almost every point has uncountably many expansions in base beta. In this talk, we will introduce a transformation, the so called random beta transformation, whose iterations produce all possible expansions in base beta. We exhibit two natural ergodic invariant measures for this transformation, give their properties and prove that these measures are mutually singular.

Ellipses are the only integrable billiard domains near the circle

When: Thu, March 12, 2015 - 2:10pm
Where: MATH 1311
Speaker: Vadim Kaloshin (UMD) - http://www2.math.umd.edu/~vkaloshi/
Abstract: The classical Birkhoff conjecture says that the only integrable convex domains are circles and ellipses. In the talk we show that this conjecture is true for small perturbations of the circle. This is joint work with A. Avila and J. De Simoi.

Flows on irreducible parabolic geometries

When: Thu, March 26, 2015 - 2:10pm
Where: MATH 1311
Speaker: Karin Melnick (UMD) - http://www.math.umd.edu/~kmelnick/
Abstract: Irreducible parabolic geometries are a family of geometric structures including conformal semi-Riemannian structures, projective structures, and many more. Automorphisms of these structures need not be linearizable around a fixed point, in contrast to semi-Riemannian isometries or affine transformations of a connection. I will discuss normal forms around a fixed point for automorphisms of some geometries in this family, focusing on how the dynamics of nonlinearizable automorphisms leads to rigidity results.

Measure rigidity for products of surface diffeomorphisms

When: Thu, April 2, 2015 - 2:10pm
Where: MATH 1311
Speaker: Aaron Brown (University of Chicago) -
Abstract: Motivated by recent work on the rigidity of stationary measures for affine actions (Benoist-Quint, Eskin-Mirzakhani), we consider a compact surface S, a compactly supported measure on the group of surface diffeomorphisms, and a corresponding stationary probability measure on S. Assuming the stationary measure is hyperbolic and admits no almost-surely invariant, measurable line fields, we show that the measure is either finitely supported or has the SRB property. In the case that almost every diffeomorphism preserves a common volume, it follows that any hyperbolic stationary measure that is not finitely supported and has no invariant measurable line fields coincides with an ergodic component of the volume. This is joint work with Federico Rodriguez Hertz.

Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure

When: Thu, April 9, 2015 - 2:10pm
Where: MATH 1311
Speaker: Zemer Kosloff (University of Warwick) - http://www2.warwick.ac.uk/fac/sci/maths/people/staff/zemer_kosloff/
Abstract: Markov partitions introduced by Sinai and Adler and Weiss are a tool that enables transferring questions about ergodic theory of Anosov diffeomorphisms into questions about topological Markov shifts and Markov chains. This talk will be about a reverse reasoning, that gives a construction of C^1 conservative (satisfying Poincare recurrence) Anosov diffeomorphism of T^2 without a Lebesgue absolutely continuous invariant measure. By a theorem of Gurevic and Oseledec, this can't happen if the map is C^{1+a}. Our method relies on first choosing a nice toral automorphism with a nice Markov partition and then constructing a bad conservative Markov measure on the symbolic space given by the Markov partition. We then push this measure back to the torus to obtain a bad measure for the toral automorphism. The final stage is to find, by smooth realization, a conjugating map H such that HFH^{-1} with Lebesgue measure is metrically equivalent to F with the bad measure.

Projection theorems for the counting and mass dimensions in Z^d

When: Thu, April 16, 2015 - 2:10pm
Where: MATH 1311
Speaker: Daniel Glasscock (Ohio State) - http://people.math.osu.edu/glasscock.4/
Abstract: The (upper) mass and counting dimensions for sets in the integer lattice Z^d capture the (maximal) polynomial rate of growth of the set on larger and larger cubes with and without a fixed center. They may be used to understand "fractal" subsets of the integer lattice; for example, the set of integers expressible in base 3 using only the digits 0 and 2 has mass and counting dimension equal to log 2 / log 3.

Both dimensions satisfy an analogue of Marstrand's theorem from geometric measure theory. In this talk, I will introduce the mass and counting dimensions, develop specific examples, and outline the proofs of these projection theorems. I will also hint at the way in which these discrete projection theorems may be used to recover Marstrand's original theorem.

This work builds on recent work of Y. Lima and C. G. Moreira; they introduced the counting dimension for subsets of Z and proved a Marstrand-type theorem for it.

Invariant measures for distal minimal flows

When: Thu, April 23, 2015 - 2:10pm
Where: MATH 1311
Speaker: Juho Rautio (University of Oulu) -
Abstract: In his 1961 paper "Strict ergodicity and transformation of the torus" Furstenberg constructed a minimal distal system on the two-dimensional torus with multiple invariant measures. The cardinality of the set of all invariant measures on this system is necessarily the continuum, but are there more invariant measures on the universal minimal distal system (with integers as the acting group)? This is indeed so. We can prove this by forming a large product of systems similar to Furstenberg's and then defining a large collection of product measures made of invariant measures on the constituent systems. Similar ideas can be used in the setting where the acting group is the reals. The main technical problem is the issue of ensuring that the product flows are minimal. Thus, we must study multiple disjointness. We can characterize the multiple disjointness of any collection of minimal distal flows in terms of the maximal equicontinuous factors of the flows in question.

Pure point spectrum in the regime of zero Lyapunov exponents

When: Thu, April 30, 2015 - 2:10pm
Where: MATH 1311
Speaker: David Damanik (Rice University) - http://www.ruf.rice.edu/~dtd3/
Abstract: In the study of ergodic Schrodinger operators, a central role is played by the Lyapunov exponent of the associated Schrodinger cocycle. In the first part of the talk we survey the connection between cocycle dynamics and spectral properties, which has been pushed to high art in the analytic one-frequency quasi-periodic setting by Artur Avila's global theory. In that case the Lyapunov exponent typically discriminates between the regimes of localization and absolutely continuous spectrum. In the second part we address the question whether the regime of zero Lyapunov exponents can ever contain pure point spectrum. We will report on joint work with Anton Gorodetski which shows that this is indeed possible. In the third part we describe some ingredients of the proof of this result.

Unique equilibrium states for geodesic flows in non-positive curvature

When: Thu, May 7, 2015 - 2:10pm
Where: MATH 1311
Speaker: Vaughn Climenhaga (University of Houston) - http://www.math.uh.edu/~climenha/
Abstract: The geodesic flow on a negatively curved manifold is one of the classical examples of a uniformly hyperbolic (transitive Anosov) system; in particular, it has a unique measure of maximal entropy, and more generally, unique equilibrium states for Holder continuous potentials. When curvature is only assumed to be non-positive, the geodesic flow becomes non-uniformly hyperbolic and much less is known. For a rank 1 manifold of non-positive curvature, Knieper showed uniqueness of the measure of maximal entropy, but his methods do not generalize to equilibrium states for non-zero potentials.

This talk is a preliminary report on work in progress with Keith Burns, Todd Fisher, and Daniel J. Thompson, in which we use a non-uniform version of Bowen's specification property to establish existence and uniqueness of equilibrium states for a class of non-zero potentials functions. This class includes scalar multiples of the geometric potential for an interval of parameter values. Our methods also have applications to partially hyperbolic diffeomorphisms.