Dynamics Archives for Academic Year 2015

Ergodic Measures for shifts with eventually constant complexity growth

When: Thu, September 10, 2015 - 2:10pm
Where: Math 1311
Speaker: Jon Fickenscher-Princeton University - jonfick@princeton.edu

Abstract: We will consider (sub)shifts with complexity such that the difference from n to n+1 is constant for all large n. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most d/2 ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron.

Entropy, volume growth and SRB measures for Banach space mappings

When: Thu, September 17, 2015 - 2:10pm
Where: Math 1311
Speaker: Alex Blumenthal (Courant Institute) - alex@cims.nyu.edu
Abstract: (Joint with Lai-Sang Young) This talk is about extending certain results in smooth ergodic theory to mappings of Banach spaces; these pertain to the interplay between volume growth and dynamical complexity in the sense of metric entropy. Our results have applications to mappings generated by a large class of dissipative PDEs.
In the first part, we discuss a general framework and give examples of the types of systems to which our framework applies; we end the first part with a discussion of infinite-dimensional extensions of two results: (i) the Multiplicative Ergodic Theorem (due to Ruelle, Mane et al. in infinite dimensions) and (ii) the characterization of SRB measures as those for which the entropy formula holds (due to Ledrappier et al. for finite-dimensional diffeomorphisms and AB-LSY for Banach space mappings in our setting).
In the second part of the talk, we give a technical discussion of result (ii), emphasizing differences in the infinite-dimensional setting. We highlight two major differences: (a) there is no natural notion of d-dimensional volume element, and whatever notion of volume we use must not only be compatible with the MET, but should also be regular enough to support distortion estimates; and (b) the map need not be invertible away from its attractor, and contraction in stable directions may be arbitrarily strong- this has implications, e.g., for the regularity of unstable distributions.

Quasicrystals, ergodic theory and cohomology

When: Thu, September 24, 2015 - 2:10pm
Where: Math 1311
Speaker: Rodrigo Trevino (Courant Institute) - rodtrevino@gmail.com

Abstract: I will talk about some recent results on deviation of ergodic averages for systems coming from aperiodic tilings and aperiodic point sets which are self affine (the Penrose tiling is an example of these). These rely on fun interactions between ergodic theory and some cohomology theories associated with aperiodic point sets. Time permitting I will discuss applications to problems of diffraction, counting problems, and future directions. This is joint work with S. Schmieding.

On Virtual Conjugacy of Generalized Odometers

When: Thu, October 1, 2015 - 2:10pm
Where: Math 1311
Speaker: Kostya Medynets-US Naval Academy - medynets@usna.edu

Abstract: We show that generalized odometers are continuously orbit equivalent if and only if the sequences of finite-index subgroups defining the systems are virtually isomorphic. For minimal equicontinuous $\mathbb Z^d$-systems the continuous orbit equivalence implies that the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\mathbb Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^*$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable. This is a joint work with Maria Isabel Cortez.

Quasiperiodicity and Chaos

When: Thu, October 8, 2015 - 2:10pm
Where: Math 1311
Speaker: James Yorke (University of Maryland) - http://www.chaos.umd.edu/~yorke/
Abstract: TBA

Mixing properties of smooth flows on surfaces

When: Thu, November 5, 2015 - 2:10pm
Where: Math 1311
Speaker: Adam Kanigowski (Pennsylvania State University) - http://www.adkanigowski.cba.pl/en.php

Abstract: We will study a variant of Ratner property, which describes a special way of divergence of nearby points, in the class of smooth flow on surfaces. As a consequence we will be able to show that some mixing flows on surfaces are mixing of all orders (Rohlin problem). Moreover we will show that there exist a smooth mildly mixing flow on every surface genus \geq 2.

A Massera type theorem for the existence of almost automorphic solutions to some evolution equations

When: Thu, November 12, 2015 - 2:10pm
Where: Math 1311
Speaker: Gaston N'Guerekata (Morgan State University) - http://www.morgan.edu/research_and_economic_development/faculty_highlights/gaston_nguerekata.html

Abstract: Consider in $X=R^n$ or $C^n$ the Eq (1) x'(t)=A(t)x(t)+f(t), where A(t) and f(t) are T-periodic $R-->X$. Massera proved in 1950 that Eq(1) has a T-periodic solution on the whole real line if and only if it has a bounded solution on the positive semi axis. We first will give an elementary proof of this result. Then we will extend the theorem to the case where f(t) is an almost automorphic function in the sense of Bochner. Moreover, if A(t) is independent of t, then the Carleman spectrum of the almost automorphic is at most the one of f. We will also prove that the set of all almost periodic functions is a set of first category in the set of almost automorphic functions.

Hitting time statistics for random dynamical systems

When: Thu, November 19, 2015 - 2:10pm
Where: Math 1311
Speaker: Jerome Rousseau (Universidade Federal da Bahia, Brazil) - http://www.sd.mat.ufba.br/~jerome.rousseau/

Abstract: In this talk, we will give an overview of recent results on law of rare events for random dynamical systems.
We will show that for super-polynomially mixing i.i.d. random dynamical systems, we obtain an exponential law (with respect to the invariant measure of the skew-product) for hitting and return times.
For random subshifts of finite type, we analyze the distribution of hitting times with respect to the sample measures.
We prove that with a superpolynomial decay of correlations one can get an exponential law for almost every point and with stronger mixing assumptions one can get a law of rare events depending on the extremal index for every point.
(These include some joint works with Benoit Saussol and Paulo Varandas, Mike Todd, Nicolai Haydn and Fan Yang)

Towards the global bifurcation theory on the plane

When: Fri, November 20, 2015 - 2:10pm
Where: Math 1313
Speaker: Yulij Ilyashenko (Cornell University) - https://www.math.cornell.edu/m/People/bynetid/isi1

Abstract: The talk provides a new perspective of the global bifurcation theory on the plane. Theory of planar bifurcations consists of three parts: local, nonlocal and global ones. It is now clear that the latter one is yet to be created.
Local bifurcation theory (in what follows we will talk about the plane only) is related to transfigurations of phase portraits of differential equations near their singular points. This theory is almost completed, though recently new open problems occurred. Nonlocal theory is related to bifurcations of separatrix polygons (polycycles). Though in the last 30 years there were obtained many new results, this theory is far from being completed.
Recently it was discovered that nonlocal theory contains another substantial part: a global theory. New phenomena are related with appearance of the so called sparkling saddle connections. The aim of the talk is to give an outline of the new theory and discuss numerous open problems. The main new results are: existence of an open set of structurally unstable families of planar vector fields, and of families having functional invariants (joint results with Kudryashov and Schurov). These results disprove a conjecture of Arnold (1985).
NOTE: unusual time and place!

From rational billiards to unexpected families of Riemann surfaces.

When: Thu, December 3, 2015 - 2:10pm
Where: Math 1311
Speaker: Alex Wright (Stanford University) - http://web.stanford.edu/~amwright/
Abstract: The first talk will serve as a general introduction to dynamics on moduli spaces and will overview some of the recent progress in the field. The second talk will discuss joint work with Alex Eskin, Curtis McMullen, and Ronen Mukamel, in which we construct six new and unexpected orbit closures for the GL(2,R) action on moduli spaces of translation surfaces. These examples disprove a conjecture of Mirzakhani. The construction takes place in the realm of Hurwitz spaces of covers of CP^1 and involves real multiplication of Hecke type.

Exponential decay of correlations for Sinai billiard flows

When: Fri, December 4, 2015 - 3:00pm
Where: Math 1308
Speaker: Mark Demers (Fairfield University) - http://www.faculty.fairfield.edu/mdemers/

Abstract: While billiard maps for large classes of dispersing billiards are known to enjoy exponential decay of correlations, the corresponding flows have so far resisted such analysis. We describe recent results, based on the construction of function spaces on which the associated transfer operator has good spectral properties, which provide a description of the spectrum of the generator of the semi-group. This construction, together with a Dolgopyat-type cancellation argument to eliminate certain eigenvalues, prove that the generator has a spectral gap and that the billiard flow with finite horizon has exponential decay of correlations. This is joint work with V. Baladi and C. Liverani.

On deformational spectral rigidity of convex symmetric planar domains.

When: Thu, December 10, 2015 - 2:00pm
Where: Math 1311
Speaker: Vadim Kaloshin (UMD) - http://www.math.umd.edu/~vkaloshi/
Abstract: One can associate to a planar convex domain $\Omega \subset \R^2$
two types of spectra: the Laplace spectrum consisting of eigenvalues of
a Dirichlet problem and the length spectrum consisting of perimeters of
all periodic orbits of a billiard problem inside $\Omega$. The Laplace and
length spectra are closely related, generically the first determines the second.
M. Kac asked if the Laplace spectrum determines a domain $\Omega$. There
are counterexamples. During the talk we show that a planar axis symmetric
domain close to the circle can't be smoothly deformed preserving the length
spectrum unless the deformation is a rigid motion. This gives a partial answer
to a question of P. Sarnak. This is a joint work with J. De Simoi and Q. Wei.

On energy transferring in a periodic pendulum lattice with analytic weak couplings

When: Thu, February 4, 2016 - 2:10pm
Where: Math 1311
Speaker: Guan Huang (UMD) - http://math.umd.edu/~guan/
Abstract: In this talk we discuss the long time dynamics of a periodic lattice of pendulums with neighbouring interactions of certain forms. When the interaction is sufficiently weak (measured by a small parameter), the resulting system is nearly integrable. The KAM theory indicates that for "most" of the solutions, the energy of each pendulums varies inside a very narrow interval for all time: the energy do not transfer. Still, there exist some particular solutions of the system such that the corresponding energy could propagate along the chain of pendulums in any predetermined order.

Positive metric entropy arises between some KAM tori

When: Thu, February 11, 2016 - 2:10pm
Where: Math 1311
Speaker: Dong Chen (Pennsylvania State University) -
Abstract: The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still have a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens between KAM tori draws lots of attention. In this talk I will present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is C^m small (m can be arbitrarily large) but the flow has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy between some KAM tori.

Convex shapes and complex polynomials

When: Thu, February 18, 2016 - 2:10pm
Where: Math 1311
Speaker: Kathryn Lindsey (University of Chicago) - http://math.uchicago.edu/~klindsey/
Abstract: Any planar shape P can be embedded isometrically as part of a convex surface S in R^3 such that the boundary of P is the support of the curvature of S. In particular, we could take P to be a filled polynomial Julia set and the curvature distribution to be proportional to the measure of maximal entropy on the Julia set. What would the associated convex shape look like? In joint work with Laura DeMarco, we explored this question, building on discussions with Bill Thurston, Curt McMullen, and Laurent Bartholdi.

Dimension, Zeta functions and Resonances

When: Thu, February 25, 2016 - 2:10pm
Where: Math 1311
Speaker: Mark Pollicott (University of Warwick) - http://homepages.warwick.ac.uk/~masdbl/
Abstract: A pair of pants is an example of a Riemann surface associated to a convex cocompact Fuchsian group. We describe the dependence of the dimension associated limit set in the boundary of hyperbolic space via the zeros of the associated Selberg zeta function. Finally, we study the other zeros of the zeta function.

Equidistribution of Shears and Applications

When: Thu, March 3, 2016 - 2:10pm
Where: Math1311
Speaker: Alex Kontorovich (Rutgers University) - http://www.math.rutgers.edu/~ak1230/
Abstract: A "shear" is a unipotent translate of a cuspidal geodesic
ray in the quotient of the hyperbolic plane by a non-uniform discrete
group (possibly of infinite co-volume). In joint work with Dubi
Kelmer, we prove the regularized equidistribution of shears under
large translates. We give applications including to moments of GL(2)
automorphic L-functions, and to effective counting of integer points
on affine homogeneous varieties (in particular resolving a missing
case of the Eskin-McMullen/Duke-Rudnick-Sarnak machinery). No prior
knowledge of these topics will be assumed.


When: Thu, March 17, 2016 - 2:10pm
Where: Math 1311
Speaker: () -

On renormalization and rigidity for a class of nonlinear interval exchange transformations

When: Tue, March 22, 2016 - 2:00pm
Where: Math 0103
Speaker: Konstantin Khanin (University of Toronto) - https://www.math.toronto.edu/cms/khanin-konstantin/
Abstract: TBA

N^{1/3} fluctuations: how to see them in the Ising model?

When: Thu, March 31, 2016 - 2:10pm
Where: Math 1311
Speaker: Senya Shlosman (Aix Marseille Universite, France) - http://www.cpt.univ-mrs.fr/~shlosman/shlosman.htm
Abstract: I will talk about the Ising model -- the drosophila of the rigorous statistical physics. It turns out that some new phenomena which appear in modern mathematical physics, can be observed in the Ising model as well.
One example which I will focus on is the size of typical fluctuations of the extended systems. If the size of the system is N, then the usual (gaussian) fluctuations are of the order of N^{1/2}. While in the random matrix theory one sees the fluctuations of the order N^{1/3}. I will explain that one can see them in the Ising model as well -- one just needs to know where to look.
Namely, the level lines of the Ising droplet near its edge have fluctuations of the desired order. When scaled by N^{1/3}, their limiting behavior for large N is given by the Airy diffusion process.
Joint work with D. Ioffe and Y. Velenik.

Regional proximality and the Veech relation in minimal flows

When: Thu, April 7, 2016 - 2:10pm
Where: Math 1311
Speaker: Joe Auslander (University of Maryland) -

Measures maximizing the entropy for smooth surface diffeomorphisms

When: Thu, April 14, 2016 - 2:10pm
Where: Math1311
Speaker: Jerome Buzzi (Universite Paris-Sud) - http://www.math.u-psud.fr/~buzzi/
Abstract: In a joint work with Sylvain Crovisier and Omri Sarig, we extend classical results about the spectral decomposition of uniformly hyperbolic diffeomorphisms and the coding of their basic pieces to surface diffeomorphisms with positive entropy. We deduce the finite multiplicity of their measures maximizing the entropy in the C^infinity case. The proof uses Sarig's symbolic extensions, Yomdin's theory, dimensions of dynamical foliations and a version of Sard's lemma

An abstract KAM result

When: Tue, April 19, 2016 - 2:30pm
Where: Math 0103
Speaker: Livia Corsi (McMaster University) - http://ms.mcmaster.ca/~lcorsi/
Abstract: I'll discuss an abstract KAM result on the existence of invariant tori for
possibly infinite dimensional dynamical systems.
Differently from the classical Moser's approach, I'll show that in principle
there is no need to impose the second Mel'nikov conditions but only to
invert (in some appropriate norm) the linearized operator in the normal
directions: in particular this means that the serious technical difficulties
in small divisors problems are those appearing in forced cases.
The latter statement is commonly believed to be true: the main purpose
is indeed to prove it under the weakest possible assumptions.
The result is obtained in collaboration with R. Feola and M. Procesi.

Expanding Thurston maps and weak expansion properties

When: Thu, April 21, 2016 - 2:10pm
Where: Math 1311
Speaker: Zhiqiang Li (Stony Brook University) - http://www.math.stonybrook.edu/~lizq/
Abstract: Thurston maps are a class of branched covering maps on the 2-sphere that arose in W. Thurston's characterization of postcritically finite rational maps. By imposing a natural expansion condition, M. Bonk and D. Meyer investigated a subclass of Thurston maps known as expanding Thurston maps, which turned out to enjoy nice topological, metric, and dynamical properties.
In this talk we will first introduce expanding Thurston maps with a brief account of history and motivations. We then discuss some weak expansion properties of such maps. More precisely, we will sketch a proof to show that an expanding Thurston map is asymptotically $h$-expansive if and only if it has no periodic critical points, and moreover, no expanding Thurston map is $h$-expansive.
If time permits, we will discuss some applications to the distribution of periodic points with respect to some natural invariant measures, and to a question of K. Pilgrim.

New Developments in the Hilbert-Smith Conjecture

When: Thu, April 28, 2016 - 2:10pm
Where: Math 1311
Speaker: James Keesling (University of Florida) - http://people.clas.ufl.edu/kees/
Abstract: The Hilbert-Smith Conjecture says that if a compact group $G$ acts effectively on a compact (or connected) manifold, then $G$ is a Lie group. The conjecture is true for dimensions $n = 1, 2, 3$ and is unknown for higher dimensions.

This conjecture has a long history. In one way of thinking, it is what remains of Hilbert's Fifth Problem which at this point has a century of dramatic and exciting results. Most of what we know about the structure of locally compact topological groups is a result of work on Hilbert's Fifth Problem.

This talk will cover some of the history of the Hilbert-Smith Conjecture. The talk will also cover several new approaches to this problem that we have developed which show some promise. These new approaches have also led to contributions to other areas of mathematics as well.

Shrinking targets and inhomogeneous approximation

When: Thu, May 5, 2016 - 2:10pm
Where: Math 1311
Speaker: Dmitry Kleinbock (Brandeis University) - http://people.brandeis.edu/~kleinboc/
Abstract: This is a tale of two problems in metric Diophantine approximation: one easy to state and seemingly hard to solve, and another (the inhomogeneous version of the first one) a bit more involved but with an elegant solution coming from dynamics on the space of lattices. The topic is Dirichlet's Theorem and its improvability, and our approach uses exponential mixing of a certain homogeneous flow. Joint work with Nick Wadleigh.