Dynamics Archives for Fall 2021 to Spring 2022

Counterexamples to a rigidity conjecture

When: Thu, September 2, 2021 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Giovanni Forni (University of Maryland) -
Abstract: We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, C0 conjugacy implies C1 (even C∞) conjugacy. We construct examples of non-rigid diffeomorphisms on the 2-torus, which satisfy the assumptions of Khanin's (but not of Krikorian's) conjecture. We also construct examples of flows which are topologically conjugate, but not C1 conjugate, in contradiction to a natural generalization of the conjecture to flows. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions.

Strongly mixing actions of countable abelian groups are almost strongly mixing of all orders

When: Thu, September 9, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Rigoberto Zelada (University of Maryland) -
Abstract: Let $(G,+)$ be a countable discrete abelian group and let $(T_g)_{g\in G}$ be a strongly mixing measure preserving
$G$-action on a probability space $(X,\mathcal A,\mu)$.
We will present a result which states that $(T_g)_{g\in G}$ is "almost strongly mixing of all orders".
It is worth noting that this result, when applied to $\mathbb Z$-actions, offers a new way of dealing with strongly mixing transformations.
In particular, it allows us to obtain several new characterizations of strong mixing for $\mathbb Z$-actions, including a result which can
be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his
proof of Szemer{\'e}di's theorem.\\
The proofs of these results rely on a new notion of largeness for subsets of $G^d$ and utilize $\mathcal R$-limits, a notion ofconvergence based on the classical Ramsey Theorem. This talk is based on joint work with Vitaly Bergelson.

Invariant Family of Leaf measures and The Ledrappier-Young Property for Hyperbolic Equilibrium States

When: Thu, September 23, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Snir Ben Ovadia (Penn State) -
Abstract: Let $M$ be a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and let $f$ be a $C^{1+\beta}$ ($\beta>0$) diffeomorphism of $M$. Let $\varphi$ be a H\"older continuous potential on $M$. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi$), which sit on local unstable leaves.
We present two main applications. First, given an ergodic homoclinic class $H_\chi(p)$, we prove that $\varphi$ admits a local equilibrium state on $H_\chi(p)$ if and only if $\varphi$ is ``recurrent on $H_\chi(p)$" (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitutes as its conditional measures.
Second, we prove a Ledrappier-Young property for hyperbolic equilibrium states- if $\varphi$ admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi$) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic SRB measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure 0) has conditional measures on local unstable leaves which are absolutely continuous w.r.t the Riemannian leaf volume.

Actions of Lattices in Higher-Rank Semisimple Groups

When: Thu, September 30, 2021 - 2:00pm
Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
Speaker: Darren Creutz (U.S. Naval Academy) -
Abstract: Let \Gamma be an irreducible lattice in a higher-rank semisimple group G. Generalizing Margulis' Normal Subgroup theorem in the connected case, Stuck and Zimmer (1994) showed that every nonatomic probability-preserving action of \Gamma in a connected higher-rank Lie group G is essentially free. The general case was proven by the speaker and J. Peterson (2017).
However, being (very) nonamenable, not every action of such a lattice on a compact metric space admits an invariant measure, leaving open the question of whether every minimal action of such a lattice on an infinite compact metric space is (topologically) free.

I will present the complete result: every action of such a lattice is indeed (topologically) free. The main idea is to replace invariant measure by stationary measure (which always exist) and show that every stationary action is essentially free. This answers the general form of a question of Glasner and Weiss on URS's for such lattices.


Dynamics on Character Varieties

When: Thu, October 7, 2021 - 2:00pm
Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
Speaker: William Goldman (University of Maryland) -
Abstract : Many interesting dynamical systems arise from the classification of
locally homogeneous geometric structures and flat connections on manifolds.
Their classification mimics that of Riemann surfaces by the Riemann moduli
space, which identifies as the quotient of Teichmueller space of marked Riemann surfaces
by the action of the mapping class group.However, unlike Riemann surfaces,
these actions are generally chaotic.
A striking elementary example is Baues's theorem that the deformation
space of complete affine structures on the 2-torus is the plane with
the usual linear action of GL(2,Z) (the mapping class group of the torus).
We discuss specific examples of these dynamics for some simple surfaces,
where the relative character varieties appear as cubic surfaces in affine 3-space.
Complicated dynamics seems to accompany complicated topology,
which we interpret them as (possibly singular) hyperbolic structures on surfaces

Nielsen realization problem

When: Thu, October 14, 2021 - 2:00pm
Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
Speaker: Lei Chen (University of Maryland) -
Abstract: Nielsen realization problem asks whether the projection Homeo(M)\to \pi_0(\Homeo(M)) has a section or not for a manifold M. This problem relates the existence of flat structure on the universal M-bundle. In this talk, I will focus on the case when M is a surface. I will talk about what other people have done in the past and then describe my results. The proof method involves homological obstruction, fixed point theory, surface dynamics and Poincare-Birkhoff Theorem. This is partly a joint work with Markovic.

Effective counting estimates for filling closed geodesics on hyperbolic surfaces

When: Thu, October 21, 2021 - 2:00pm
Where: Kirwan Hall 3206 and via Zoom https://umd.zoom.us/j/97401191391
Speaker: Francisco Arana-Herrera (Stanford University) -
Abstract: Counting problems for closed geodesics on hyperbolic surfaces have been extensively studied since the 1950s. I will discuss a new quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length at most L on an arbitrary closed, orientable hyperbolic surface. This estimate solves an open problem alluded to in work of Mirzakhani and advertised by Wright. The proof relies on recent developments on the theory of effective mapping class group dynamics.


Flexibility of the Pressure Function

When: Thu, November 4, 2021 - 2:00pm
Where: Kirwan Hall 3206 and via Zoom https://umd.zoom.us/j/97401191391
Speaker: Tamara Kucherenko (The City College of New York) -
Abstract: We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite. This is based on joint work with Anthony Quas.


Interactions between group operations and universal minimal flows

When: Thu, November 11, 2021 - 2:00pm
Where: Kirwan Hall 3206 and via Zoom https://umd.zoom.us/j/97401191391
Speaker: Dana Bartosova (University of Florida) -
Abstract: For a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X. A subflow is a closed G-invariant subset of X with the restricted action. A G-flow is minimal if it has no proper non-empty subflows. The most complicated minimal G-flow is called the universal minimal flow of G, M(G), and it is defined by the property that every minimal flow is its quotient. Ellis showed that M(G) exists for any topological group G and it is unique up to a G-isomorphism, however it is non-trivial to determine how exactly it looks like. In this talk, we will show that universal minimal flows behave well with respect to ``nice'' extensions of or by a compact group. For example, if G is a totally disconnected locally compact SIN group and K is any open compact normal subgroup of G, then M(G) is isomorphic to the product of K with M(G/K).


Rigidity in Rank-One Systems

When: Thu, November 18, 2021 - 2:00pm
Where: Kirwan Hall 3206 and via zoom https://umd.zoom.us/j/97401191391
Speaker: Kelly Yancey (Institute for Defense Analyses) -

Abstract: Rank-one transformations may be defined by cutting parameters and spacer parameters. For a given Rank-one transformation, Gao and Hill defined the canonical sequence of these parameters and showed that, when these sequences are bounded, the transformation has trivial centralizer. That means that when these sequences are bounded, the transformation cannot be rigid.
For rank-one systems we will discuss partial rigidity and bound the partial rigidity constant away from one for a subclass of transformations that are canonically bounded. We will also discuss rigidity in the unbounded case. This is joint work with Jon Fickenscher.


Conformal groups of compact Lorentzian manifolds

When: Thu, December 2, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Karin Melnick (University of Maryland) -
Abstract: The Lorentzian Lichnerowicz Conjecture is a Lorentzian analogue of the Ferrand-Obata Theorem on conformal transformation groups of Riemannian manifolds.  I will discuss my verification of the conjecture in dimension three, for real-analytic metrics, in recent joint work with C. Frances.  The introductory part of the talk will be similar to the one I gave in the Geometry-Topology seminar two weeks ago, but then I will focus on a different case in the proof.  This case is more dynamical in flavor; I can elaborate further on it in the second part of the talk.

Exponential multiple mixing for moduli spaces of Abelian differentials

When: Thu, December 9, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Fernando Camacho Cadena (Heidelberg Institute for Theoretical Studies (HITS)) -
Abstract: This talk will be about dynamical properties of the Teichmüller geodesic flow and the SL(2,R) action on moduli spaces of Abelian differentials. Back in 2006, Avila, Gouëzel and Yoccoz proved that the Teichmüller geodesic flow is exponentially mixing. Here we are interested in not just mixing, but exponentially multiple mixing. After introducing the moduli spaces and the group action on them, I will go through the idea of the proof for exponential multiple mixing. If time permits, I will also talk about an application to a central limit theorem.

Exceptional orbits on homogeneous spaces and their applications

When: Thu, January 27, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Dmitry Kleinbock (Brandeis University) - https://people.brandeis.edu/~kleinboc/
Abstract: Today's featured dynamical systems will be $(X,F)$, where $X$ is the set of unimodular lattices in $\mathbb{R}^d$and $F$ a diagonal subsemigroup of $SL_d(\mathbb{R})$. The goal is to construct points in $X$ with $F$-trajectories eventually avoiding some interesting (and quite mysterious) compact subsets of called critical loci. I will discuss the situations when this canbe done, in particular using the technique of Schmidt games. The results arejoint with Jinpeng An, Lifan Guan and Anurag Rao.

Exponential mixing

When: Thu, February 3, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Dima Dolgopyat (UMD) - https://www.math.umd.edu/~dolgop/
Abstract: Exponential mixing is one of the strongest statistical properties for smooth systems. In this talk I review recent progress and describe open problems related to exponential mixing. In the second part I describe the proof of the fact that exponentially mixing systems are Bernoulli proven in a recent joint work with Adam Kanigowski and Federico Rodriguez Hertz.

Prime number theorem for Anzai skew-products

When: Thu, February 10, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Maksym Radziwill (Cal Tech) - http://www.its.caltech.edu/~maksym/
Abstract: I will discuss recent work with Kanigowski and Lemanczyk in which
we establish the pointwise convergence of ergodic averages over primes for
uniquely ergodic analytic Anzai skew products. I will explain the motivation for
considering this problem and the method of proof.

On local rigidity of linear abelian actions on the torus

When: Thu, February 17, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Bassam Fayad (University of Maryland) -

Abstract: In which cases and ways can one perturb the action on the torus of a commuting pair of SL(n,Z) matrices? Two famous manifestations of local rigidity in this context are: 1) KAM-rigidity of simultaneously Diophantine torus translations (Moser) and 2) smooth rigidity of hyperbolic or partially hyperbolic higher rank actions (Damjanovic and Katok). To complete the study of local rigidity of affine abelian actions on the torus one needs to address the case of actions with parabolic generators. In this talk, I will review the two different mechanisms behind the rigidity phenomena in 1) and 2) above, and show how blending them with parabolic cohomological stability and polynomial growth allows to address the rigidity problem in the parabolic case. This is a joint work with Danijela Damjanovic and Maria Saprykina.

Floer homology and the smooth closing lemma for area-preserving maps of surfaces

When: Thu, February 24, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Daniel Cristofaro-Gardiner (UMD) -
Abstract: I will talk about some recent joint work showing that a generic smooth area-preserving diffeomorphism of a closed surface has a dense set of periodic points. In the C^1 topology, these kind of results were proved by Pugh and Robinson and are generally called “closing lemmas”; finding closing lemmas in higher regularity is the subject of Smale’s 10th problem. A Weyl law recovering the average rotation through the actions of periodic points plays a central role in the proof. Our work uses techniques from Floer homology; the first half of my talk will be a gentle introduction to Floer homology through the lens of Weinstein’s conjecture concerning the existence of periodic orbits for Reeb flows.

Trace: the final frontier

When: Thu, March 3, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Rodrigo Treviño (UMD) - trevino.cat
Abstract: Abstract: In this talk I'll talk about types of operators that go under the name of "discrete Schrodinger operators". In the first half I'll give an overview of their study, while in the second half I'll focus on the case where the underlying material has an aperiodic atomic structure. I'll sketch how equidistribution estimates on a certain system gives information on the spectrum (more precisely the integrated density of states) of these operators using some mysterious objects called traces. Some of this work is joint with S. Schmieding.

Measures of maximal u-entropy for maps that factor over Anosov

When: Thu, March 10, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Fan Yang (Michigan State University) -
Abstract: In this talk, we will discuss the existence, uniqueness and statistical properties of the measures of maximal u-entropy for diffeomorphisms that factor over an Anosov diffeomorphism on a torus. Furthermore, we will use those measures to construct transverse measures of the unstable foliation, and prove that a natural dynamical averaging converges exponentially fast to the transverse measures. This is a joint work with Marcelo Viana, Raul Ures and Jiagang Yang.

Dynamics on nilpotent character varieties

When: Thu, March 17, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Sean Lawton (George Mason) -
Abstract: Let N be a finitely generated nilpotent group and let G be a compact connected Lie group. Then Out(N) acts on the conjugation orbit space X(N,G)=Hom(N,G)/G by precomposition. In collaboration with J-P Burelle, we show that there exists a finite Out(N)-invariant measure with full support on X(N,G) and whenever Out(N) has at least one hyperbolic element, the action of Out(N) on the identity component of X(N,G) is strong mixing with respect to this measure (arXiv:2111.11922). In the first hour, I will illustrate this theorem with a concrete example, and in the second hour I will sketch the proof of this theorem and discuss related topics.

A Riemannian metric on hyperbolic components of moduli space of rational maps

When: Thu, April 7, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Yan Mary He (University of Oklahoma) -
Abstract: In this talk, we introduce a Riemannian metric on certain hyperbolic components of the moduli space of rational maps, which is conformal equivalent to the pressure metric. As an application, we show that the Hausdorff dimension function has no local maximum on any hyperbolic component. Along the way, we introduce multiplier functions for invariant probability measures on Julia sets, which is a key ingredient in the construction of our metric. This is joint work with Hongming Nie.

Physical and u-Gibbs measures for partially hyperbolic skew products

When: Thu, April 14, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Davi Obata (University of Chicago) -

For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist.

It is expected that a "typical" system with enough hyperbolicity (such as partial hyperbolicity) should have such measures.

In this talk, we will see a new example of open sets of partially hyperbolic systems with two dimensional center having a unique physical measure. One of the key features for these examples is a rigidity result for the so-called u-Gibbs measures, which allows us to conclude the existence of physical measures.

Weyl laws in contact and symplectic geometry

When: Thu, April 21, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Dan Cristofaro-Gardiner (UMD) - https://dancg.sites.ucsc.edu
Abstract: My talk will be about some asymptotic formulas that recover some "classical" quantities such as the volume of symplectic four-manifolds from the lengths of certain sets of periodic orbits of canonical vector fields. The relevant periodic orbits are selected via Floer homology, and I will explain a bit about how this works; no previous knowledge of Floer homology will be assumed. I will also discuss some applications of these Weyl laws in two seemingly different contexts: a) counting periodic orbits of Reeb vector fields; b) better understanding the structure of homeomorphism groups of surfaces.

Speed of mixing of Anosov diffeomorphisms from their action on cohomology

When: Thu, April 28, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Daniele Galli (Bologna)
Abstract: My talk will be focused on some techniques to investigate speed of mixing and theRuelle-Pollicott asymptotic for the correlation functions forAnosov diffeomorphisms. I will show that some information about these issues can be obtained studying the action of the dynamics on DeRham cohomology, both in linear and nonlinear cases.

The Structure of the Spectrum of a Dynamically Defined Schrodinger Operator

When: Thu, May 5, 2022 - 2:00pm
Where: Kirwan Hall 3206
Speaker: David Damanik (Rice) -
Abstract: We consider Schr"odinger operators whose potentials are defined by sampling the orbits of a homeomorphism of a compact metric space with a continuous function. Motivated by the phenomenon of spectral pseudo-randomness we discuss mechanisms that allow one to show that the gap structure of such a spectrum is very simple under suitable assumptions.