Dynamics Archives for Fall 2018 to Spring 2019

C*-algebras, topological dynamics, and noncommutative dimensions

When: Thu, August 30, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jianchao Wu (Penn State University) - http://www.personal.psu.edu/jxw710/
Abstract: C*-algebras are a kind of operator algebras tailored to describe noncommutative (i.e., quantum) topological spaces via functional analytical means. A major source of examples throughout the history of C*-algebra theory lies in the construction of crossed products from topological dynamical systems. This bridge between operator algebras and dynamics, valid also in the measure-theoretical setting, has proven immensely fruitful. On the other hand, the dimension theory of C*-algebras, which studies analogs of classical dimensions for topological spaces, is young but has been gaining momentum lately thanks to the pivotal role played by the notion of finite nuclear dimension in the classification program of simple separable nuclear C*-algebras. The confluence of these two themes leads to the question: What type of topological dynamical systems give rise to crossed product C*-algebras with finite nuclear dimension? I will present some recent work on this problem.

Homogeneous Structures from Topological Flows and Applications in Dynamics

When: Thu, September 6, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Kurt Vinhage (University of Chicago) - http://home.uchicago.edu/~kvinhage/
Abstract: A standard way of building a homogeneous structure on a smooth manifold X is to find a family of finite-dimensional subspace of vector fields which span the tangent space of X at every point and are closed under taking Lie brackets. We will describe a more topological approach to building a homogeneous structure, which has applications to smooth dynamical systems which have associated structures which are not a priori smooth.
In the first hour, I will discuss basic definitions and sketch an application of these tools to local rigidity of abelian group actions. In the second hour, I will discuss the details of the proof and possible future directions. Joint with Zhenqi Wang.

Regular and semi regular minimal flows

When: Thu, September 13, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Joe Auslander (UMD) -
Abstract: Let (X,T) be a minimal flow and let x,y in X. An obvious necessary condition for there to be an automorphism f of (X,T) with f(x)=y is that (x,y) be an almost periodic point of the product flow The flow (X,T) is said to be regular if this is always the case. Regular minimal flows are the minimal left ideals of the enveloping semigroup of a flow.

A further necessary condition for the existence of an automorphism is given in terms of the automorphism group of the universal minimal flow, namely that if (m,n) projects to (x,y) and g(m)=n, then g must be in the normalizer of the Ellis group of (X,T). When this always occurs (X,T) is said to be semi regular. Every minimal flow has a semi regular one in the same proximal class.

Dynamics of slow-fast Hamiltonian Systems

When: Thu, September 20, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Thibaut Castan (UMD) -
Abstract: The presence of gaps in the distribution of asteroids in the asteroid belt between Mars and Jupiter is far from being well understood. A common physical explanation is the presence of resonance in a system Sun-Jupiter-Asteroid, however this phenomenon is not fully understood yet. In 1983, this problem was restated as the study of a slow-fast Hamiltonian system, and allowed to understand the chaotic motion observed in simulations at the 3:1 resonance.
More precisely, when the fast system has a hyperbolic fixed point, the motion for initial condition close to the separatrices has yet to be understood. When close to the separatrix, the usual averaging of the fast system does not hold anymore, leading to some chaotic behavior for the slow system.
We will study how to understand the seemingly chaotic behavior of this Hamiltonian system.

Stochastic diffusive behavior at Kirkwood gaps

When: Thu, September 27, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Vadim Kaloshin (UMD) - https://www.math.umd.edu/~vkaloshi/
Abstract: One of well known indications of instability in the Solar system is presence of Kirkwood gaps in the Asteroid belt. They are caused by interaction of Asteroids with Jupiter. We study the restricted Sun-Jupiter-Asteroid problem. One mechanism of creation for the 3:1 Kirkwood gap was discovered by Wisdom and indepently by Neishtadt. We propose another mechanism, based on apriori chaotic underlying dynamical structure, and exhibit chaotic behavior. As an indication of chaos we show that eccentricity behaves like a stochastic diffusion process. More exactly, we establish probability measures such that the distributions of the projection onto eccentricity under the push forward in a proper scale weakly converges to a stochastic diffusion process on the line. This is a joint work with M. Guardia, P. Martin, and P. Roldan.

Cohomological equations and Ruelle resonances

When: Thu, October 11, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Giovanni Forni (UMD) -
Abstract: We summarize results on the cohomological equation for translation flows on translation surfaces (myself, Marmi, Moussa and Yoccoz, Marmi and Yoccoz) and apply these results to the asymptotic of correlations for pseudo-Anosov maps, which were recently obtained by a direct method by Faure, Gouezel and Lanneau. In this vein, we consider the generalization of this asymptotic to generic Teichmueller orbits (pseudo-Anosov maps correspond to periodic Teichmueller orbits) and to (partially hyperbolic) automorphisms of Heisenberg nilmanifolds (from results on the cohomological equation due to L. Flaminio and myself).

Area-preserving maps of surfaces: complex and PL versions

When: Thu, October 18, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Simion Filip (Institute for Advanced Study and Harvard) - http://www.math.ias.edu/~sfilip/
Abstract: I will start by introducing K3 surfaces, a class of (compact, algebraic) surfaces with automorphisms which preserve a natural area form. Cantat & Dupont showed that if the volume form is also the measure of maximal entropy, then the examples must come from linear maps of tori. I will explain a different and much shorter proof of this result, joint with Valentino Tosatti, using Ricci-flat metrics. After that, I will discuss what happens in a 1-parameter family of such automorphisms, as the surface degenerates. One gets PL maps of real 2-dimensional spheres which inherit many of the structures available in complex geometry.

Preperiodic points of polynomials

When: Tue, October 23, 2018 - 2:00pm
Where: MTH 0104
Speaker: Laura DeMarco (Northwestern) - http://www.math.northwestern.edu/~demarco/
Abstract: I will discuss results about the geometry of the periodic (or pre-periodic) points for rational maps on P^1. In new work with Holly Krieger and Hexi Ye, we give a uniform bound on the number of shared preperiodic points for two distinct complex polynomials of the form f(z) = z^2 + c. The question was inspired by analogous results/questions in arithmetic geometry (and the topic of my Number Theory seminar), and its proof involves p-adic analysis and heights -- combined with features of the Julia sets of the quadratic polynomials.

Simple Low-dimensional Paradigms for higher-dimensional chaos

When: Thu, October 25, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: James Yorke (UMD) - http://www.chaos.umd.edu/~yorke/
Abstract: The most frequently studied dynamical systems are low dimensional and all the periodic orbits in a chaotic set have the same number of unstable dimensions, but this property seems to fail in high dimensional systems.

In this talk I continue this theme. Here we define a property called ``hetero-chaos'', in which, along with the usual properties of chaos, there is a dense set of k-dimensionally unstable periodic orbits, and this holds for more than one k. We provide examples including a piecewise linear generalized baker map. I will explain how to prove ergodicity for our baker-like skew-product maps in 3D. This talk summarizes joint work with Shuddho Das, Yoshi Saiki, Miguel Sanjuan, and Hiroki Takahashi.

Multiscale substitution schemes and Kakutani sequences of partitions.

When: Tue, October 30, 2018 - 2:00pm
Where: MTH 0104
Speaker: Yotam Smilansky (Hebrew University of Jerusalem) -
Abstract: Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.

Zimmer's conjecture for co-compact lattices in simple complex Lie groups

When: Thu, November 1, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Zhiyuan Zhang (Institute of Advanced Study, Princeton) - https://sites.google.com/site/homepageofzhiyuanzhang/
Abstract: We will discuss Zimmer's conjecture on the actions of lattices in higher rank Lie groups on manifolds of small dimension. This conjecture is recently proved by Aaron Brown, David Fisher and Sebastian Hurtado for split simple real Lie groups. Then we discuss a recent generalisation of their result to simple complex Lie groups. This is a joint work with Jinpeng An and Aaron Brown.

Dynamics on abelian covers

When: Thu, November 8, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Wenyu Pan (Penn State) - https://wenyupanblog.wordpress.com
Abstract: In the first part, I will present a measure rigidity result on abelian covers. A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. I will present a joining classification result of horocycle flows on hyperbolic surfaces of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface. In the second part, I will talk further about ergodic properties of the geodesic flow/ frame flow on a general abelian cover. In particular, I will present the local mixing property of the geodesic flow/ frame flow, which is introduced to substitute the well-known strong mixing property in infinite volume setting. Part of the talk is based on the joint work with Hee Oh.

Diffraction from mathematical quasicrystals

When: Thu, November 15, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Alan Haynes (University of Houston) - https://www.math.uh.edu/~haynes/
Abstract: In the first part of this talk we will discuss the mathematics behind diffraction and explain some basic examples, such as diffraction from lattices. Next we will introduce cut and project sets, which are dynamically defined models for physical materials called quasicrystals. We will explain the classical approach to calculating the diffraction patterns seen from these objects, using the Poisson summation formula. In the second part of the talk, which is joint work with Michael Baake, we will take a more practical point of view and attempt to quantify how much the diffraction patterns observed from finite patches of cut and project sets deviate from the infinite models. Our methods, which are explicit and geared towards numerical computation, demonstrate the importance of Diophantine approximation to accurately determining complex phases and amplitudes of diffraction patterns produced by finite patches.

Coding geodesic flows and various continued fractions

When: Tue, November 27, 2018 - 2:00pm
Where: Kirwan Hall 1308
Speaker: Claire Merriman (UIUC) - https://faculty.math.illinois.edu/~emerrim2/
Abstract: I will connect continued fractions with even or odd partial quotients to geodesic flows on modular surfaces. The connection between geodesics on the modular surface PSL(2,Z)\H and regular continued fractions was established by Series, and we extend this to the odd and grotesque continued fractions and even continued fractions. This is joint work with Florin Boca.

Dynamics on Clebsch's Diagonal Cubic Surface.

When: Thu, November 29, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: William Goldman (UMD) - https://www.math.umd.edu/~wmg/
Abstract: We describe dynamical systems arising from the classification of
locally homogeneous geometric structures on manifolds.
Their classification mimics the classification of Riemann surfaces
by the Riemann moduli space --- the quotient of Teichmueller space
by the properly discontinuous action of the mapping class group.

However, this action is misleading:
mapping class groups generally act chaotically on character varieties.
For fundamental examples, these varieties appear as affine cubics,
and we relate the projective geometry of cubic surfaces to dynamical
properties of the action.

Rigidity of joining for certain parabolic dynamical systems

When: Thu, February 7, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Changguang Dong (UMD) -
Abstract: We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma$ with $\Gamma$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded type Heisenberg nilflows. This is joint work with A. Kanigowski and D. Wei.

On the Moebius disjointness conjecture and a class of three-interval exchange maps

When: Thu, February 21, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Davit Karagulyan (UMD) -
Abstract: I will start by surveying on the Möbius disjointness conjecture of P.Sarnak, which was introduced in 2010 and has initiated many studies. Then, I will discuss a result of J. Bourgain, which establishes the conjecture for a class of interval exchange maps. I will present a result, where we estimate the measure of the parameter set of this class. As a consequence we show that it has positive, but not full Hausdorff dimension.

Templates and dimension games with Diophantine targets

When: Thu, March 7, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tushar Das (University of Wisconsin) - https://sites.google.com/a/uwlax.edu/tdas/research
Abstract: We describe our recent program (ongoing with Lior Fishman, David Simmons and Mariusz Urbański) to resolve certain conjectures and questions regarding systems of linear forms in metric Diophantine approximation. The reduction of various problems to questions about certain combinatorial objects that we call <<templates>> along with a variant of Schmidt's game allows us to answer some of these problems, while leaving plenty that remain open. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry; in the hope of inspiring/luring such minds to a deeper study of this predominantly unexplored yet incredibly verdant mathematical landscape.

Limit laws for almost Anosov flows.

When: Tue, March 12, 2019 - 2:00pm
Where: CHM 0128
Speaker: Henk Bruin (University of Vienna (Austria)) - https://www.mat.univie.ac.at/~bruin/
Abstract: Almost Anosov flows are Anosov flows where a (single) periodic
orbit is perturbed to become a neutral saddle.
In this joint work with Dalia Terhesiu and Mike Todd,
we derive stable laws/Gaussian & non-Gaussian laws as well as
particular results on thermodynamic formalism.
One focus of this talk is how to transform an almost Anosov map given
by an ODE into a suspension flow and retain/obtain the tail estimates
for an induced version.

On entropy of nonuniformly hyperbolic measures

When: Thu, March 14, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dominik Kwietniak (Jagiellonian University in Krakow (Poland)) - http://www2.im.uj.edu.pl/DominikKwietniak/
Abstract: There are two main constructions of nonuniformly hyperbolic measures: the first was introduced by Gorodetski et al. and the second was given by Bochi et al. We study the Kolmogorov-Sinai entropy of these measures.
Together with Martha Lacka, we show that measures defined by Gorodetski et al. always have zero entropy and are Kakutani equivalent to an ergodic group rotation (in other words, these measures are always loosely Kronecker).
In cooperation with Christian Bonatti and Lorenzo Diaz, we prove that in a robustly transitive partially hyperbolic setting there always exists an ergodic nonhyperbolic measure with full support and positive entropy. The novelty of this result is that we address all four conditions (robustness, ergodicity, positive entropy, and full support) together, while previous works dealt only with a subset of these conditions. For the proofs, we introduce and study a new tool: the Feldman-Katok
pseudometric fk-bar, which leads to a new notion of convergence for invariant measures.

Proximality of generalized B-free systems

When: Thu, April 4, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Aurelia Dymek (Nicolaus Copernicus University (Torun, Poland)) -
Abstract: For any subset of integers B by B-free numbers we call the set of all integers with no factor in B. A B-free system is the orbit closure of the characteristic function of B-free numbers under the left shift. In 2010 Sarnak proposed to study the square-free system, i.e. the B-free system where B is the set of all squares of primes. As he postulates this system is proximal.
In the joint paper with Kasjan, Kulaga-Przymus and Lemanczyk we showed that a B-free system is proximal if and only if B contains an infinite pairwise coprime subset.
The topic of my talk is the proximality of generalized B-free systems in the case of number fields and lattices. Our main results are the similar characterization of proximality in case of number fields and some lattices, which are generalization of multidimensional B-free systems studied by Cellarosi, Vinogradov, Baake and Huck. We will give an example that such theorem fails in case of general lattices.

A new criterion for the physical measures of partially hyperbolic systems

When: Tue, April 9, 2019 - 2:00pm
Where: CHM 0128
Speaker: Fan Yang (University of Oklahoma) -
Abstract: In dynamical systems, an invariant measure is called physical if its basin has positive volume. In this talk, we will provide a new criterion on the existence and finiteness of physical measures using partial entropy along the unstable foliation, and discuss the application of this result on $C^{1+\alpha}$ partially hyperbolic systems with one-dimensional center, and on certain $C^1$ diffeomorphisms. This is a joint work with Yongxia Hua and Jiagang Yang.

Prime number theorem for analytic skew products.

When: Thu, April 25, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Adam Kanigowski (UMD) -
Abstract: Let T(x,y)=(x+\alpha, y+g(x)) for \alpha irrational and g\in C^\omega(\T). We show that if T is uniquely ergodic, then for every (x,y)\in \T^2, \{T^p(x,y)\}_{p- prime} is equidsitributed. We will also recall other systems for which equidisitrubtion along primes holds for every point.

Positive Lyapunov exponents and quenched correlation decay for Lagrangian flow in fluid mechanics models.

When: Thu, May 2, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/
Abstract: Given a model of an incompressible fluid on a compact domain M, the Lagrangian flow $\phi^t$ is a volume-preserving flow of diffeomorphisms on M describing the motion of a passive particle (e.g., a dust particle) advected by the fluid. It is anticipated that when the fluid is subject to "stirring" that $\phi^t$ should be chaotic in the sense of a positive Lyapunov exponent and exponential correlation decay. I will present a recent joint work with Jacob Bedrossian (U Maryland) and Sam Punshon-Smith (Brown U) in which we rigorously verify these chaotic properties for various incompressible and stochastically forced fluid models on the periodic box, including stochastic 2D Navier-Stokes and stochastic hyperviscous 3D Navier-Stokes.