Dynamics Archives for Fall 2021 to Spring 2022
Exponential mixing of 3D Anosov flows
When: Thu, September 3, 2020 - 2:00pmWhere: Zoom
Speaker: Zhiyuan Zhang (CNRS, Universite Paris 13) - https://sites.google.com/site/homepageofzhiyuanzhang/home
Abstract: We show that a topologically mixing $C^\infty$ Anosov flow on a 3-dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.
Spectral gaps for a class of random products of bounded linear operators
When: Thu, September 17, 2020 - 2:00pmWhere: Zoom
Speaker: Joseph Horan (Department of Mathematics and Statistics, University of Victoria) -
Abstract: For a primitive Markov chain (represented by a non-negative matrix where a power has all positive entries), we may use the classical Perron-Frobenius theorem to conclude that the Markov chain has a unique stationary distribution and all initial states converge to that distribution with some exponential decay rate determined by the largest and second largest eigenvalues of the matrix. What happens when we allow random products (cocycles) of matrices, or of bounded linear operators? In the first part of this talk, we set up and present a new cocycle Perron-Frobenius theorem, and give a couple of interesting applications at a high level, including the situation of a cocycle of ”paired tent maps”. In the second part of this talk (after the break), we give an idea of the proof of the theorem and elaborate on the paired tent maps example.
The video for the talk can be found here: https://umd.zoom.us/rec/share/eo4LSKlEdHSIaYrvB8ZkIz3hp-hsusuIxxMGOYwS7Nd80qdhDH-pvGsBV9-mR_-z.9_MONl1MX3enVk9k
Effective equidistribution of horospherical flows in infinite volume
When: Thu, September 24, 2020 - 2:00pmWhere: Zoom
Speaker: Nattalie Tamam (UC San Diego) - https://www.math.ucsd.edu/~natamam/
Abstract: We want to provide effective information about averages of orbits of the horospherical subgroup acting on a hyperbolic manifold of infinite volume. We start by presenting the setting and results for manifolds with finite volume. Then, discuss the difficulties that arise when studying the infinite volume setting, and the measures that play a crucial role in it. This is joint work with Jacqueline Warren.
The video for the talk can be found here: https://umd.zoom.us/rec/share/XZao71i2C4E_HIcJXPZdHdl4_FUaJZCAkMCo87E6SaKRA_wbjM6w-t5Sf0hVxAze.T2Rc998HpDJsUqOZ
Stabilizers in group Cantor actions and measures
When: Thu, October 1, 2020 - 2:00pmWhere: Zoom
Speaker: Olga Lukina (University of Vienna) - https://sites.google.com/view/olgalukina
Abstract: Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has full measure. On the other hand, there are many examples of group actions, where every point has a non-trivial stabilizer. In this talk, we introduce an analog of the notion of an essentially free action for such highly non-free actions, using the concept of holonomy. For equicontinuous actions of countable groups on Cantor sets, we answer the following question: under what conditions there exists a subgroup H of G, such that the stabilizers of almost all points in X are conjugate to H? These conditions are that the action is locally quasi-analytic and locally non-degenerate. We discuss applications of this work to the study of the properties of the invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Groeger.
The video for the talk can be found here: https://umd.zoom.us/rec/share/HnhA7kV6wAR8qJHb36ohsQbia_LQKZJZWTG3jgekMuletS-5zMkI3NszXaBvbwHA.IIfOLN8Itmqo6JFb
Classification and statistics of cut-and-project sets
When: Thu, October 8, 2020 - 2:00pmWhere: Online
Speaker: Barak Weiss (Tel Aviv University) - http://www.math.tau.ac.il/~barakw/
Abstract: Cut and project sets are a well-studied model of aperiodic order, with connections to diverse topics in crystallography, mathematical physics, ergodic theory, and geometry of discrete sets.We introduce a class of so-called Ratner-Marklof-Strombergsson measures. These are probability measures supported on parameter spaces of cut-and-project sets in $R^d$ which are invariant and ergodic for the action of the groups $ASL(d,R)$ or $SL(d,R)$. We classify the measures that can arise in terms of data of algebraic groups. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. With this in hand we derive results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.
Joint work with Rene Ruehr and Yotam Smilansky.
The video for the talk can be found here: https://umd.zoom.us/rec/share/7Xp1Vaxv8BqDSNBa9ohAV_to41vq-ojww4o02CXSwKQzWQMLd4ub6aZl3Kwu_X63.uQ3uImM3h9amliA3
Thermodynamics of smooth models of pseudo-Anosov homeomorphisms
When: Thu, October 15, 2020 - 2:00pmWhere: Online
Speaker: Dominic Veconi (Penn State) - http://dominic.veconi.com/
Abstract: We discuss the thermodynamic and ergodic properties of a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the pseudo-Anosov map is uniformly hyperbolic outside of a neighborhood of a set of singularities, and the trajectories are slowed down so the differential is the identity at the singularities. Using Young towers, we prove existence and uniqueness of equilibrium measures for geometric $t$-potentials. This family of equilibrium measures includes a unique smooth SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.
The video for the talk can be found here: https://umd.zoom.us/rec/share/Hg5XzKKFbyIntuH_z2V_UWvqPsICaTBJRS-xqOoT-YNDu4pwC5oMwpOQTiWtrS6N.qYoDBloWbmYkllor
Large intersection classes for pointwise emergence
When: Thu, October 22, 2020 - 2:00pmWhere: Online
Speaker: Agnieszka Zelerowicz (UMD) -
Abstract: Recently, P. Berger introduced a concept of metric emergence to quantitatively study the non-uniqueness of statistics (such as Newhouse or KAM phenomena). Later, S. Kiriki, Y. Nakano, and T. Soma introduced a concept of pointwise emergence to quantitatively study non-existence of averages, and constructed a residual subset of the full shift with high pointwise emergence. In this talk I will consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. I will present my joint work with Yushi Nakano, where we show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property.
The video for the talk can be found here: https://umd.zoom.us/rec/share/l_9KxEJQyjaY37_KKuatfQbn4rFyJXq89TjtBOvEkiBm8tujkQntI5GJrLshPwfU.VP31-rdQqsBurokM
Quasi-periodic attractors for dissipative systems
When: Thu, October 29, 2020 - 2:00pmWhere: Online
Speaker: Alessandra Celletti (University of Rome Tor Vergata) - http://www.mat.uniroma2.it/celletti/
Abstract: We consider the existence of invariant attractors for the specific case of dissipative systems known as conformally symplectic systems, which are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter. We provide a KAM theorem in an a-posteriori format: assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby. The theorem does not assume that the system is close to integrable.
This method can be also used to get different results: (i) a breakdown criterion for invariant attractors; (ii) an efficient algorithm to generate the solution, which can be implemented successfully in model problems; (iii) the existence of whiskered tori for conformally symplectic systems, (iv) the analyticity domains of the quasi-periodic attractors in
the symplectic limit.
The content of this talk refers to works in collaboration with R. Calleja and R. de la Llave.
The recorded video for the talk can be found here: https://umd.zoom.us/rec/share/kKQ2GSxJLFOt5X9caTpAlimdg8NFpkeDxUhbwg3_LhSar0aOLahkXcCniX4fNx02.bQDTym7jft59hb-u
Multiscale substitution tilings
When: Thu, November 5, 2020 - 2:00pmWhere: Zoom
Speaker: Yotam Smilansky (Rutgers University) - https://sites.math.rutgers.edu/~ys755/
Abstract: Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.
Entropy rate of product of independent processes
When: Thu, November 12, 2020 - 2:00pmWhere: Online
Speaker: Joanna Kulaga-Przymus (Nicolaus Copernicus University in Toruń) - https://www-users.mat.umk.pl/~joasiak/
Abstract:
Let $\mb{X}$ and $\mb{Y}$ be two-sided stationary processes taking values in $\{0,1\}$. Let $\mb{M}$ be their coordinatewise product: $M_i=X_i \cdot Y_i$. During my talk I will discuss the entropy rate of $\mb{M}$.
The motivation is twofold. Furstenberg in his seminal paper asked when $\mb{X}$ can be recovered from $\mb{X}+\mb{Y}$ and showed that this is the case whenever the corresponding dynamical systems are disjoint. I will discuss an analogous problem for $\mb{X}+\mb{Y}$. As we admit zero as a value, we cannot use logarithm to reduce our problem to Furstenberg's setting. We show that if $\mb{Y}$ is of zero entropy rate than for a large class of positive entropy rate processes $\mb{X}$, the entropy rate of $\mb{M}$ is strictly lower than that of $\mb{X}$ and thus $\mb{X}$ cannot be recovered from $\mb{M}$.
Another reason we are interested in the entropy rate of $\bm{M}$ comes from dynamics. More precisely, Mariusz Lemanczyk, Benjamin Weiss and me left some open problems related to invariant measures for $\mathscr{B}$-free systems. The most prominent example of such a system is the subshift generated by the square of the Moebius function.
During my talk I will show how these two settings are related to each other and I will discuss the following questions:
a) is there a general formula for the entropy rate of $\mb{M}$?
b) when is $\mb{M}$ of positive entropy?
c) what is the relation between the entropy rates of $\mb{M}$ and $\mb{X}$?
The talk is based on joint work with Michał Lemańczyk https://arxiv.org/pdf/2004.07648.pdf
Linear repetitive Delone sets beyond Abelian groups
When: Thu, November 19, 2020 - 2:00pmWhere: Zoom
Speaker: Felix Pogorzelski (Universität Leipzig) - http://www.math.uni-leipzig.de/~pogorzelski/
Abstract: There is no rigorous mathematical definition of a quasicrystal. In spaces with some group translation the latter term usually refers to well-scattered point sets (Delone sets) that are not periodic but display long range symmetries. Classes of these sets such as model sets have already been studied by Meyer in the 70's, i.e. some time before Shechtman's discovery of physical alloys with non-periodic molecular structure in 1982 (Nobel prize for chemistry in 2011). In this talk we focus on non-periodic point sets in (possibly non-Abelian) lcsc groups that are not too far from crystals in a dynamical sense. The main focus will be on the generalization of the concept of linearly repetitive Delone sets known from Euclidean space. Although they are not found easily, non-periodic, linearly repetitive Delone sets exist in many non-Abelian groups as well. Going beyond a result from Lagarias and Pleasants for R^d, we explain how to prove unique ergodicity for the associated dynamical systems for a class of Lie groups of polynomial volume growth. Joint work with Siegfried Beckus and Tobias Hartnick.
On the dimension drop conjecture for diagonal flows on the space of lattices
When: Thu, December 3, 2020 - 2:00pmWhere: Online
Speaker: Shahriar Mirzadeh (Michigan State) - Abstract:
Consider the set of points in a homogeneous space $X = G/\Gamma$ whose ${g_t}$-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when has real rank 1. \\
In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: $G=\SL_{m+n}(\mathbb{R})$, $\Gamma= \SL_{m+n}(\mathbb{Z})$, and $g_t=\diag (e^{t/m}, \cdots, e^{t/m},e^{-t/n}, \cdots, e^{-t/n})$. We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.
A recording of the talk can be found here: https://umd.zoom.us/rec/share/u4hHGBJh2LFW5lHcugurXLyAEusjgaud9oSfdBuL3zuvbn83AJBNnv6AOFxdmqRC.ZgxclQqiTb-MQ6jb
The nature of equations and the equations of nature
When: Thu, February 4, 2021 - 2:00pmWhere: Probably online
Speaker: Jim Yorke (UMD) - http://yorke.umd.edu
Abstract: Systems of N equations in N unknowns are ubiquitous in mathematical modeling. Thesesystems, often nonlinear, are used to identify equilibria of dynamical systemsin ecology, genomics, control, and many other areas. Structured systems, wherethe variables that are allowed to appear in each equation are pre-specified,are especially common. For modeling purposes, there is a great interest indetermining circumstances under which physical solutions exist, even if thecoefficients in the model equations are only approximately known.
Thestructure of a system of equations can be described by a directed graph G that reflects the dependence of one variable onanother, and we can consider the family F(G) of systems that respect G.
We define a solution X of F(X)=0 to be robust if for each continuous F∗ sufficiently close to F, a solution X∗ exists. Robust solutions are those that are expected tobe found in real systems. There is a useful concept in graph theory called"cycle-coverable". We show that if G is cycle-coverable, then for "almost every" F∈F(G) in the sense of prevalence, every solutionis robust. Conversely, when G fails to be cycle-coverable, each system F∈F(G) has no robust solutions.
Failureto be cycle-coverable happens precisely when there is a configuration of nodesthat we call a "bottleneck," a criterion that can be verified fromthe graph. A "bottleneck" is a direct extension of what ecologistscall the Competitive Exclusion Principle, but we apply this idea from theequations of nature to describe the nature of almost all structured systems.Sana Jahedi, Timothy Sauer, James A. YorkePosted on arXiv Jan 2021 A recording of the talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7a19d4c3-c70b-4e34-bbd3-acc5015366ab
Dimension spectra and thermodynamic expansions of conformal fractal limit sets
When: Thu, February 11, 2021 - 1:50pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Tushar Das (Department of Mathematics & Statistics, University of Wisconsin-La Crosse) - https://www.uwlax.edu/profile/tdas/
Abstract:
Studying the fine geometric-measure-theoretic properties of dynamical limit sets is often an endeavor beset with myriad challenges. In this vein we focus on the dimension-theoretic study of continued fraction Cantor sets, a rich seam inaugurated by the work of Besicovitch and Jarník in the 1920s. I will report on two projects about such elementary yet still fascinating fractals. The first considers small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original; and establishes a formula that may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. The second resolves two recent questions posed by Chousionis, Leykekhman, and Urbański regarding the dimension spectrum of a CIFS (i.e. the set of all Hausdorff dimensions of its various subsystem limit sets) and provokes fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra. There are plenty of interesting directions left to explore in the wake of our results, and we hope that a non-trivial set of Maryland mathematicians will be inspired to develop further ramifications of our techniques!
A recording of the talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=0b6bf19a-dae5-4c5b-be91-accc01504732
The slides for the talk can be found here: https://umd.box.com/s/a0dh2ziz9ia08l66ljsy4a12nmz19frm
The classification problem for diffeomorphisms of manifolds
When: Thu, February 18, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Matthew Foreman (UC Irvine) -
Abstract:
First part: Mathematics is uniquely capable of producing impossibility results. The most famous include the impossibility of proving the parallel postulate, solving the quintic polynomial or the word problem for finitely generated groups. To show such a result you need rules for what constitutes a solution. In the case of the word problem for finitely generated groups, you must use \emph{inherently finite} (recursive) techniques.
Descriptive Set theory gives techniques for stronger impossibility results: \emph{it is impossible to do $x$ using inherently countable techniques}. The context of these results are complete separable metric spaces and the notion of inherently countable is \emph{Borel}.
The focus will be on equivalence relations and Borel reducibility. The latter gives a natural hierarchy of complexity. The main talk will use this hierarchy in the context of \emph{measure equivalence} and \emph{topological equivalence} of diffeomorphisms of manifolds.
Second part: Descriptive Set Theory gives rigorous tools to show \emph{impossibility results}. This talk shows that it is impossible to classify diffeomorphisms of smooth manifolds up to either topological or measure equivalence. It also exhibits an explicit diffeomorphism that is isomorphic to its inverse if and only if the Riemann Hypothesis holds, as well as a different diffeomorphism for which the question of $T\cong T^{-1}$ is independent of the the axioms for mathematics (ZFC). (The talk include joint work with B. Weiss and A. Gorodetski.)
The slides for the first part can be found here: https://umd.box.com/s/s7nbbgu1ibo61yydywxzfnug4fuibaqf
The slides for the second half can be found here: https://umd.box.com/s/kydu87djrrbe0dm8gpkwpxoggs1zn7h0
Entropy, orbit equivalence, and sparse connectivity
When: Thu, February 25, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: David Kerr (University of Munster) - https://www.math.tamu.edu/~kerr/
Abstract: It was shown by Tim Austin that if an orbit equivalence between probability-measure-preserving actions of finitely generated amenable groups is integrable then it preserves entropy. I will discuss some joint work with Hanfeng Li in which we show that the same conclusion holds for the maximal sofic entropy when the acting groups are countable and sofic and contain an amenable w-normal subgroup which is not locally virtually cyclic, and that it is moreover enough to assume that the Shannon entropy of the cocycle partitions is finite (what we call Shannon orbit equivalence). It follows that two Bernoulli actions of a group in the above class are Shannon orbit equivalent if and only if they are conjugate. I will also describe a topological version of our measure entropy invariance result, along with an application to the construction of simple C*-simple groups whose von Neumann algebras have property Gamma.
A recording of this talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=8a640a1a-dd3f-418d-b71c-acda0161b51c
Quantitative weak mixing for interval exchange transformations
When: Thu, March 4, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Pedram Safaee (Zurich) -
Abstract: An interval exchange transformation (IET) is an orientation preserving piecewise isometry of the interval $[0,1]$. These transformations are low complexity systems that exhibit interesting spectral properties; They are never mixing, typically uniquely ergodic, typically rigid, and typically weakly mixing. Weak mixing is equivalent to having the Cesàro averages of correlations tend to zero. In this talk, we will focus on the decay rate of the Cesàro averages of correlations for sufficiently regular observables for typical IETs. We show that a (rather unexpected) dichotomy holds for this decay depending on whether the IET is of rotation class or not. In the former case, we provide logarithmic lower and upper bounds for the decay of Cesàro averages whereas we provide polynomial upper bounds in the latter case. This is joint work with Artur Avila and Giovanni Forni.
A recording of the talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=448ca9bd-5568-4451-a205-ace1016248e7
The flow group of rooted abelian or quadratic differentials
When: Thu, March 11, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Rodolfo Gutierrez-Romo (CMM - Universidad de Chile) - http://dim.uchile.cl/~rgutierrez/
Abstract:
A rooted abelian or quadratic differential is such a differential (defined on a Riemann surface) together with a choice of a horizontal unit tangent vector. The space of rooted differentials is a manifold and is naturally stratified by the orders of the zeros of the differential. An interesting geodesic flow, known as the Teichmüller flow, preserves this stratification.
Fix a connected component of a stratum of rooted differentials C and a contractible set U in C. The flow group of C with respect to U is the subgroup of the fundamental group of C generated by "almost-flow loops", that is, loops that follow the Teichmüller flow, up to a small perturbation occurring only inside U. We show that, for any U, the flow group of C is equal to its fundamental group. In other words, the topology of the component is completely detected by the dynamics of the Teichmüller flow: any loop is homotopic to a concatenation of almost-flow loops.
Moreover, C admits a natural Lebesgue-class measure, and the vector bundle with base C and a suitable homology group as the fibre admits a linear cocycle known as the Kontsevich–Zorich cocycle. Using our result on the flow group, we show that the Lyapunov spectra of this cocycle is simple for any C, meaning that all Lyapunov exponents are distinct. In this way, we generalise earlier work by Avila–Viana for the abelian case, and by myself for the quadratic case.
This is joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre and Saul Schleimer.
A recording of this talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=bd7725b2-bd3b-4e63-80c4-ace8016a27bf
Topological stability of boundary actions
When: Thu, April 1, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Kathryn Mann (Cornell) - https://e.math.cornell.edu/people/mann/index.html
Abstract:
This talk is about a new rigidity result for a large class of group actions on spheres. These include the actions of thefundamental groups of compact, negatively curved manifolds on the "sphere at infinithy" of the universal cover of the manifold, and more generally, all Gromov hyperbolic groups with sphere boundary. In recent works with Jonathan Bowden and with Jason Manning , we show that these actions are structurally stable: any nearby action of that group by homeomorphisms of the sphere is semi-conjugate to the original action.Bowden and I also applied this perspective to actions coming from skew-Anosov flows, using it to solve an old problem about flows on hyperbolic 3-manifolds. My talk will be a general introduction to this family of problems and techniques, with a focus on some of the large scale geometric ingredients in the proofs. A recording of the talk can be found here: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=93533f54-0a9f-45ff-9e7e-acfd014bc7de
Central Limit Theorem and geodesic flow——a dynamical point of view
When: Thu, April 8, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Tianyu Wang (Ohio State) - https://sites.google.com/view/wangty823/
Abstract: Central limit theorem (CLT) of certain equilibrium measures (i.e. measures reflecting global orbital information) is a heavily studied statistical property in dynamical systems with hyperbolic behaviors. In the first half of the talk, I will briefly introduce some common strategies to study CLT in classic settings, including the example of geodesic flow on compact negatively curved manifold. I will also give a brief idea on how a strongly uniform version of transitivity, which is called specification, can be used to derive an asymptotic version of CLT. In the second half, I will show how this approach works in the case of geodesic flow on non-positively curved rank-one manifold, which is non-uniformly hyperbolic. The method is first introduced by Denker, Senti and Zhang and the result is based on a recent joint work with Dan Thompson.
Locating Ruelle-Pollicott resonances
When: Thu, April 15, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Oliver Butterley (Roma II) - https://www.mat.uniroma2.it/butterley/
Abstract: Our aim is to obtain precise information on the asymptotic
behaviour of various dynamical systems by an improved understanding the
discrete spectrum of the associated transfer operators. I'll discuss the
general principle that has come to light in recent years and which often
allows us to obtain substantial spectral information. I'll then describe
several settings where this approach applies, including affine expanding
Markov maps, monotone maps, hyperbolic diffeomorphisms. (Joint work
with: Niloofar Kiamari & Carlangelo Liverani.)
When are equilibrium states Rajchman?
When: Thu, April 22, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Tuomas Sahlsten (University of Manchester) - https://personalpages.manchester.ac.uk/staff/tuomas.sahlsten/
Abstract: A probability measure in the Euclidean space is a Rajchman measure if its Fourier transform decays to zero in the high frequency limit. Depending on the rate of decay, Rajchman measures share many properties to the Lebesgue measure and they appear to have useful consequences on various structural properties of the support of the measure, which I will briefly review. In the context of invariant measures for dynamical systems, Rajchman property seems to capture some chaotic nature of the system. In this talk we will use contraction theorems of complex transfer operators and discretised sum product bounds to prove Rajchman property of positive dimensional equilibrium states for expanding Markov maps when the corresponding contractive iterated function system (IFS) of inverse branches is not C^2 cohomologous to a self-similar IFS. We will also discuss the recent progress in the case of self-similar IFSs.
Based on joint works with Thomas Jordan (Bristol), Jialun Li (Zürich) and Connor Stevens (Manchester)
Eigenvalues of constant length $\mathcal{S}$-adic shifts
When: Thu, April 29, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Paulina Cecchi (CMM - Universidad de Chile) - https://sites.google.com/view/paulinacb/home
Abstract: $\mathcal{S}$-adic shifts are shift spaces obtained by performing an infinite composition of morphisms $\sigma_n:\mathcal{A}_{n+1}\to \mathcal{A}_n$ defined over possible different finite alphabets $\mathcal{A}_{n+1}$. They are a generalization of substitution shifts. Being $\mathcal{S}$-adic is indeed a way to represent a minimal shift: any minimal shift can be represented in an $\mathcal{S}$-adic way. In this talk I will speak about spectral properties of $\mathcal{S}$-adic shifts. In the first part of the talk, I will introduce $\mathcal{S}-adic$ shifts and I will recall some known results about eigenvalues of substitution shifts, with particular emphasis on a theorem by B. Host which characterizes continuous and measurable eigenvalues of substitution shifts. I will also present a result which extends this theorem in some particular cases of $\mathcal{S}-adic$ shifts. In the second part, I will sketch the proof and present some examples. This is a joint work with Valérie Berthé and Reem Yassawi.
Dynamics of piecewise isometries
When: Thu, May 6, 2021 - 2:00pmWhere: Online: https://umd.zoom.us/j/91572187498
Speaker: Ana Rodrigues (University of Exeter) - http://empslocal.ex.ac.uk/people/staff/ar409/
Abstract: In this talk I will discuss some features of the dynamics of Piecewise isometries (PWIs) which are higher dimensional generalizations of one dimensional IETs, defined on higher dimensional spaces and Riemannian manifolds. In particular, I will introduce the concept of embedding of an IET into a PWI, some particular renormalization scheme and if time allows, the proof of existence of invariant curves for PWIs.