Departement Mathematik

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Ergodic theory and dynamical systems

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Herbstsemester 2020

Datum / Zeit Referent:in Titel Ort
21. September 2020
15:00-16:15 		Prof. Dr. Manfred Einsiedler
ETHZ
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Ergodic theory and dynamical systems seminar

Titel Rigidity for commuting toral automorphisms
Referent:in, Affiliation Prof. Dr. Manfred Einsiedler, ETHZ
Datum, Zeit 21. September 2020, 15:00-16:15
Ort Y27 H 28
Abstract In the long awaited joint work with Elon Lindenstrauss we partially classify positive entropy measures for higher rank commutative actions on tori and solenoids. This can be applied to show that abstract measurable factors are in fact always arising from algebraic constructions.
Rigidity for commuting toral automorphismsread_more
Y27 H 28
28. September 2020
15:00-16:15 		Prof. Dr. Roland Zweimüller
University of Vienna
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Ergodic theory and dynamical systems seminar

Titel Return-times and hitting-times of small sets (e-seminar)
Referent:in, Affiliation Prof. Dr. Roland Zweimüller, University of Vienna
Datum, Zeit 28. September 2020, 15:00-16:15
Ort Y27 H 28
Abstract The asymptotic behavior of return-times and hitting-times of small sets when their size tends to zero (rare events) in ergodic dynamical systems has undergone some intense research in the past twenty years, the probabilistic laws governing their occurrence being one central aspect. I will review basic facts and present some new developments, mentioning, in particular, the use of classical abstract ergodic theory to obtain Poisson limit theorems under surprisingly soft conditions, spatiotemporal limit theorems and local limit theorems.
Return-times and hitting-times of small sets (e-seminar)read_more
Y27 H 28
5. Oktober 2020
15:00-16:15 		Dr. Frank Trujillo
Universität Zürich
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Ergodic theory and dynamical systems seminar

Titel Persistence of degenerate lower dimensional invariant tori in Hamiltonian systems
Referent:in, Affiliation Dr. Frank Trujillo, Universität Zürich
Datum, Zeit 5. Oktober 2020, 15:00-16:15
Ort Y27 H 28
Abstract The classical KAM theory establishes the persistence, under sufficiently small perturbations, of most of the n-dimensional invariant tori of non-degenerate integrable Hamiltonians with n degrees of freedom. The surviving tori are those carrying a quasi-periodic motion by a Diophantine vector and, in particular, their restricted dynamics is minimal. On the other hand, such systems also admit n-dimensional invariant tori whose restricted dynamics is not minimal. These tori, which we call resonant, are foliated by invariant tori whose dimension is smaller than the number of degrees of freedom of the system. In this talk I will present a criterion for the persistence of at least one of the lower dimensional invariant tori associated to a resonant invariant torus.
Persistence of degenerate lower dimensional invariant tori in Hamiltonian systemsread_more
Y27 H 28
12. Oktober 2020
15:00-16:15 		Dr. Henna Koivusalo
University of Bristol
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Ergodic theory and dynamical systems seminar

Titel Linear repetitivity in polytopal cut and project sets (e-seminar)
Referent:in, Affiliation Dr. Henna Koivusalo, University of Bristol
Datum, Zeit 12. Oktober 2020, 15:00-16:15
Ort Y27 H 28
Abstract Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study the number and repetition of finite patterns. From this perspective, sets with patterns repeating linearly often, called linearly repetitive sets, can be viewed as the most ordered aperiodic sets. Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In an earlier work, joint with Haynes and Walton, we showed that for cut and project sets with a cube window, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the cut and project set has minimal number of different finite patterns (minimal complexity), and (ii) the irrational slope satisfies a certain Diophantine condition (badly approximable condition). In a new joint work with Jamie Walton, we give a generalisation of this result to all polytopal windows satisfying a mild geometric condition. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.
Linear repetitivity in polytopal cut and project sets (e-seminar)read_more
Y27 H 28
19. Oktober 2020
15:00-16:15 		Malo Jézéquel
LPSM
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Ergodic theory and dynamical systems seminar

Titel Trace formulae and dynamical determinants for Anosov flows (e-seminar)
Referent:in, Affiliation Malo Jézéquel, LPSM
Datum, Zeit 19. Oktober 2020, 15:00-16:15
Ort Y27 H 28
Abstract In order to study fine statistical properties of an Anosov flow, one may introduce the notion of Ruelle resonances. These resonances are complex numbers, with spectral theoretic interpretation, that are in principle hard to compute. However, these resonances are known to be the zeros of a dynamical determinant (an entire function defined from the periodic data of the flow). We will discuss the validity of a morally stronger relationship between Ruelle resonances and periodic orbits (a trace formula conjectured by Dyatlov and Zworski), and explain how it is related to complex analytic properties of the dynamical determinant.
Trace formulae and dynamical determinants for Anosov flows (e-seminar)read_more
Y27 H 28
21. Oktober 2020
10:30-11:30 		Dr. Irene Pasquinelli
Institut de Mathématiques de Jussieu
Details

Ergodic theory and dynamical systems seminar

Titel Deligne-Mostow lattices: fundamental domains and line arrangements
Referent:in, Affiliation Dr. Irene Pasquinelli, Institut de Mathématiques de Jussieu
Datum, Zeit 21. Oktober 2020, 10:30-11:30
Ort Y27 H 28
Abstract We will talk about lattices in the group PU(n,1) of holomorphic isometries of complex hyperbolic space. A well known class is that of the Deligne-Mostow lattices. I will describe different interpretations of these lattices and explain how one can use Thurston's interpretation to build fundamental domains for them. Then I will explain how Hirzebruch's line arrangements construction of the Deligne-Mostow lattices is related to the fundamental domains. Finally I will tell you how we used this connection to go a step forward towards giving a complex equivalent to the hybrid construction. Part of these results are in a joint work with Elisha Falbel.
Deligne-Mostow lattices: fundamental domains and line arrangementsread_more
Y27 H 28
26. Oktober 2020
15:00-16:15 		Timothée Bénard
ENS Paris
Details

Ergodic theory and dynamical systems seminar

Titel Radon stationary measures for a random walk on T^d x R (e-seminar)
Referent:in, Affiliation Timothée Bénard, ENS Paris
Datum, Zeit 26. Oktober 2020, 15:00-16:15
Ort Y27 H 28
Abstract I will present the content of my paper [B20] classifying the Radon stationary measures for a random walk on T^d x R. This walk is realised by a random action of SL_d(Z) on the T^d component, coupled with a translation on the R component. I will explain under assumptions of irreducibility and recurrence, the rigidity and homogeneity of Radon ergodic stationary measures. References [B20] Timothée Bénard. Radon stationary measures for a random walk on T^d x R. arXiv preprint arXiv:2006.05742, 2020.
Radon stationary measures for a random walk on T^d x R (e-seminar)read_more
Y27 H 28
2. November 2020
15:00-16:15 		Dr. Alba Marina Malga Sabogal
University of Lorraine
Details

Ergodic theory and dynamical systems seminar

Titel Generic Dynamics of Infinite measure translation surfaces (e-seminar)
Referent:in, Affiliation Dr. Alba Marina Malga Sabogal, University of Lorraine
Datum, Zeit 2. November 2020, 15:00-16:15
Ort Y27 H 28
Abstract A staircase surface is an infinite genus translation surface obtained by gluing rectangles in a way similar to steps of a staircase. An Ehrenfest wind-tree model is a mathematical billiard played upon the complement of some arbitrarily placed pairwise parallel square scatterers. In this talk I will introduce parameterized families of wind-trees and staircases, then I'll explain why for a G_delta dense sets of parameters the corresponding system has nice dynamics (minimality, ergodicity, etc). This is a joint work with Serge Troubetzkoy.
Generic Dynamics of Infinite measure translation surfaces (e-seminar)read_more
Y27 H 28
9. November 2020
15:00-16:15 		Minju Lee
Yale University
Details

Ergodic theory and dynamical systems seminar

Titel Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends (e-seminar)
Referent:in, Affiliation Minju Lee, Yale University
Datum, Zeit 9. November 2020, 15:00-16:15
Ort Y27 H 28
Abstract This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in SO(d,1) acting on the space \( \Gamma \) \ SO(d,1), assuming that the associated hyperbolic manifold M= \(\Gamma \) \ \( H^d \) is a convex cocompact manifold with Fuchsian ends. For d = 3, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any k \(\geq\) 1, (1) the closure of any k-horosphere in M is a properly immersed submanifold; (2) the closure of any geodesic (k+1)-plane in M is a properly immersed submanifold; (3) an infinite sequence of maximal properly immersed geodesic (k+1)-planes intersecting core(M) becomes dense in M.
Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends (e-seminar)read_more
Y27 H 28
16. November 2020
15:00-16:15 		Prof. Dr. Dimitry Dolgopyat
University of Maryland
Details

Ergodic theory and dynamical systems seminar

Titel Multiple Borel Cantelli Lemma and MultiLog Law for Recurrence (e-seminar)
Referent:in, Affiliation Prof. Dr. Dimitry Dolgopyat, University of Maryland
Datum, Zeit 16. November 2020, 15:00-16:15
Ort Y27 H 28
Abstract A classical Borel Cantelli Lemma gives condition for an infinitely many unlikely events to occur. A multiple Borel Cantelli Lemma asks how often we have k such events occurring at the same scale. In this talk I describe a version of multiple Borel Cantelli Lemma for smooth exponentially mixing systems and present applications to recurrence problems. The talk is based on a joint work with Bassam Fayad and Sixu Liu.
Multiple Borel Cantelli Lemma and MultiLog Law for Recurrence (e-seminar)read_more
Y27 H 28
23. November 2020
15:00-16:15 		Prof. Dr. Carlos Matheus Silva Santos
CNRS
Details

Ergodic theory and dynamical systems seminar

Titel Speed of mixing of geodesic flows on certain non-positively curved surfaces (e-seminar)
Referent:in, Affiliation Prof. Dr. Carlos Matheus Silva Santos, CNRS
Datum, Zeit 23. November 2020, 15:00-16:15
Ort Y27 H 28
Abstract In this talk, we discuss the speed of mixing on certain surfaces whose curvatures are negative except along a closed geodesic. This is a joint work in progress with Y. Lima and I. Melbourne.
Speed of mixing of geodesic flows on certain non-positively curved surfaces (e-seminar)read_more
Y27 H 28
30. November 2020
15:00-16:15 		Dr. Jialun Li
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Exponential mixing of geodesic flow for geometrically finite manifolds with cusps
Referent:in, Affiliation Dr. Jialun Li, Universität Zürich
Datum, Zeit 30. November 2020, 15:00-16:15
Ort Y27 H 28
Abstract Let H^n be the hyperbolic n-space and D be a geometrically finite discrete subgroup in Isom_+(H^n) with cusps. In the joint work with Wenyu Pan, we establish exponential mixing of the geodesic flow over the unit tangent bundle T^1(D \ H^n). Previously, such results were proved by Stoyanov for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for D with large critical exponent. We obtain our result by constructing a nice coding for the geodesic flow and then prove a Dolgopyat-like spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding, which is partly inspired by the works of Lai-Sang Young and Burns-Masur-Matheus-Wilkinson. I will also discuss the application of obtaining a resonance-free region for the resolvent on D\ H^n.
Exponential mixing of geodesic flow for geometrically finite manifolds with cuspsread_more
Y27 H 28
7. Dezember 2020
15:00-16:15 		Prof. Dr. Omri Sarig
Weizmann Institute of Science
Details

Ergodic theory and dynamical systems seminar

Titel Measures of maximal entropy for smooth surface diffeomorphisms (e-seminar)
Referent:in, Affiliation Prof. Dr. Omri Sarig, Weizmann Institute of Science
Datum, Zeit 7. Dezember 2020, 15:00-16:15
Ort Y27 H 28
Abstract Thirty years ago, Sheldon Newhouse proved that every C^\infty diffeomorphism on a compact manifold has at least one measure of maximal entropy. I will explain the proof of the following result (joint with J. Buzzi and S. Crovisier): If the diffeomorphism has positive topological entropy, and if the dimension of the manifold is equal to two, then there are at most finitely many different ergodic measures of maximal entropy, and in the topologically transitive case -- exactly one (joint work with J. Buzzi and S. Crovisier).
Measures of maximal entropy for smooth surface diffeomorphisms (e-seminar)read_more
Y27 H 28
14. Dezember 2020
15:00-16:15 		Dr. Taylor McAdam
Yale University
Details

Ergodic theory and dynamical systems seminar

Titel Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R) (e-seminar)
Referent:in, Affiliation Dr. Taylor McAdam, Yale University
Datum, Zeit 14. Dezember 2020, 15:00-16:15
Ort Y27 H 28
Abstract Inspired by the work of Sarnak and Ubis [1] in SL(2,Z)\SL(2,R), we prove that almost-prime times (i.e. integer times having fewer than a fixed number of prime factors) in maximal horospherical orbits of generic points in SL(3,Z)\SL(3,R) are dense in the whole space, where the number of prime factors allowed in the almost-primes is independent of the basepoint. This is in contrast to previous work [2] in which the number of prime factors depends on a Diophantine property of the basepoint. The proof involves a case-by-case analysis of the different ways in which a basepoint can fail the Diophantine property. If a basepoint fails to equidistribute rapidly in the whole space with respect to the continuous time flow, then there exists a sequence of nearby periodic orbits of increasing volume that approximate the original orbit up to larger and larger time scales, and which equidistribute in the whole space as the volume grows. Given an open set, one can find a large enough approximating orbit such that almost-primes of a fixed order in the approximating orbit land inside that set, and this property can then be transported to the nearby orbit of the original basepoint. This is joint work-in-progress with Manuel Luethi. [1] Sarnak, Peter, and Adrián Ubis. "The horocycle flow at prime times." Journal de mathématiques pures et appliquées 103.2 (2015): 575-618. [2] McAdam, Taylor. "Almost-prime times in horospherical flows on the space of lattices." Journal of Modern Dynamics 15 (2019): 277.
Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R) (e-seminar)read_more
Y27 H 28

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