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Autumn Semester 2022

Date / Time Speaker Title Location
26 September 2022
13:30-14:30 		Prof. Dr. Mark Pollicott
University of Warwick
Details

Ergodic theory and dynamical systems seminar

Title Random matrices and iterated function schemes: Lyapunov exponents and dimensions
Speaker, Affiliation Prof. Dr. Mark Pollicott, University of Warwick
Date, Time 26 September 2022, 13:30-14:30
Location Y27 H 28
Abstract We will set the scene by considering Cantor sets in the real line and generated by simple iterated function schemes and estimates on the value of their Hausdorff dimension. This has applications to problems related to the Zaremba conjecture and Lagrange spectra. We will then focus on the problem of estimating the (top) Lyapunov exponent for random matrix products. By way of an application, we will be interested in the value of the drift for Fuchsian groups and implications for the harmonic measure.
Random matrices and iterated function schemes: Lyapunov exponents and dimensionsread_more
Y27 H 28
3 October 2022
13:30-14:30 		Dr. Reynold Fregoli
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Title Multiplicatively badly approximable vectors
Speaker, Affiliation Dr. Reynold Fregoli, Universität Zürich
Date, Time 3 October 2022, 13:30-14:30
Location Y27 H 28
Abstract The Littlewood Conjecture states that for all pairs of real numbers \((\alpha, \beta)\) the product \(\mid q\mid \mid q\alpha+p_1\mid\mid q\beta+p_2\mid\) becomes arbitrarily close to \(0\) when the vector \((q, p_1, p_2)\) ranges in \( \mathbb{Z}^3 \) and \( q \neq 0\). To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood conjecture, where the factor \(\mid q\mid\) is replaced by an increasing function \(f(\mid q\mid).\) More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions \(f\) for which the above product and its higher dimensional generalisations stay bounded away from \(0\) for at least one pair \((\alpha, \beta) \in\, \mathbb{R}^2.\) This problem happens to be intimately connected with the equidistribution rate of certain segments on the expanding torus in \( SL_3(\mathbb{R})/SL_3(\mathbb{Z})\) under the action of the full diagonal group.
Multiplicatively badly approximable vectorsread_more
Y27 H 28
10 October 2022
13:30-14:30 		Andreas Wieser
Einstein Institute, Hebrew University of Jerusalem
Details

Ergodic theory and dynamical systems seminar

Title Birkhoff genericity for points on curves in expanded horospheres
Speaker, Affiliation Andreas Wieser, Einstein Institute, Hebrew University of Jerusalem
Date, Time 10 October 2022, 13:30-14:30
Location Y27 H 28
Abstract Let \(\{a(t):t \in \mathbb{R}\}\) be a diagonalizable subgroup of \(SL(d,\mathbb{R})\) for which the expanded horosphere \(U\) is abelian. By the Birkhoff ergodic theorem, for any point \(x \in SL(d,\mathbb{R})/SL(d,\mathbb{Z})\) and almost every \(u \in U\) the point \(ux\) is Birkhoff generic for the flow \(a(t)\). One may ask whether the same is true when the points in \(U\) are sampled with respect to a measure singular to the Lebesgue measure. In this talk, we discuss work with Omri Solan proving that almost every point on an analytic curve within U is Birkhoff generic when the curve satisfies a non-degeneracy condition.
Birkhoff genericity for points on curves in expanded horospheresread_more
Y27 H 28
17 October 2022
13:30-14:30 		Dr. Samantha Fairchild
University of Osnabrück
Details

Ergodic theory and dynamical systems seminar

Title Shrinking rates of horizontal gaps for generic translation surfaces
Speaker, Affiliation Dr. Samantha Fairchild, University of Osnabrück
Date, Time 17 October 2022, 13:30-14:30
Location Y27 H 28
Abstract A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.
Shrinking rates of horizontal gaps for generic translation surfacesread_more
Y27 H 28
27 October 2022
13:30-14:30 		Prof. Dr. Gabriel Paternain
University of Cambridge
Details

Ergodic theory and dynamical systems seminar

Title No ETDS seminar this week
Talk by Gabriel Paternain in the PDEs and Mathematical Physics Seminar
Speaker, Affiliation Prof. Dr. Gabriel Paternain, University of Cambridge
Date, Time 27 October 2022, 13:30-14:30
Location Y27 H 28
No ETDS seminar this week
Talk by Gabriel Paternain in the PDEs and Mathematical Physics Seminar
Y27 H 28
31 October 2022
13:30-14:30 		Wooyeon Kim
ETHZ
Details

Ergodic theory and dynamical systems seminar

Title Divergent on average trajectories for higher rank actions
Speaker, Affiliation Wooyeon Kim, ETHZ
Date, Time 31 October 2022, 13:30-14:30
Location Y27 H 28
Abstract Let \(A\) be the group of positive diagonal \(d\times d\) matrices on \(SL_d(\mathbb{R})\) and \(U\cong \mathbb{R}^{d-1}\) be an abelian expanding horospherical group in \(SL_d(\mathbb{R})\), where \(d\ge 2\). Denote by \(A^{+}\) the expanding cone in \(A\) associated to \(U\). We say that \(x\in SL_d(\mathbb{R})/SL_d(\mathbb{Z})\) is \(A^{+}\)-divergent on average if for any compact set \(K\) the orbit \(A^{+}x\) escapes \(K\) on average. One may ask how large the set of points which are \(A^{+}\)-divergent on average is. In this talk, I will discuss upper and lower bounds for the Hausdorff dimension of the set of points which are \(A^{+}\)-divergent on average.
Divergent on average trajectories for higher rank actionsread_more
Y27 H 28
7 November 2022
13:30-14:30 		Prof. Dr. Pascal Hubert
Aix-Marseille Université
Details

Ergodic theory and dynamical systems seminar

Title Interval exchange transformations with discrete spectrum
Speaker, Affiliation Prof. Dr. Pascal Hubert, Aix-Marseille Université
Date, Time 7 November 2022, 13:30-14:30
Location Y27 H 28
Abstract We know from a result by Avila and Forni that almost every interval exchange transformation is weakly-mixing. Until very recently, no (non trivial) example with discrete spectrum was known. In this talk, I will describe two families of interval exchange transformations which are conjugate to rotations of the two dimensional torus.
Interval exchange transformations with discrete spectrumread_more
Y27 H 28
14 November 2022
13:30-14:30 		Prof. Dr. Artur Avila
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Title Newhouse phenomenon in the complex Hénon family, and homoclinic bifurcations
Speaker, Affiliation Prof. Dr. Artur Avila, Universität Zürich
Date, Time 14 November 2022, 13:30-14:30
Location Y27 H 28
Newhouse phenomenon in the complex Hénon family, and homoclinic bifurcations
Y27 H 28
21 November 2022
13:30-14:30 		Prof. Dr. François Ledrappier
CNRS and Sorbonne Université
Details

Ergodic theory and dynamical systems seminar

Title Exact dimension of Oseledets measures
Speaker, Affiliation Prof. Dr. François Ledrappier, CNRS and Sorbonne Université
Date, Time 21 November 2022, 13:30-14:30
Location Y27 H 28
Abstract This talk will report on an ongoing joint work with Pablo Lessa (Montevideo). We consider a random walk on a group of matrices. Under suitable assumptions, Oseledets Theorem yields numbers (the Lyapunov exponents) and a random splitting into so-called Oseledets subspaces. This splitting defines a (random) point in a product of Grassmannians. Our Main result is that the distribution of this point is an exact-dimensional measure. The dimension has a geometric interpretation in terms of the exponents and some partial entropies. The talk will present the statement and the main partial results. As an example, we also discuss the case when the random walk is supported by an Anosov 3-dimensional representation of a surface group.
Exact dimension of Oseledets measuresread_more
Y27 H 28
28 November 2022
13:30-14:30 		Prof. Dr. Gabriela Weitze-Schmithüsen
Universität des Saarlandes
Details

Ergodic theory and dynamical systems seminar

Title Veech groups of origami that are congruence groups
Speaker, Affiliation Prof. Dr. Gabriela Weitze-Schmithüsen, Universität des Saarlandes
Date, Time 28 November 2022, 13:30-14:30
Location Y27 H 28
Abstract Origamis are special translation surfaces which are obtained as patchwork from finitely many Euclidean squares. An important algebraic invariant is their Veech group which is a finite index subgroup of \(SL(2,\mathbb{Z})\). It is unknown which subgroups of \(SL(2,\mathbb{Z})\) occur as Veech groups. However for congruence groups of prime level there is a constructive approach which shows that all but five special cases occur. We find four of the missing five cases and study the question whether the result can be generalized to general congruence groups. The question is intimately related to the study of orbit spaces of \(SL(2,\mathbb{Z}/n\mathbb{Z})\).
Veech groups of origami that are congruence groupsread_more
Y27 H 28
5 December 2022
13:30-14:30 		Colin Guillarmou
CNRS and Université Paris-Saclay
Details

Ergodic theory and dynamical systems seminar

Title Lens rigidity for manifolds of Anosov type
Speaker, Affiliation Colin Guillarmou, CNRS and Université Paris-Saclay
Date, Time 5 December 2022, 13:30-14:30
Location Y27 H 28
Abstract I will discuss recent results on the problem of determining a Riemannian metric on a compact manifold with boundary from the set of lengths of geodesics with endpoints on the boundary and their tangent vectors at the boundary (the so-called lens data). Based on joint works with Bonthonneau, Cekic, Lefeuvre, Jezequel.
Lens rigidity for manifolds of Anosov typeread_more
Y27 H 28
12 December 2022
13:30-14:30 		Prof. Dr. Sylvain Crovisier
CNRS and Université Paris-Saclay
Details

Ergodic theory and dynamical systems seminar

Title Existence of physical measures for smooth surface diffeomorphisms
Speaker, Affiliation Prof. Dr. Sylvain Crovisier, CNRS and Université Paris-Saclay
Date, Time 12 December 2022, 13:30-14:30
Location Y27 H 28
Abstract Marcelo Viana has conjectured that a smooth diffeomorphism admits a physical measure if the Lyapunov exponents of its orbits in a full volume set do not vanish. I will explain how a technique controlling the continuity of Lyapunov exponents allows to prove this conjecture in the case of smooth surface diffeomorphisms. This is a joint work with Jérôme Buzzi and Omri Sarig.
Existence of physical measures for smooth surface diffeomorphismsread_more
Y27 H 28

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