awb1472 – Page 2 – Midwest dynamics and group actions seminar

03/08: Thang Nguyen

Sub-Riemannian dynamics and local rigidity of higher hyperbolic rank

Abstract: Symmetric and locally symmetric spaces are important classes of Riemannian manifolds of nonnegative curvatures. We study a rigidity question whether a less symmetric geometric condition leads to symmetric and locally symmetric space structures. The condition we consider is that every geodesic in the universal cover is contained in a totally geodesic constant curvature hyperbolic plane (such condition is called higher hyperbolic rank). We are able to obtain a local rigidity for perturbations of locally symmetric spaces. The novelty of the work is a new idea in using geometric analysis on sub-Riemannian manifolds to study dynamics of geodesic flows. We also obtain a local rigidity for some boundary actions using this new idea. This is based on a joint work with Chris Connell and Ralf Spatzier.

Slides

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03/01: Bassam Fayad

On KAM rigidity of parabolic affine actions on the torus

Abstract: A celebrated result of KAM theory is Arnold’s proof that a perturbed Diophantine rotation of the circle is reducible if the rotation number is preserved.

We say that an action is KAM rigid if any of its small perturbations is reducible, under a preservation of some Diophantine data, even when the model is not necessarily a toral translation.

In earlier work with Danijela Damjanovic, we proved that an affine $\mathbb Z^2$ action on the torus that has a higher rank linear part except for a rank one factor that is Identity is KAM rigid. That result combined the mechanisms of KAM rigidity with the mechanisms of local rigidity of partially hyperbolic higher rank affine actions on tori proved by Damjanovic and Katok.

For affine actions with parabolic generators the situation is completely different due to the absence of cohomological stability above an individual parabolic element, except for the step 2 case as shown by Katok.

We will see that KAM rigidity holds for typical abelian actions by step 2 unipotent matrices on the torus, and in some cases even when only one generator is of step 2.

This is a joint work with Danijela Damjanovic and Maria Saprykina.

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04/12: Semyon Dyatlov

Ruelle zeta at zero for nearly hyperbolic 3-manifolds

Abstract: For a compact negatively curved Riemannian manifold $(\Sigma,g)$ , the Ruelle zeta function $\zeta_{\mathrm R}(\lambda)$ of its geodesic flow is defined for $\Re\lambda\gg 1$ as a convergent product over the periods $T_{\gamma}$ of primitive closed geodesics

$\displaystyle \zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}})$

and extends meromorphically to the entire complex plane. If $\Sigma$ is hyperbolic (i.e. has sectional curvature $-1$ ), then the order of vanishing $m_{\mathrm R}(0)$ of $\zeta_{\mathrm R}$ at $\lambda=0$ can be expressed in terms of the Betti numbers $b_j(\Sigma)$ . In particular, Fried proved in 1986 that when $\Sigma$ is a hyperbolic 3-manifold,

$\displaystyle m_{\mathrm R}(0)=4-2b_1(\Sigma).$

I will present a recent result joint with Mihajlo Cekić, Benjamin Küster, and Gabriel Paternain: when $\dim\Sigma=3$ and $g$ is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely

$\displaystyle m_{\mathrm R}(0)=4-b_1(\Sigma).$

This is in contrast with dimension 2 where $m_{\mathrm R}(0)=b_1(\Sigma)-2$ for all negatively curved metrics. The proof uses the microlocal approach of expressing $m_{\mathrm R}(0)$ as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott–Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations.

Slides

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02/22: Lei Chen

Actions of Homeo and Diffeo groups on manifolds

Abstract: In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

Slides

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02/15: Nattalie Tamam

Effective equidistribution of horospherical flows in infinite volume

Abstract: Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space.In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.

Slides

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02/08: Serge Cantat

Stationary measures on real projective surface

Abstract: Consider a real projective surface $X(\mathbb R)$ , and a group $\Gamma$ acting by algebraic diffeomorphisms on $X(\mathbb R)$ . If $\nu$ is a probability measure on $\Gamma$ , one can randomly and independently choose elements $f_j$ in $\Gamma$ and look at the random orbits $x$ , $f_1(x)$ , $f_2(f_1(x))$ , … How do these orbits distribute on the surface? This is directly related to the classification of stationary measures on $X(\mathbb R)$ . I will describe recent results on this problem, all obtained in collaboration with Romain Dujardin. The main ingredients will be ergodic theory, notably the work of Brown and Rodriguez-Hertz, algebraic geometry, and complex analysis. Concrete geometric examples will be given.

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02/01: Kurt Vinhage

Entropy Rigidity for Anosov Flows in Dimension Three

Abstract: In the 80’s, A. Katok proved that for a geodesic flow on a negatively curved surface, coincidence of its entropy with respect to the Liouville measure with its topological entropy is equivalent to that surface being hyperbolic. The Katok Entropy conjecture states that similar conclusions should hold in higher dimensions as well. In this talk, I will discuss recent work, joint with Jacopo de Simoi, Martin Leguil and Yun Yang, which extends the scope of the rigidity phenomenon to Anosov flows in three dimensions. Time permitting, I will discuss ongoing progress in the dual question of entropy flexibility

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01/11: Thomas Hille

Distribution of Values of Irrational Forms at Integral Points and Spherical Averages

Abstract: Let $Q$ be a non-degenerate indefinite quadratic form in $n$ variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if $n \geq 3$ and $Q$ is not proportional to a rational form, then the set of values of $Q$ at integral points is dense in $\mathbb{R}$ . In some cases, homogeneous forms of higher degree exhibit the same behavior if the number of variables is large enough in terms of the degree and if the group preserving the form is large enough, then the set of values at integral points can be studied from the point of view of homogeneous dynamics.

In this talk we will discuss the problem of effective and quantitative distribution of values of certain forms at integral points. A central and recurrent theme revolves around (asymptotic) estimates of certain spherical averages going back to the work of Eskin, Margulis and Mozes.

This talk is based on the one hand on joint work with P. Buterus, F. Götze and G. Margulis and on the other hand with E. Fromm and H. Oh.

Slides

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01/25: Lewis Bowen

Title: A New Infinite-Dimensional Multiplicative Ergodic Theorem

Abstract: In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. This will be a gentle introductory-style talk; no knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (Ben-Gurion U.).

Slides

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11/30: Stephen Cantrell

Counting limit theorems for dominated representations

Abstract: In this talk we will discuss how to use symbolic dynamics and thermodynamic formalism to study dominated (or Anosov) representations. These are certain representations of finitely generated groups into general linear groups that can be viewed as higher rank analogies of cocompact, isometric group actions on the hyperbolic plane. We will show how to use our dynamical view point to study the statistics of random matrix products in this setting. This talk is based on joint work with Rhiannon Dougall, Italo Cipriano and Cagri Sert.