February 2021 – Midwest dynamics and group actions seminar

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.