Rigidity phenomena in dynamics and spectral theory

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Schedule

	Monday (Y15G40)	Tuesday (Y21F70)	Wednesday (Y15G40)	Thursday (Y15G40)
09:00 - 10:30	Lefeuvre 1	Nonnenmacher 1	Rivière 2	Sorrentino 2
10:30 - 11:00	coffee / tea	coffee / tea	coffee / tea	coffee / tea
11:00 - 12:30	Rivière 1	Sorrentino 1	Lefeuvre 2	Nonnenmacher 2
12:30 - 14:00	lunch	lunch (end 13:45)	lunch	lunch
14:00 - 15:00	Sadanand	Koval (start 13:45)	Vinhage	Slipantschuk
15:00 - 15:30	coffee / tea	Leclerc (start 14:50)	free	coffee / tea
15:30 - 16:30	Chaubet	Leclerc (end 15:50)	free	Tsodikovich
17:30 - 19:00		apéro / meze
18:30 - 21:00				social dinner

	Friday (Y15G40)
09:00 - 10:00	Ghazouani
10:00 - 10:30	coffe / tea
10:30 - 11:30	Jézéquel
11:30 - 12:30	Leguil
12:30 - 14:00	lunch

The venue is always at the Irchel campus of the University of Zurich. On Monday, Wednesday, Thursday, and Friday we will be in Y15-G-40 (building Y15, lecture hall G40). Only
on Tuesday we are in Y21-F-70.

Coffee/tea breaks will be in Dozentenfoyer Y22-G-74. On Tuesday afternoon we will not
have the afternoon coffee/tea break and we end at 16:00. Lunch breaks will be in the canteen
(building Y21).

The social dinner will be held in restaurant Luigia (city centre) and the ap´ero/meze on
campus (Green Kitchen Lab).

Mini Courses

Thibault Lefeuvre (CNRS and Sorbonne): “Marked length spectrum rigidity of Anosov surfaces”.

Abstract: Anosov surfaces are Riemannian surfaces with Anosov geodesic flow, such as those with negative curvature. The purpose of this minicourse is to prove that two Anosov surfaces with same marked length spectrum (ie same lengths of closed geodesics marked by the free homotopy of the manifold) are isometric. The proof relies on the microlocal notion of wavefront set (of distributions); I will take some time to review this concept and explain its basic properties.

Stéphane Nonnenmacher (Paris-Saclay): “The Fractal Uncertainty Principle, and applications to Quantum Chaos”.

Abstract: The idea of a fractal uncertainty principle (FUP) arose in the study of quantum chaotic scattering systems (say, the scattering of waves by several disjoint convex obstacles), more precisely when studying the high-frequency distribution of quantum resonances of such quantum systems. In the corresponding classical systems, the set of trapped rays (”trapped set”) forms a fractal hyperbolic repeller; this fractal geometry was known to influence the wave decay at high frequency, and the corresponding distribution of the quantum resonances. Dyatlov-Zahl conjectured a form of uncertainty principle adapted to such a fractal trapped set in the context of hyperbolic co-convex surfaces, with the aim to show a resonance free strip, with consequence a uniform wave decay at high frequency. This Fractal Uncertainty Principle was then proved by Bourgain-Dyatlov, providing resonance free strips for those hyperbolic surfaces.

The usual uncertainty principle provides a constraint between the ”essential support” of a function on R and the ”essential support” of its Fourier transform: if the essential support of the former has length L, then the essential support of the latter must have length at least 1/L. As a novelty, the FUP does not only take into account the lengths of the essential supports, but also their fractal geometry. By ”fractal”, it is understood that those sets have ”holes” at all scales within a certain interval of scales. Roughly speaking, the FUP states that the two essential supports cannot both be ”fractal”, in appropriate intervals of scales. I plan to provide some ideas of the proof of the FUP, which uses tools of harmonic analysis and some elementary fractal ”geometry”. I will also sketch how this FUP is used to prove the resonance gap in chaotic scattering systems. Time permitting, I will mention (but not prove) a second application of the FUP, concerning the spatial distribution of the eigenmodes of the Laplacian on a compact hyperbolic surface.

Gabriel Rivière (Nantes): “Analytical study of Poincaré series”.

Abstract: I will start by discussing the growth of lattice points in the dilation of a convex subset of the Euclidean space. This problem naturally gives rise to a ”length spectrum” to which we can associate Poincaré series. My goal is to explain first on this simple geometric model how the study of these zeta functions (or of the related couting problem) can be understood as a problem in dynamical systems. More precisely, they can be transferred into the analytic study of certain flows acting on the unit tangent bundle of the torus. Once this first problem is discussed, I will focus on the extension of these questions to negatively curved manifolds and discuss the relation of Poincaré series with certain topological invariants (Euler characteristic, linking number of knots for instance).

Alfonso Sorrentino (Rome Tor Vergata): “On the rigidity of Birkhoff billiards and symplectic twist maps”.

Abstract: A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain. While it is evident how the shape determines the dynamics, a more subtle and difficult question is how the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this minicourse, we shall address some of these questions, with particular attention to the integrability of Birkhoff billiards (i.e., the so-called Birkhoff conjecture) and their length-spectral rigidity (“Can you hear the shape of a billiard?”). Moreover, we shall also discuss some related questions in the more general framework of symplectic twist maps.

Talks

Yann Chaubet (Nantes): “Nonlinear chaotic Vlasov equations”.

Abstract: The Vlasov equation describes the evolution of a distribution of interacting
particles. I will present some recent results about the long time dynamics of nonlinear
Vlasov equations on a negatively curved manifold, with smooth interaction kernel. I will
explain why small and smooth initial data supported far from the null section give rise to
solutions which weakly converge to an equilibrium state at exponential rate. This is a joint
work with Daniel Han-Kwan and Gabriel Rivi`ere.

Selim Ghazouani (UCL): “Horocycle averages and quantum chaos”.

Abstract: I will start by reminding the audience of the classical connection between the Schrödinger equation and the geodesic flow on a Riemannian manifold, and the breakdown of this approximation due to interference effects at the so-called Ehrenfest time. I will then put forward a (partially conjectural) connection between interference effects in the quantum world and ergodic averages of horocycle flows, when the dynamics of the geodesic flow is Anosov.

Malo Jézéquel (CNRS and Brest): “Zeta functions for smooth pseudo-Anosov flows”.

Abstract: To an Anosov flow (i.e. a smooth uniformly hyperbolic flow on a closed manifold), one may associate a zeta function This is a meromorphic function defined in term of the periodic orbits of the flow, that can be used for instance to count these orbirs. After recalling this definition, I will explain how to extend it to certain “smooth pseudo-Ansov flows” on 3-manifolds, a class of flows that looks like Anosov flows except for a finite number of singular orbits. The motivation for studying zeta functions for these flows comes from topology. In particular, I will discuss a version of Fried’s conjecture for smooth pseudo-Anosov flows. This is a joint work with Jonathan Zung.

Illya Koval (IST Austria): “Silent Orbits and Cancellations in the Wave Trace”.

Abstract: We show that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. To do so, we construct large families of billiard tables for which there exist multiple periodic billiard orbits having the same length but different Maslov indices. Using the microlocal Balian-Bloch-Zelditch parametrix for wave invariants we solve a system of equations for the boundary curvature jets, which leads to the required cancellations. Such cancellations show that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned. This is a joint project with V. Kaloshin and A. Vig.

Gaétan Leclerc (Sorbonne): “On the oscillations of a temporal distance function”.

Abstract: Consider a smooth, area-preserving, Axiom A diffeomorphism on a surface. Fix a smooth roof function, and consider the associated suspension flow. In the (rare) case when the temporal distance function for this flow is smooth, then the usual Dolgopyat’s method can be adapted to prove exponential mixing (under some condition on the roof function). But what happens when the associated temporal distance function is not smooth? The goal of this talk is to describe briefly a rigidity phenomenon that occurs in this setting, allowing us to find oscillations ”everywhere and at all scales” for the temporal distance function. This talk is inspired by some recent unpublished work that can be found in my PhD manuscript (Chapter 5), and in a recent paper by Tsujii-Zhang (Annals of Maths, 2023).

Martin Leguil (Ècole polytechnique): “Rigidity for conjugacies of dissipative Anosov flows in dimension 3”.

Abstract: In an ongoing project with A. Gogolev and F. Rodriguez Hertz, we study when two transitive Anosov flows in dimension 3 which are topologically conjugated are actually smoothly conjugated. By the work of de la Llave, Marco and Moriyón from the 80s, a necessary and sufficient condition for that is that stable and unstable eigenvalues at corresponding periodic points match. In our work we show that in many cases, the latter condition is redundant, as it is already implied by the existence of a topological conjugacy. In particular, we show that the conjugacy is smooth, unless one of the flows is a suspension, or the conjugacy swaps positive and negative SRB measures of the two flows. This extends a recent work of Gogolev-Rodriguez Hertz in the volume preserving case. I will try to explain how this rigidity problem is connected with other notions, in particular, the Foulon-Hasselblatt cocycle, and the so-called templates introduced by Tsujii and Zhang to study the regularity of stable and unstable distributions.

Chandrika Sadanand (Bowdoin): “You can hear the shape of a polygonal billiard table”.

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

Julia Slipantschuk (Bayreuth and Warwick): “Resonances for hyperbolic maps”.

Abstract: I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.

Daniel Tsodikovich (IST Austria and Tel Aviv): “Locally maximizing orbits in billiards”.

Abstract: The set of locally maximizing orbits is an important feature of exact twist maps, using which we can study the integrability of the system. In this talk, we give a condition for two generating functions of the same twist map to have the same sets of locally maximizing orbits. We show that the billiard system satisfies this condition. As an application, we give a quantitative version of the rigidity result for integrability of billiards by Bialy and Mironov. Joint work with Misha Bialy.

Kurt Vinhage (Utah): “Entropy rigidity in dimension 3”.

Abstract: Entropy rigidity is a phenomenon which arises from “equidistribution of complexity.” The measure of maximal entropy concentrates its mass at the parts of the system with highest complexity, so when that measure is a volume, the system should be very special. In dimension 3, we show that Anosov flows who measure of maximal entropy is a volume are either constant-time suspensions of linear Anosov diffeomorphisms or geodesic flows on hyperbolic surfaces. This generalizes results of Katok and Foulon, who showed this in the setting of geodesic flows and contact flows, respectively. Joint with M. Leguil, J. de Simoi and Y. Yang.