# Pacific Dynamics Seminar

Organizers: Jayadev Athreya (UW), Lior Silberman (UBC)

• Meetings: Thursdays, 14:00-15:30 PDT (UTC-7)
• Format:
• In most talks, the first 45 minutes of each talk are suitable for graduate students with some interest in dynamics; after a short break the next 45 minutes are technical.
• In some talks the slides and video are posted ahead of time, with the scheduled meeting reserved for discussion.
• Thanks to the Pacific Institute for the Mathematical Sciences for hosting videos from past talks on its MathTube.

If you are interested in our seminar you might also be interested in the following seminars:

## Agenda

1. July 22, 2021 16:00 PDT: Random walks on Gromov hyperbolic spaces and Teichmüller spaces
Speaker: Inhyeok Choi (KAIST)
[abstract] In this talk, I will discuss random walks on Gromov hyperbolic spaces. Due to the hyperbolicity of the spaces, random walks exhibit behaviors that differ from the classic (Euclidean) ones. These behaviors include the escape to infinity, central limit theorems when centered at the escape rate, and geodesic tracking. I will explain how one can sharpen these behaviors based on the recent observations by Gouëzel and Baik-Choi-Kim. If time allows, I will also explain how one can implement this theory on (non-hyperbolic) Teichmüller spaces.
2. May 27, 2021: The extremal length systole of the Bolza surface
[abstract] Extremal length is a conformal invariant that plays an important role in Teichmüller theory. For each essential closed curve on a Riemann surface, it furnishes a function on the Teichmüller space. The extremal length systole of a Riemann surface is defined as the infimum of extremal lengths of all essential closed curves. Its hyperbolic analogue is the hyperbolic systole: the infimum of hyperbolic lengths of all essential closed curves. While the latter has been studied profusely, the extremal length systole remains widely unexplored. For example, it is known that in genus 2, the hyperbolic systole has a unique global maximum: the Bolza surface. In this talk we introduce the extremal length systole and show that in genus two it attains a strict local maximum at the Bolza surface, where it takes the value square root of 2. This is joint work with Maxime Fortier Bourque and Franco Vargas Pallete.
3. May 20, 2021: The Manhattan curve and rough similarity rigidity
Speaker: Ryokichi Tanaka (Tohoku University)
[abstract] For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993), associated to any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant.

I would like to explain how it is related to central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to multifractal structure of Patterson-Sullivan measures.
4. May 13, 2021: Towards optimal spectral gaps in large genus
Speaker: Michael Lipnowsky (McGill University)
[abstract] I'll discuss recent joint work with Alex Wright (arXiv: 2103.07496) showing that typical large genus hyperbolic surfaces have first Laplacian eigenvalue at least 3/16−ϵ.
5. May 6, 2021 10:00 PDT: Random hyperbolic surfaces via flat geometry
Speaker: Aaron Calderon (Yale University)
[abstract] Mirzakhani gave an inductive procedure to build random hyperbolic surfaces by gluing together smaller random pieces along curves. She proved that as the length of the gluing curve grows, these families equidistribute in the moduli space of hyperbolic surfaces. In this talk, I’ll explain how the conjugacy (exposited in James’s talk) between the earthquake and horocycle flows provides a template for translating equidistribution results for flat surfaces into equidistribution results for hyperbolic ones. Using this correspondence, we address Mirzakhani’s twist torus conjecture and exhibit new limiting distributions for hyperbolic surfaces built out of symmetric pieces. This is joint work (in progress) with James Farre.
6. April 29, 2021 10:00 PDT: Conjugating flows on the moduli of hyperbolic and flat surfaces
Speaker: James Farre (University of Heidelberg)
[abstract] A measured geodesic lamination on a hyperbolic surface encodes the horizontal trajectory structure of certain quadratic differentials. Thurston’s earthquake flow along such a lamination induces a dynamical system on the moduli space of hyperbolic surfaces sharing many properties with the classical Teichmüller horocycle flow. Mirzakhani gave a dynamical correspondence between the earthquake and horocycle flows, defined Lebesgue-almost everywhere. In this talk, we extend Mirzakhani’s conjugacy and define an extension of the earthquake flow to an action of the upper triangular group P in PSL(2,R) mapping certain flow lines to Teichmüller geodesics. We classify the P-invariant ergodic probability measures as those coming from affine invariant measures on quadratic differentials and show that our map is a measurable isomorphism between P actions with respect to these measures. This is joint work with Aaron Calderon.
7. March 11, 2021, 14:30 PDT: Ergodic theorems along trees
Speaker: Anush Tserunyan (University of Illinois at Urbana-Champaign)
[abstract] In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in the forward orbit of the point $x$. In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. For free group actions, this strengthens the best known result in this vein due to Bufetov (2000). After reviewing the subject history and discussing the statements of our theorems in the first half of the talk, we will highlight some ingredients of proofs in the second half.
8. January 21 (14:30), January 28 (14:00), February 4 (14:00), February 11 (14:30), February 18 (14:00), 2021
Note variable starting times
Five-lecture series: Large orbit closures of translation surfaces are strata or loci of double covers
Speakers: Paul Apisa (University of Michigan) and Alex Wright (University of Michigan)
[abstract]
Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.

In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.

We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.

The five lectures will introduce novel and broadly applicable techniques, organized as follows:
1. An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
2. Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
3. Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
4. An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
5. Completion of the proof of the main theorem.
9. July 16, 2020: Stationary measure and orbit closure classification for random walks on surfaces
Speaker: Ping Ngai (Brian) Chung (The University of Chicago)
Slides
[abstract] We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces.

We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the \Out(F_2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.
10. July 2, 2020: Quantitative weak mixing for random substitution tilings
Speaker: Rodrigo Treviño (Maryland)
Slides
[abstract] "Quantitative weak mixing" is the term used to bound the dimensions of spectral measures of a measure-preserving system. This type of study has gained popularity over the last decade, led by a series of results of Bufetov and Solomyak for a large class of flows which include general one-dimensional tiling spaces as well as translation flows on flat surfaces, as well as results on quantitative weak mixing by Forni. In this talk I will present results which extend the results for flows to higher rank parabolic actions, focusing on quantitative results for a broad class of tilings in any dimension. The talk won't assume familiarity with almost anything, so I will define all objects in consideration.
(talk postponed from June 25)
11. June 18, 2020: Counting social interactions for discrete subsets of the plane
Speaker: Samantha Fairchild (Washington)
[abstract] Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will spend the first 40 minutes discussing how these sets arise and counting results arise from the study of concrete translation surfaces. The following 40 minutes will be spent highlighting the proof strategy used to obtain these results, and advertising the generality and strength of this argument that arises from the computation of all higher moments of the Siegel--Veech transform over quotients of SL(2,R) by non-uniform lattices.
12. June 11, 2020: There exists a weakly mixing billiard in a polygon
Speaker: Jon Chaika (Utah)
[abstract] This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to Lebesgue measure on the unit tangent bundle to the billiard. This strengthens Kerckhoff, Masur and Smillie's result that there exists ergodic billiard flows in polygons. The existence of a weakly mixing billiard follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues. This talk will explain the problem, related results, and approach. The talk will not assume familiarity with translation surfaces. This is joint work with Giovanni Forni.
13. June 4, 2020, 13:00 PDT: Arithmetic and geometric properties of planar self-similar sets
Speaker: Pablo Shmerkin (T. Di Tella University and Conicet)
[abstract] Furstenberg's conjecture on the dimension of the intersection of x2,x3-invariant Cantor sets can be restated as a bound on the dimension of linear slices of the product of x2,x3-Cantor sets, which is a self-affine set in the plane. I will discuss some older and newer variants of this, where the self-affine set is replaced by a self-similar set such as the Sierpinski triangle, Sierpinski carpet or (support of) a complex Bernoulli convolution. Among other things, I will show that the intersection of the Sierpinski carpet with circles has small dimension, but on the other hand the Sierpinski carpet can be covered very efficiently by linear tubes (neighborhoods of lines). The latter fact is a recent result joint with A. Pyörälä, V. Suomala and M. Wu.
14. May 28, 2020: Almost-Prime Times in Horospherical Flows
[abstract] There is a rich connection between homogeneous dynamics and number theory. Often in such applications it is desirable for dynamical results to be effective (i.e. the rate of convergence for dynamical phenomena are known). In the first part of this talk, I will provide the necessary background and relevant history to state an effective equidistribution result for horospherical flows on the space of unimodular lattices in R^n. I will then describe an application to studying the distribution of almost-prime times (integer times having fewer than a fixed number of prime factors) in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture. In the second part of the talk I will describe some of the ingredients and key steps that go into proving these results.
15. May 21, 2020: A Bratteli-Vershik model for Z^2 actions, or how cohomology can help us make dynamical systems
Speaker: Ian Putnam (UVic)
[abstract] The Bratteli-Vershik model is a method of producing minimal actions of the integers on a Cantor set. It was given by myself, Rich Herman and Chris Skau, building on seminal ideas of Anatoly Vershik, over 30 years ago. Rather disappointingly and surprisingly, there isn't a good version for Z^2 actions. I'll report on a new outlook on the problem and recent progress with Thierry Giordano (Ottawa) and Christian Skau (Trondheim). The new outlook focuses on the model as an answer to the question: which cohomological invariants can arise from such actions? I will not assume any familiarity with either the original model or the cohomology. The first half of the talk will be a gentle introduction to the Z-case and the second half will deal with how to adapt the question to get an answer for Z^2.
16. May 14, 2020: Gaps of saddle connection directions for some branched covers of tori
Speaker: Anthony Sanchez (UW)
[abstract] Translation surfaces given by gluing two identical tori along a slit have genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of trajectories called saddle connections that are the straight lines trajectories between cone points. We can associate a holonomy vector in the plane to each saddle connection whose components are the horizontal and vertical displacement of the saddle connection. How random is the planar set of holonomy of saddle connections? We study this question by computing the gap distribution for slopes of saddle connections for these and other related classes of translation surfaces.
17. May 7, 2020: Factors of Gibbs measures on subshifts
Speaker: Sophie MacDonald (UBC)
[abstract] Classical results of Dobrushin and Lanford-Ruelle establish, in rough terms, that for a local energy function on a subshift without too much long-range order, the translation-invariant Gibbs measures are precisely the equilibrium measures. There are multiple definitions of a Gibbs measure in the literature, which do not always coincide. We will discuss two of these definitions, one introduced by Capocaccia and the other used by Dobrushin-Lanford-Ruelle, and outline a proof (available at arxiv.org/abs/2003.05532) that they are equivalent.

We will also discuss forthcoming work, in which we show that Gibbsianness is preserved by pushforward through a certain kind of almost invertible factor map. As an application in one dimension, we show that for a sufficiently regular potential, any equilibrium measure on an irreducible sofic shift is Gibbs. As far as we know, this is the first reasonably general result of the Lanford-Ruelle type for a class of subshifts without the topological Markov property.

Joint work with Luísa Borsato, with extensive advice from Brian Marcus and Tom Meyerovitch.
18. April 30, 2020: Quantum Unique Ergodicity
Speaker: Lior Silberman (UBC)
[abstract] In the first half I'll give a colloquium-style introduction to the equidistribution problem for Laplace eigenfucntions on Riemannian manifolds, with emphasis on the locally symmetric spaces. I will introduce positive results for exact eigenfunctions (with and without reference to the number-theoretic symmetries of the manifold), and negative results for approximate eigenfunctions. I will present results (independenlty) joint with A. Venkatesh, N. Anantharaman, and S. Eswarathasan. In the second half I'll answer questions and provide details as requested by the audience.
19. April 23, 2020: Effective equidistribution of horospherical flows in infinite volume
Speaker: Natalie Tamam (UCSD)
Videorecording on MathTube
[abstract] The horospherical flow on finite-volume hyperbolic surfaces is well-understood. In particular, effective equidistribution of non-closed horospherical orbits is known. New difficulties arise when studying the infinite-volume setting. We will discuss the setting in finite- and infinite-volume manifolds, and the measures that play a crucial role in the latter. This is joint work with Jacqueline Warren.

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