a zoom mini-conference

August 3-6, 2020

The online conference Dynamics on your screen" will cover a broad range of topics in dynamical systems and will feature 3 talks each day using the platform zoom. All are welcome to participate. The invitations to zoom talks will be distributed by email to registered participants. Available slides and lecture notes will be posted on this page after the conference concludes. However, we don't plan to video record the lectures.

Organizers: Livio Flaminio (Lille), Andrey Gogolev (Ohio State), Federico Rodriguez Hertz (Penn State)

#### Registration

We ask all participants to register and use their real names while participating in the conference. Registration is free. Please follow this link to register.

### Invited speakers

Danijela Damjanovic (KTH Royal Institute of Technology)
Giovanni Forni (University of Maryland)
Catalina Freijo (Federal University of Minas Gerais)
Colin Guillarmou (Paris-Saclay University)
Asaf Katz (University of Chicago)
Wenyu Pan (University of Chicago)
Yakov Pesin (Pennsylvania State University)
Jana Rodriguez Hertz (Southern University of Science and Technology)
Çağrı Sert (University of Zurich)
Michele Triestino (University of Burgundy)
Kurt Vinhage (Pennsylvania State University)

#### Daily schedule (Monday to Thursday)

 First talk 9:00-9:55am (New York time) = 3:00-3:55pm (Paris time) Second talk 10:05-10:50am (New York time) = 4:05-4:50pm (Paris time) Coffee/tea break 11:00-11:15am (New York time) = 5:00-5:15pm (Paris time) Third talk 11:15-12:00am (New York time) = 5:15-6:00pm (Paris time)

### Schedule

 Monday 8/3 Tuesday 8/4 Wednesday 8/5 Thursday 8/6 First talk Yakov Pesin Danijela Damjanovic Jana Rodriguez Hertz Colin Guillarmou Second talk Çağrı Sert Michele Triestino Asaf Katz Wenyu Pan Coffee/tea break Third talk Olga Paris-Romaskevich Catalina Freijo Kurt Vinhage Giovanni Forni 2-3pm/8-9pm Open problem session

### Open problem session

David Fisher
Classification of smooth group actions with uniform hyperbolicity

Miaohua Jiang
Does the entropy functional of an Anosov map on a torus with respect to its SRB measure have any non-trivial local extrema?

Clark Butler
Classification problem for Anosov diffeomorphisms in dimension 6

Alistair Windsor
Smooth realization of a gaussian process

### Abstracts

Danijela Damjanovic
Global rigidity of some partially hyperbolic abelian actions (slides)

I will discuss global rigidity theme for abelian actions in the context of partially hyperbolic higher-rank actions and present several results and challenges in this direction.This is joint work with Amie Wilkinson and Disheng Xu.

Giovanni Forni
On decay of correlations for parabolic flows (slides)

I will survey recent progress and open questions on mixing, decay of correlations and spectral properties for several classes of smooth parabolic flows given by time-changes of homogeneous unipotent flows and translation flows. In particular, I will discuss joint work on speed of mixing of smooth time-changes of horocycle flows (with Ulcigrai) of Heisenberg nilflows (with Kanigowski), on mixing of smooth time-changes of general ergodic nilflows (with Avila, Ravotti and Ulcigrai) and on the countable Lebesgue spectral type of a class of Kochergin flows (with Fayad and Kanigowski).

Catalina Freijo
Lyapunov exponents for non-uniformly fiber-bunched cocycles (slides)

We consider a fixed hyperbolic dynamic in the base and study how the Lyapunov exponents vary as functions of the cocycle. In this context, Backes, Brown and Butler have proved that the Lyapunov exponents are continuous when restricted to the cocycles with both stable and unstable uniform holonomies. A conjecture of Marcelo Viana states that this condition on the holonomies can be relaxed asking only for an holonomy that variates continuously with the cocycle. We provide evidence that the conjecture is true by proving some partial results in this direction over non- uniformly fiber-bunched cocycles. This is a joint work with Karina Marin (UFMG).

Colin Guillarmou
The Ruelle-Taylor joint spectrum for Anosov actions (slides)

We define a notion of joint spectrum for Anosov R^k actions using Koszul complexes (and the so-called Taylor spectrum) on certain functional spaces called anisotropic Sobolev spaces. We show that this joint spectrum is an intrinsic discrete subset of C^k and study the particular value $\lambda=0$. This leads to some new SRB type measures and the question of mixing can be interpreted spectrally using this theory. Joint work with Bonthonneau, Hilgert and Weich.

Asaf Katz
Measure rigidity of Anosov flows via the factorization method (slides)

Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a negatively curved surface. In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows. Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold. In the talk I will discuss the factorization method, the relations to previous works (Eskin-Mirzakhani, Eskin-Lindenstrauss) and the result together with some examples and applications.

Wenyu Pan
Exponential mixing of geodesic flow for geometrically finite manifolds with cusps (slides)

Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with cusps. In the joint work with Jialun Li, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$. Previously, such results were proved by Stoyanov for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for $\Gamma$ with large critical exponent. We obtain our result by constructing a nice coding for the geodesic flow and then prove a Dolgopyat-like spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding, which is partly inspired by the works of Lai-Sang Young and Burns-Masur-Matheus-Wilkinson. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.

Slow holomorphic dynamics and polynomial entropy (slides)

The maps with chaotic dynamics represent a great interest and have been studied in detail, in various contexts. In this talk, I will be interested in the maps living on the opposite edge of the spectrum — maps with the dynamics of low complexity. They can be defined, for example, as those that have zero topological entropy. To quantify the complexity of such maps, one can consider a slower, polynomial entropy. I will give an introduction to polynomial entropy, with a focus on the low complexity holomorphic dynamics, following my recent work with Serge Cantat.

Yakov Pesin
The Smooth Realization Problem: Area Preserving Diffeomorphisms with Polynomial Decay of Correlations (slides)

Given any smooth compact connected and oriented surface, I present a construction of an area preserving diffeomorphism with non-zero Lyapunov exponents, which is Bernoulli and has polynomial decay of correlations, satisfies the Central Limit Theorem and has the Large Deviation Property. In addition, this diffeomorphism possesses a unique hyperbolic Bernoulli measure of maximal entropy with respect to which it has exponential decay of correlations. This is a joint work with Samuel Senti and Farruh Shahidi.

Jana Rodriguez Hertz
Stable ergodicity beyond partial hyperbolicity (slides)

The first known mechanism implying stable ergodicity of the volume measure is hyperbolicity. It took time to obtain new examples: partially hyperbolic diffeomorphisms. Pugh and Shub conjectured that partial hyperbolicity generically implied stable ergodicity, a conjecture that was proved by Avila, Crovisier and Wilkinson. Non-partially hyperbolic stably ergodic diffeomorphisms were also known to exist, but so far all the known examples were DA-diffeomorphisms. We will discuss different mechanisms other than partial hyperbolicity that could generically imply stable ergodicity with respect to the volume measure. In fact, we conjecture that very little hyperbolicity, namely, positive metric entropy, generically implies stable ergodicity. While this conjecture still seems far to be solved, we show new examples of stably ergodic diffeomorphisms. For instance, in 3-dimensional manifolds, there is a stably ergodic non-partially hyperbolic diffeomorphism in the isotopy class of each partially hyperbolic diffeomorphism. In particular, there are such examples in 3-manifolds other than the 3-torus. On the other hand, we conjecture that the existence of a minimal invariant expanding foliation generically implies stable ergodicity. We have proven this conjecture for 3-dimensional manifolds, and there are some advances for higher dimensional manifolds. This is part of different joint works with G. Núñez and D. Obata.

Cagri Sert
Equilibrium measures of affine fractals (slides)

We will start by giving an overview of dimension theory and equilibrium measures of similarity and affine fractals making connections with repellers of expanding maps. In a second part, we will explain a dimension gap result yielding in particular a sort of converse to a classical result of Hutchinson: under an irreducibility assumption, an affine fractal admitting a Bernoulli equilibrium measure with maximal dimension is a similarity fractal. In the last part, I will explain some ways to turn around this gap by consideration of subsystems. Based on joint works with Ian Morris.

Michele Triestino
Actions of locally moving groups of homeomorphisms of the real line (slides)

In a recent project in collaboration with Joaquín Brum, Nicolás Matte Bon, and Cristóbal Rivas, we investigate the structure of the space of actions on the real line of locally moving groups. A group of homeomorphisms of the real line is locally moving if, for every open interval, there exists a subgroup acting on it without global fixed points, and acting trivially in restriction to its complement. Classical examples are provided by Thompson's group F, but even by large groups, such as the group of compactly supported homeomorphisms of the real line. As a consequence of classical work of Rubin, every isomorphism between two locally moving groups is realized by a homeomorphism, and in particular every locally moving group admits a unique locally moving action. What about the other actions?

Kurt Vinhage
Complete classification of transitive, totally Cartan actions (slides)

I will discuss recent progress in rigidity theory for actions of abelian groups. After several results on local rigidity of lattice and abelian group actions in the 80s and early 90s, Katok and Spatzier conjectured that in fact *all* hyperbolic actions without rank one factors are smoothly conjugate to an algebraic model. We verify the conjecture for all transitive, totally Cartan actions: ones for which there are many hyperbolic elements, and for which a dynamical splitting is simple. This is the first time the conjecture can be proven without strong ergodicity or perturbative assumptions. We furthermore classify all transitive, totally Cartan actions, even those with rank one factors. They are all built from direct products of Anosov flows and diffeomorphisms and homogeneous actions. Joint with Ralf Spatzier.

### Participants

1. Aaron Brown
7. Agnieszka Zelerowicz
8. Albert Artiles
9. Alejandro Kocsard
10. Alejo García
11. Alena Erchenko
12. Alessandro Arsie
13. Alex Furman
14. Alex Zamudio
15. Alexander Arbieto
16. Alexander Cantoral Vilchez
17. Alexander Gorodnik
18. Alfonso Artigue
19. Ali Tahzibi
20. Alisson Pereira
21. Alistair Windsor
22. Alp Uzman
23. Amie Wilkinson
24. Amin Talebi
25. Ana Carolina Ramos
26. Anastasios Stylianou
27. Anders Karlsson
28. Andre Oliveira
29. Andreas Wieser
30. Andrés Bellei
31. Andres Koropecki
32. Angel Pardo
33. Anibal Velozo
34. Anindya Chanda
35. Aritro Pathak
36. Artem Dudko
37. Asgar Jamneshan
38. Aygul Galimova
39. Barak Weiss
40. Bárbara Núñez
41. Benedict Sewell
42. Benjamin Call
43. Bernardo Carvalho
44. Biao MA
45. Boris Hasselblatt
46. Boris Kalinin
47. Boris Kruglikov
48. Brayan Mauricio Rodriguez
49. Bruce Peckham
50. Bruno Yemini
51. C Cheng
52. Cagri Sert
53. Carllos Eduardo Holanda
54. Carlos H. Vasquez
55. Carlos Matheus
56. Carsten Peterson
57. Cássio Morais
58. Cesar Romero Mora
59. Changguang Dong
60. Charles Walkden
61. Christoforos Neofytidis
62. Christophe Golé
63. Clark Butler
64. Cláudia Ferreira
65. Connor Jackman
66. Constantin Kogler
67. Corinna Ulcigrai
68. Cristina Lizana
69. Cristóbal Rivas
70. Daniel Alvey
71. Daniel Ingebretson
72. Daniel Reis de Oliveira
73. Danijela Damjanovic
74. Danny Stoll
75. Danyu Zhang
76. Daren Wei
77. Dave Constantine
78. Davi Obata
79. David Fisher
80. Débora de Oliveira
81. Deniel Correa
82. Deokwon Sim
84. Diaaeldin Taha
85. Diego Lugo
86. Dmitri Scheglov
87. Dong Chen
88. el Houcein el Abdalaoui
89. Elias Rego
90. Elizabeth S Flores
91. Elon Lindenstrauss
92. Elmer Beltran
93. Emilio Corso
94. Erez Nesharim
95. Eric Albers
96. Eric Cabezas
97. Erwan Lanneau
98. Fabrizio Bianchi
99. Feliks Przytycki
100. Ferrán Valdez
101. Farruh Shahidi
102. Fernando Lenarduzzi
103. Fernando Micena
104. Filiz Dogru
105. Francois Ledrappier
106. Frank Trujillo
107. Freddy Castro
108. Freddy Vicente
109. Gerhard Knieper
110. Gil Astudillo
111. Grace Work
112. Greg Davis
113. Guido Marinoni
114. Habibeh Pourmand
115. Hector Sanchez
116. Hellen de Paula
117. Henrik Kreidler
119. Hiba Fayoumi
120. Homin Lee
121. Hongkun Zhang
122. Hugo Araújo
123. Hugo Carrillo
124. Ibai Aedo
125. Inti C. Diaz
126. Ivan Gonzalez
127. Jack Burkart
128. Jacob Mazor
129. Jacqueline Warren
131. Jairo Bochi
132. Jamerson Bezerra
133. Jana Rodriguez Hertz
134. Jane Hawkins
135. Jaqueline Siqueira
136. Javier Saenz
137. Jean Lafont
138. Jennyffer Bohorquez
139. Jialun LI
140. Jianlong Liu
141. Jiaqi Yang
142. Jieun Seong
143. Jimi Daniel Gomez
144. Jimmy Santamaria
145. Jimmy Tseng
146. Jing Zhou
147. Jiyoung Hanv
148. Joan Gimeno
149. Joao Rijo
150. João Lopes Dias
151. Joaquín brum
153. Joaquín Lejtreger
154. Joaquín Lema Perez
155. John Hynes
156. Jon Chaika
157. Jonathan DeWitt
158. Jorge Gonzalez
159. Josh Southerland
160. Juan Carlos Morelli Ramírez
161. Juan Carlos Salcedo Sora
162. Juan Paucar
163. Juan Tolosa
164. Kari Kuester
165. Karin Reinhold
166. Karina Marin
167. Karl Petersen
168. Keith Burns
169. Keivan Mallahi-Karai
170. Kelly Yancey
172. Kiho Park
173. Konstantin Khanin
174. Kriti Sehgal
175. Kursat Yilmaz
176. LASSALLE Philippe
177. Lautaro Cilenti
178. Lin Shu
179. Lei Yang
180. Leonardo Dinamarca
181. Leydiane Campos
183. Mahdi Teymuri Garakani
184. Mahesh Nerurkar
185. Maheshan Ekanayaka
186. Maisam Hedyehloo
187. Manfred Denker
188. Manuel Jesús Saavedra Jiménez
190. Marco López
191. María Isabel Cortez
192. Mark Bell
193. Mark Pollicott
194. Marlies Gerber
195. Martha Łącka
196. Martin Leguil
197. Martyna Górska
198. Mary He
199. Mauricio Genta
200. Mauro Artigiani
201. Meysam Nassiri
202. Miaohua Jiang
203. Michael Jakobson
204. Michele Triestino
205. Mihajlo Cekić
206. Minsung Kim
207. Mitul Islam
208. Nattalie Tamam
209. Nawaf Alansari
210. Nayana Wanasingha
211. Nelda Jaque
212. Nestor Nina
213. Nguyen-Thi Dang
215. Nicolas Monod
216. Nikolai Edeko
217. Noy Soffer Aranov
218. Nyima Kao
219. Odylo Costa
221. Olga Lukina
222. Osama Khalil
223. Ozkan Demir
224. Pablo D. Carrasco
225. Pallavi Panda
226. Paulo Varandas
227. Peng Sun
228. Philipp Kunde
229. Ping Ngai (Brian) Chung
230. Polina Vytnova
231. Porfirio Toledo
232. Predrag Punosevac
233. Rachel McEnroe
235. Rafael de la Llave
236. Rafael Pereira
237. Rafael Potrie
238. Rainer Nagel
239. Ralf Spatzier
240. Raul Ures
241. Renato Velozo
243. Rhiannon Dougall
244. Richard Buckalew
245. Richard Montgomery
246. Richard Sharp
247. Robert Leek
248. Robert Niemeyer
250. Rodrigo Bissacot
251. Rodrigo Treviño
252. Roland Gunesch
254. Ruxi Shi
255. Salman Siddiqi
256. Sam Dodds
257. Samuel Everett
258. Santiago Martinchich
259. Santiago Montouliu
260. Satbir Malhi
261. Saúl Quispe
262. Sebastian Burgos
264. Sebastian Ramirez
265. Sébastien Labbé
266. Sejal babel
267. Seth Chaiken
268. ShengYuan ZHAO
269. Shilpak Banerjee
271. SHREYASI DATTA
272. Simion Filip
273. Simon Baker
274. Sinisa Slijepcevic
275. Sixu Liu
276. Siyuan Tang
277. Solly Coles
278. Sonia Pinto-de-Carvalho
279. Stefano Luzzatto
280. Steve Hurder
281. Sumana Kundu
282. Svetlana Katok
283. Tanja Eisner
284. Tejbir Lohan
285. Thang Nguyen
286. Thomas Barthelmé
287. Thomas Richards
288. Tsviqa Lakrec
289. Tulio Ferreira
290. Valéria Santos
291. Victor Castillo
292. Victor Kleptsyn
293. Victor Vilaça Da Rocha
295. Vincent Delecroix
296. Vincent Pecastaing
297. Weisheng Wu
298. Welington Cordeiro
299. Wenyu Pan
300. Yakov Pesin
301. Yang Yang
302. Yi PAN
303. Ying Sue Huang
304. Yu-Wei Fan
305. Yulia Meshkova
306. Yun Yang
307. Yuping Ruan
308. Yuting Fang