SCHEDULE OF TALKS. Semi-annual Workshop in Dynamical Systems and Related Topics Pennsylvania State University, October 4-7, THURSDAY, October 4

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1 SCHEDULE OF TALKS Semi-annual Workshop in Dynamical Systems and Related Topics Pennsylvania State University, October 4-7, 2018 THURSDAY, October 4 1:00-1:50 Registration (in 114 McAllister) 1:50-2:00 Opening Remarks (in 114 McAllister) 2:00-2:50 Manfred Einsiedler (in 114 McAllister) Rigidity of higher rank actions and transport of entropy 3:00-3:30 DEPARTMENTAL TEA 3:35-4:25 Department of Mathematics Colloquium (in 114 McAllister) Giovanni Forni On mixing and spectral properties of smooth parabolic flows 4:40-5:30 Svetlana Katok (in 114 McAllister) Coding of geodesics via continued fractions and their generalizations FRIDAY, October 5 9:00-9:50 Benjamin Weiss (in 114 McAllister) On the Approximation by Conjugation Method 10:00-10:30 COFFEE BREAK 10:30-11:20 Alex Eskin (in 114 McAllister) On stationary measure rigidity and orbit closures for actions of non-abelian groups 11:30-12:20 Helmut Hofer (in 114 McAllister) Feral pseudoholomorphic curves as a bridge between dynamics and topology 12:30-2:00 LUNCH BREAK 2:00-2:50 Ralf Spatzier (in 113 Carnegie) On some rigidity problem in geometry and dynamics 3:00-3:50 Bassam Fayad (in 113 Carnegie) Instabilities in analytic quasi-periodic Hamiltonian dynamics 4:00-4:25 COFFEE BREAK 4:25-4:55 Special Session: Inauguration of Anatole Katok Center for Dynamical Systems and Geometry (in 113 Carnegie) 5:00-5:50 Boris Hasselblatt (in 113 Carnegie) Anatole Katok a half-century of dynamics 7:00-9:00 Banquet at Atherton Hotel

2 SATURDAY, October 6 9:00-9:50 Amie Wilkinson (in 102 Thomas) Pathology and asymmetry 10:00-10:30 COFFEE BREAK 10:30-11:20 Raphael Krikorian (in 102 Thomas) On the divergence of Birkhoff Normal Forms 11:30-12:20 Hee Oh (in 102 Thomas) Orbit closures of the SL(2, R) action on hyperbolic manifolds 12:30-2:00 LUNCH BREAK 2:00-2:50 Peter Sarnak (in 102 Thomas) Integer points on Markoff type cubic surfaces and dynamics Special Session: Michael Brin Dynamical Systems Prize for Young Mathematicians 3:10-3:30 Award Ceremony (in 102 Thomas) 3:30-4:00 COFFEE BREAK 4:00-4:50 Talk 1 (in 102 Thomas) 5:00-5:50 Talk 2 (in 102 Thomas) SUNDAY, October 7 9:00-9:50 Hillel Furstenberg (in 114 McAllister) Strong proximality and strong distality of dynamical systems 10:00-10:50 Danijela Damjanovic (in 114 McAllister) On Katok-Spatzier global rigidity conjecture for abelian Anosov actions and related questions 11:00-11:30 COFFEE BREAK 11:30-12:20 Jean-Paul Thouvenot (in 114 McAllister) The generic extension of a Bernoulli shift is relatively mixing 12:30-1:20 Elon Lindenstrauss (in 114 McAllister) Joinings of higher rank diagonalizable actions

3 TITLES AND ABSTRACTS Danijela Damjanovic, Royal Institute of Technology (KTH), Sweden Title: On Katok-Spatzier global rigidity conjecture for abelian Anosov actions and related questions Abstract: Katok and Spatzier conjectured in the 90 s that up to smooth conjugacy the only occurrence of Anosov discrete abelian group actions (modulo degenerate essentially rank-one situations) is algebraic: in (affine) centralizers of hyperbolic automorphisms of infranilmanifolds (in particular, tori). The conjecture is completely resolved by Rodriguez Hertz and Wang when the underlying manifold is infranilmanifold. We will discuss some new findings in case of a general compact manifold, and some related results for certain classes of partially hyperbolic actions. This is joint work with D. Xu. Manfred Einsiedler, ETH Zürich, Switzerland Title: Rigidity of higher rank actions and transport of entropy Abstract: We consider higher rank actions on irreducible arithmetic quotients of SL 2 (R). If the quotient is compact, positive entropy of an ergodic invariant measure µ implies algebraicity of µ with semisimple stabiliser. For non-compact quotients more possibilities appear. The main novelty is that the acting group does not have to be maximal or in a special position. The main new idea is to use a quantitative recurrence phenomenon to transport positivity of entropy for one acting element to another. This is joint work with Elon Lindenstrauss. Alex Eskin, University of Chicago Title: On stationary measure rigidity and orbit closures for actions of non-abelian groups Abstract: I will describe joint work in progress with Aaron Brown, Federico Rodriguez-Hertz and Simion Filip. Our aim is to find some analogue, in the context of smooth dynamics, of Ratner s theorems on unipotent flows. This would be a (partial) generalization of the results of Benoist-Quint and my work with Elon Lindenstrauss in the homogeneous setting, the results of Brown and Rodriguez-Hertz in dimension 2, and my results with Maryam Mirzakhani in the setting of Teichmuller dynamics. Bassam Fayad, CNRS, IMJ-PRG, France Title: Instabilities in analytic quasi-periodic Hamiltonian dynamics Abstract: We introduce a new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom. Using this mechanism, we obtain the first examples of real analytic Hamiltonians that have a Lyapunov unstable non-resonant elliptic equilibrium. We also give examples of real analytic invariant quasi-periodic tori of arbitrary frequency vectors that are Lyapunov unstable but have a polynomial Birkhoff normal form. Giovanni Forni, University of Maryland Title: On mixing and spectral properties of smooth parabolic flows Abstract: We will survey several recent results on mixing, decay of correlations, and spectral properties of several examples of parabolic flows, in particular time-changes of nilflows, horocycle flows and flows on surfaces. Most of these results were directly inspired by questions or conjectures of Anatole Katok and motivated by his vision of a parabolic paradigm.

4 Hillel Furstenberg, Hebrew University of Jerusalem, Israel Title: Strong proximality and strong distality of dynamical systems Abstract: The notions of proximality, strong proximality, and distality are fairly well developed. For many groups one can identify all strongly proximal actions, and in some sense one can describe explicitly all distal actions for any group. The missing notion is strong distality. A possible definition is that whenever one probability measure is in the orbit closure (in weak topology) of another measure, then the second is in the orbit closure of the first. This implies distality, but a simple example shows that it isn t equivalent to distality. We discuss the question: Does strong distality imply equicontinuity? Boris Hasselblatt, Tufts University Title: Anatole Katok a half-century of dynamics Abstract: The breadth of mathematics Anatole Katok practiced is remarkable, and even more so is his outsize impact as he influenced, shaped and promoted dynamical systems and its practitioners. Helmut Hofer, Institute for Advanced Study Title: Feral pseudoholomorphic curves as a bridge between dynamics and topology Abstract: Theories like Symplectic Field Theory use periodic orbits to build symplectic invariants for odd-dimensional manifolds with a stable Hamiltonian structure and symplectic cobordisms between them. Having a stable Hamiltonian structure is a rather strong condition, but one knows that without it, periodic orbits might not exist, i.e. the building blocks for the theory are gone. It is, of course, hard to believe that a symplectic cobordism suddenly ceases to have any meaningful symplectic properties. In this talk we present strong evidence thatthere isstill a lot of structure which interestingly is related to important dynamical questions. A new class of feral pseudoholomorphic curves relates symplectic properties to more general closed invariant subsets. As one of theapplications we answer a question raised by M. Herman during his 1998 ICM talk by showing that a compact regular Hamiltonian energy surface in R 4 has a proper closed invariant subset. This is joint work with Joel W. Fish, UMB. Svetlana Katok, Pennsylvania State University Title: Coding of geodesics via continued fractions and their generalizations Abstract: I will discuss a method of coding of geodesics on quotients of the hyperbolic plane by Fuchsian groups using boundary maps and reduction theory. For the modular surface these maps are related to a family of (a, b)-continued fractions, and for compact surfaces they are generalizations of the Bowen-Series maps, also studies by Adler and Flatto. The boundary maps are given by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuities the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure to which (almost) every point is mapped after finitely many iterations. These two properties belong to the list of notions of good reduction algorithms, equivalence or implications between which were suggested by Don Zagier. I will also explain how the geodesic flow can be represented symbolically as a special flow over a cross-section of reduced geodesics parametrized by the corresponding attractor, and give some applications. The talk is based on joint works with Ilie Ugarcovici and Adam Abrams.

5 Raphael Krikorian, University of Cergy-Pontoise, France Title: On the divergence of Birkhoff Normal Forms Abstract: An analytic hamiltonian system (or a symplectic diffeomorphism) admitting an elliptic fixed point is always formally conjugated to a formal integrable normal form, the Birkhoff Normal Form. It is know since Siegel (1954) that the formal conjugacy cannot in general converge and H. Eliasson asked whether the Birkhoff Normal Form itself could be divergent. Perez-Marco (2001) proved that for any given frequency vector at the origin, one has the following dichotomy: either the BNF always converges or it generically diverges and Gong (2012) exhibited a divergent example with Liouville frequency vector. I will explain in this talk the proof of the following theorem: given any diophantine frequency vector at the origin, the BNF is generically divergent. Elon Lindenstrauss, Hebrew University of Jerusalem, Israel Title: Joinings of higher rank diagonalizable actions Abstract: Higher rank diagonalizable actions have subtle rigidity properties which are quite hard to understand. One aspects where the current state of knowledge is quite satisfactory is the study of joinings of such actions, where Einsiedler and I have a rather general classification of ergodic joinings. This classification has several striking applications, I will describe two: the work of Aka, Einsiedler, and Shapira studying joint distribution of integer points on a two dimensional sphere and the shape of its orthogonal lattice and recent work of Khayutin on orbits of the class group on pairs of CM points. Hee Oh, Yale University Title: Orbit closures of the SL(2, R) action on hyperbolic manifolds Abstract: We will discuss the action of SL(2, R) on the quotient space X = Γ \ SL(2, C) for a discrete subgroup Γ < SL(2, C). More precisely, let Γ < SL(2, C) be a convex cocompact acylindrical Kleinian group, and let F be the minimal open SL(2, R)-invariant subset of X above the interior of the convex core of the hyperbolic manifold Γ \ H 3. We classify all possible closures of SL(2, R) orbits in F. An immediate consequence is the classification of all possible closures of geodesic planes in the interior of the core of Γ \ H 3. By Mostow rigidity, there are only countably many lattices in SL(2, C) up to conjugation and in those cases, these results were proved by Ratner and Shah independently almost 30 years ago. Our results present the first quasi-isometry invariant family of uncountably many Kleinian manifolds for which a strong topological rigidity of geodesic planes is established. This talk is based on joint work with McMullen and Mohammadi. Peter Sarnak, Princeton University and Institute for Advanced Study Title: Integer points on Markoff type cubic surfaces and dynamics Abstract: Markoff cubic surfaces have an action of a group of affine morphisms defined over Z, which allows one to study the integral points on the surface. We will examine the Hasse Principle and strong approximation (joint with Ghosh and with Bourgain and Gamburd).

6 Ralf Spatzier, University of Michigan Title: On some rigidity problem in geometry and dynamics Abstract: I will discuss rank rigidity results in dynamics and geometry, and some of the underlying geometric and dynamical tools and ideas, in particular measurable normal forms and holonomy. These results and ideas are closely related to the framework of geometric rigidity formulated by Anatole Katok in the 1980s. Jean-Paul Thouvenot, LPSM Sorbonne Université, France Title: The generic extension of a Bernoulli shift is relatively mixing Abstract: It has been shown, as a relative version of a theorem of Halmos, by E. Glasner and B. Weiss, that the generic extension of an ergodic transformation is relatively weakly mixing. (An earlier, less precise, statement had been shown by M. Schnurr). The theorem of Halmos that mixing is rare does not relativize. Benjamin Weiss, Hebrew University of Jerusalem, Israel Title: On the Approximation by Conjugation Method Abstract: In one of the first papers that Tolya published with Dmitry Anosov he introduced a powerful technique of constructing diffeomorphisms with prescribed dynamical properties that he later called the Approximation by Conjugation Method. This method was used in the 1970 s by Albert Fathi and Michael Herman to construct minimal and strictly ergodic diffeomorphisms on a wide class of compact manifolds. The method evolved into a very flexible tool and was has been widely used. I will give a brief survey of some of these developments, including recent joint work with Matt Foreman on classic problems raised by John von Neumann in his foundational paper of Amie Wilkinson, University of Chicago Title: Pathology and asymmetry Abstract: A foliation F of a compact manifold M is pathological is there is a full volume subset of M that meets every local leaf of F in finitely many points. Anatole Katok first showed that a diffeomorphism can have pathological invariant foliations, and Mike Shub and I discovered pathological foliations that persist under perturbations of the dynamics. I will discuss pathological foliations and some subsequent work I have done in collaboration with Avila, Viana, Damjanovic and Xu relating pathology with rigidity and asymmetry.