We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms x↦ bx 1 , where the integer b is large. We obtain estimates of the fibered Lyapunov exponents and show that in the limit as b tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on b.

It is well known that a real analytic symplectic diffeomorphism of the $2d$ -dimensional disk ( $d\geq 1$ ) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when $d=1$ the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

AbstractLet f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

We introduce a class of real analytic "peaky" potentials for which the corresponding quasiperiodic 1D-Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

This is the second of two volumes which celebrate the memory of Jean-Christophe Yoccoz. These volumes present research articles on various aspects of the theory of dynamical systems and related topics that were dear to him.

We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems. In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic. The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

We introduce a class of real analytic "peaky" potentials for which the corresponding quasi-periodic 1D Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is C ∞ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the C ∞ -topology.

Our aim in this note is to present some of the crucial contributions of Jean-Christophe Yoccoz to the theory of circle diffeomorphisms. We start with a short historical account before exposing Yoccoz' work. Then we give a brief description of the main conceptual and technical tools of the theory, with a focus on describing Yoccoz' work and contributions.

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 SL2(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”. 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. 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 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. 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. 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. 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). 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 nor

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori.We show that an analytic invariant torus T-0 with Diophantine frequency omega(0) is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T-0 satisfies a Rassmann transversality condition, the torus T-0 is accumulated by Kolmogorov-Arnold-Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least d + 1 that is foliated by analytic invariant tori with frequency omega(0).For frequency vectors coo having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian H satisfies a Kolmogorov nondegeneracy condition at T-0, then T-0 is accumulated by KAM tori of positive total measure.In four degrees of freedom or more, we construct for any omega(0) is an element of R-d, C-infinity (Gevrey). Hamiltonians H with a smooth invariant torus T-0 with frequency omega(0) that is not accumulated by a positive measure of invariant tori.

We develop a “local theory” of multidimensional quasiperiodic \({\mathrm {SL}}(2,{\mathbb R})\) cocycles which are not homotopic to a constant. It describes a \(C^1\)-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of \({\mathrm {SL}}(2,{\mathbb R})\) cocycles homotopic to a constant. Our analysis is centered around the notion of monotonicity with respect to some dynamical variable. For such monotonic cocycles, we obtain a sharp rigidity result, minimality of the projective action, typical nonuniform hyperbolicity, and a surprising result of smoothness of the Lyapunov exponent (while no better than Hölder can be obtained in the case of cocycles homotopic to a constant, and only under arithmetic restrictions). Our work is based on complexification ideas, extended “à la Lyubich” to the smooth setting (through the use of asymptotically holomorphic extensions). We also develop a counterpart of this theory centered around the notion of monotonicity with respect to a parameter variable, which applies to the analysis of \({\mathrm {SL}}(2,{\mathbb R})\) cocycles over more general dynamical systems and generalizes key aspects of Kotani Theory. We conclude with a more detailed discussion of one-dimensional monotonic cocycles, for which results about rigidity and typical nonuniform hyperbolicity can be globalized using a new result about convergence of renormalization.

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω 0 is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω 0 has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω 0 . This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.

We develop a new KAM scheme that applies to SL(2, R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

In this paper we introduce a new technique that allows us to investigate reducibility properties of smooth SL(2, R)-cocycles over irrational rotations of the circle beyond the usual Diophantine conditions on these rotations.For any given irrational angle on the base, we show that if the cocycle has bounded fibered products and if its fibered rotation number belongs to a set of full measure Sigma(alpha), then the matrix map can be perturbed in the C(infinity) topology to yield a C(infinity)-reducible cocycle. Moreover, the cocycle itself is almost rotations-reducible in the sense that it can be conjugated arbitrarily close to a cocycle of rotations. If the rotation on the circle is of super-Liouville type, the same results hold if instead of having bounded products we only assume that the cocycle is L(2)-conjugate to a cocycle of rotations.When the base rotation is Diophantine, we show that if the cocycle is L(2)-conjugate to a cocycle of rotations and if its fibered rotation number belongs to a set of full measure, then it is C(infinity)-reducible. This extends a result proven in [5].As an application, given any smooth SL(2, R)-cocycle over a irrational rotation of the circle, we show that it is possible to perturb the matrix map in the C(infinity) topology in such a way that the upper Lyapunov exponent becomes strictly positive. The latter result is generalized, based on different techniques, by Avila in [1] to quasiperiodic SL(2, R)-cocycles over higher-dimensional tori.Also, in the course of the paper we give a quantitative version of a theorem by L. H. Eliasson, a proof of which is given in the Appendix. This motivates the introduction of a quite general KAM scheme allowing to treat bigger losses of derivatives for which we prove convergence.

We present a proof of Herman's Last Geometric Theorem asserting that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F-invariant smooth curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable feature of this theorem is that it does not require any twist assumption.

In this paper we investigate the exponential growth of products of two matrices . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than nα, 0 ⩽ α < 1/2 matrices B are uniformly bounded from below by γn for some γ > 1.