arXiv subjects
Publications and source records attributed to Amie Wilkinson.
In this paper, we prove centralizer rigidity for any non-trivial diagonal map in the compact homogeneous space SL_n\mathbb{R}/\Gamma, n\ge 5, under the assumption that the perturbed volume-preserving diffeomorphism has a regular isomorphic centralizer.
Let $M$ be a compact manifold without boundary equipped with a Riemannian metric $g$ of negative curvature. In this paper, we introduce the marked Poincar\'e determinant (MPD), a homothety invariant of $g$ depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in $M$ a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local MPD rigidity result in dimension 3: if $g$ is sufficiently close to a hyperbolic metric $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.
Let Aff(X) be the group of affine diffeomorphisms of a closed homogeneous manifold X=G/B admitting a G-invariant Lebesgue-Haar probability measure $\mu$. For $f_0\in$ Aff(X), let $Z^\infty(f_0)$ be the group of $C^\infty$ diffeomorphisms of X commuting with $f_0$. This paper addresses the question: for which $f_0\in$ Aff(X) is $Z^\infty(f_0)$ a Lie subgroup of $Diff^\infty(X)$? Among our main results are the following. (1) If $f_0\in$ Aff(X) is weakly mixing with respect to $\mu$, then $Z^\infty(f_0)<$ Aff(X), and hence is a Lie group. (2) If $f_0\in$ Aff(X) is ergodic with respect to $\mu$, then $Z^\infty(f_0)$ is a (necessarily $C^0$ closed) Lie subgroup of $Diff^\infty(X)$ (although not necessarily a subgroup of Aff(X)). (3) If $f_0\in$ Aff(X) fails to be a K-system with respect to $\mu$, then there exists $f\in$ Aff(X) arbitrarily close to $f_0$ such that $Z^\infty(f)$ is not a Lie group, containing as a continuously embedded subgroup either the abelian group $C^\infty_c((0,1))$ (under addition) or the simple group $Diff^\infty_c((0,1))$ (under composition). (4) Considering perturbations of $f_0$ by left translations, we conclude that $f_0$ is stably ergodic if and only if the condition $Z^\infty<$ Aff(X) holds in a neighborhood of $f_0$ in Aff(X). (Note that by BS97, Dani77, $f_0\in$ Aff(X) is stably ergodic in Aff(X) if and only if $f_0$ is a K-system.) The affine K-systems are precisely those that are partially hyperbolic and essentially accessible, belonging to a class of diffeomorphisms whose dynamics have been extensively studied. In addition, the properties of partial hyperbolicity and accessibility are stable under $C^1$-small perturbation, and in some contexts, essential accessibility has been shown to be stable under smooth perturbation. Considering the smooth perturbations of affine K-systems, we outline a full program for (local) centralizer rigidity.
We study the topological properties of expanding invariant foliations of $C^{1+}$ diffeomorphisms, in the context of partially hyperbolic diffeomorphisms and laminations with $1$-dimensional center bundle. In this first version of the paper, we introduce a property we call *s-transversality* of a partially hyperbolic lamination with $1$-dimensional center bundle, which is robust under $C^1$ perturbations. We prove that under a weak expanding condition on the center bundle (called *some hyperbolicity*, or "SH"), any s-transverse partially hyperbolic lamination contains a disk tangent to the center-unstable direction (Theorem C). We obtain several corollaries, among them: if $f$ is a $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center, and the (strong) unstable foliation $W^{uu}$ of $f$ is minimal, then $W^{uu}$ is robustly minimal under $C^1$-small perturbations, provided that the stable and strong unstable bundles are not jointly integrable (Theorem B). Theorem B has applications in our upcoming work with Eskin, Potrie and Zhang, in which we prove that on ${\mathbb T}^3$, any $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center has a minimal strong unstable foliation, and has a unique $uu$-Gibbs measure provided that the stable and strong unstable bundles are not jointly integrable. In a future work, we address the density (in any $C^r$ topology) of minimality of strong unstable foliations for $C^{1+}$ partially hyperbolic diffeomorphisms with $1$-dimensional center and the SH property.
We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension $f_0\colon M\to M$ of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any $f\in \mathrm{Diff}^\infty(M)$ sufficiently $C^1$-close to $f_0$ has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then $f$ is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we further classify the centralizer of any partially hyperbolic diffeomorphism on a $3$-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually $\mathbb Z\times \mathbb T$.
We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with $1$-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -- including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually $\mathbb Z^\ell$ or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered in [2,3] for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25]. We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.
We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Mañé. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
In this paper, we study a natural class of groups that act as affine transformations of $\mathbb T^N$. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on $\mathbb T^N$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $C^r$ diffeomorphims can be conjugated to the affine action by $C^{r-ε}$ conjugacy. Next, we show that in any dimension, $C^1$ small perturbations can be conjugated to an affine action via $C^{1+ε}$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.
We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $ν$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hatν$ is any lift of $ν$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat ν$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $Σ$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.
We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.
We use an effective Hamiltonian to characterize particle dynamics and find escape rates in a periodically kicked Hamiltonian. We study a model of particles in storage rings that is described by a chaotic symplectic map. Ignoring the resonances, the dynamics typically has a finite region in phase space where it is stable. Inherent noise in the system leads to particle loss from this stable region. The competition of this noise with radiation damping, which increases stability, determines the escape rate. Determining this `aperture' and finding escape rates is therefore an important physical problem. We compare the results of two different perturbation theories and a variational method to estimate this stable region. Including noise, we derive analytical estimates for the steady-state populations (and the resulting beam emittance), for the escape rate in the small damping regime, and compare them with numerical simulations.
This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichmüller theory, and the spectral theory of one-frequency Schrödinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention.
What is a blender? In six illustrated pages we define the construction of a blender and the role it plays in the study of smooth dynamical systems.
We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space $T^1S$ and rescaling the flow $φ_t$.
We show that the rate of mixing of the Weil-Petersson flow on non-exceptional (higher dimensional) moduli spaces of Riemann surfaces is at most polynomial.
We investigate transverse Hölder regularity of some canonical leaf conjugacies in partially hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.