Anosov Actions Research Articles

SRB measures for Anosov actions

SRB measures for Anosov actions

Ruelle–Taylor resonances of Anosov actions

Combining microlocal methods and a cohomological theory developed by J. Taylor, we define for Anosov \mathbb{R}^{\kappa} -actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle–Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of \mathbb{C}^{\kappa} , with \lambda=0 being always a leading resonance. The joint resonant states at 0 give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows.

EQUILIBRIUM STATES FOR CENTER ISOMETRIES

AbstractWe develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations is also given. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

Mini-Workshop: $(Anosov)^3$

Three different active fields are subsumed under the keyword Anosov theory : Spectral theory of Anosov flows, dynamical rigidity of Anosov actions, and Anosov representations. In all three fields there have been dynamic developments and substantial breakthroughs in recent years. The miniworkshop brought together researchers from the three different communities and sparked a joint discussion of current ideas, common interests, and open problems.

Geometrical constructions of equilibrium states

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic systems—center isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on symbolic dynamics) and even provide new information in the classical case.

On classification of higher rank Anosov actions on compact manifold

We prove global smooth classification results for Anosov ℤk actions on general compact manifolds, under certain irreduciblity conditions and the presence of sufficiently many Anosov elements. In particular, we remove all the uniform control assumptions which were used in all the previous results towards the Katok-Spatzier global rigidity conjecture on general manifolds. The main idea is to create a new mechanism labelled nonuniform redefining argument, to prove continuity of certain dynamically-defined objects. This leads to uniform control for higher-rank actions and should apply to more general rigidity problems in dynamical systems.

Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

Global smooth and topological rigidity of hyperbolic lattice actions

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $\Gamma$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $\rho$ of $\alpha$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $\alpha$ and $\rho$, on a finite-index subgroup of $\Gamma$. If $\alpha$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a corollary, we obtain $C^\infty$ global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank $2$ or higher. We also obtain global rigidity of Anosov actions of $\mathrm{SL}(n,\mathbb Z)$ on $\mathbb T^n$ for $ n\geq 5$ and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

Nil-Anosov actions

We consider Anosov actions of a Lie group G of dimension k on a closed manifold of dimension \(k+n\). We introduce the notion of Nil-Anosov action of G (which includes the case where G is nilpotent) and establishes the invariance by the entire group G of the associated stable and unstable foliations. We then prove a spectral decomposition Theorem for such an action when the group G is nilpotent. Finally, we focus on the case where G is nilpotent and the unstable bundle has codimension one. We prove that in this case the action is a Nil-extension over an Anosov action of an abelian Lie group. In particular: if \(n \ge 3,\) then the action is topologically transitive, if \(n=2,\) then the action is a Nil-extension over an Anosov flow.

Exponential mixing and smooth classification of commuting expanding maps

We show that genuinely higher rank expanding actions of abelian semigroups on compact manifolds are \begin{document}$C.{\infty}$\end{document} -conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to Holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semigroup actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.

On the work of Rodriguez Hertz on rigidity in dynamics

This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝ d - or ℤ d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ) d /Γ and are a step in the direction of the “top-degree cohomology” part.

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livšic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite-volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livšic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.

Algebraic Anosov actions of nilpotent Lie groups

In this paper we classify algebraic Anosov actions of nilpotent Lie groups on closed manifolds, extending the previous results by P. Tomter (1970, 1975) [17,18]. We show that they are all nil-suspensions over either suspensions of Anosov actions of Zk on nilmanifolds, or (modified) Weyl chamber actions. We check the validity of the generalized Verjovsky conjecture in this algebraic context. We also point out an intimate relation between algebraic Anosov actions and Cartan subalgebras in general real Lie groups.

Global rigidity of higher rank Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are C ∞ C^{\infty } -conjugate to affine actions.

Totally nonsymplectic Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.

On integrable codimension one Anosov actions of $\RR^k$

In this paper, we consider codimension one Anosov actions of $\RR^k, k\geq 1,$ on closed connected orientable manifolds of dimension $n+k$ with $n\geq 3$. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one $1$-parameter subgroup of $\RR^k$ admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension $n$. As a byproduct, generalizing a Theorem by Ghys in the case $k=1$, we show that, under some assumptions about the smoothness of the sub-bundle $E^{ss}\oplus E$uu , and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of $\mathbb{Z}^k$ on $\TT^{n}$.

Transitivity of codimension-one Anosov actions of ℝk on closed manifolds

AbstractWe consider Anosov actions of ℝk, k≥2, on a closed connected orientable manifold M, of codimension one, i.e. such that the unstable foliation associated to some element of ℝk has dimension one. We prove that if the ambient manifold has dimension greater than k+2, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension-one Anosov flows.

A λ-lemma for foliations

We show that a C 0 codimension one foliation with C 1 leaves F of a closed manifold is minimal if there are a foliation G transverse to F , and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G . We use this result to study of Anosov actions on closed manifolds.