This paper proves various results concerning non-ergodic actions of locally compact groups and particularly Borel cocycles defined over such actions. The general philosophy is to reduce the study of the cocycle to the study of its restriction to each ergodic component of the action, while being careful to show that all objects arising in the analysis depend measurably on the ergodic component. This allows us to prove a version of the superrigidity theorems for cocycles defined over non-ergodic actions.
We prove that certain coinduced actions for an inclusion of finitely generated commensurated subgroups with relative one end are continuous cocycle superrigid actions. We also show the necessity for the relative end assumption.
These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of ``rigid'' groups) satisfy the assumptions of Popa's theorem.
We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $Λ$ is a subgroup of a countable group $Γ$, we consider a probability measure preserving action $Λ\curvearrowright X_0$ and let $Γ\curvearrowright X$ be the coinduced action. Assume either that $Γ$ has property (T) or that $Λ$ is amenable and $Γ$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $Γ\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is coho…
Chung and Jiang showed that, if a one ended group contains an infinite order element, then every continuous cocycle over the full shift on that group, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one ended groups, so that the hypothesis of admitting an infinite order element may be omitted.
In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological version of the well known Popa's measurable cocycle superrigidity theorem. For the proof of sufficient condition, we introduce a new specification property for shifts over general groups which play a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method using homoclinic equivalenc…
Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $α\colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$α$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $α$ itself is cohomologous to a homomorphism (modulo a compact group).
We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$. In particular, our results apply to SP(1,n) and $F_4^{-20}$ and lattices in those groups. The main technical step is to prove a very general result concerning existence of harmonic maps into infinite dimensional spaces, namely a class of simply connected, homogeneous, aspherical Hilbert manifolds. This builds on previous work of Corlette-Zimmer and Korevaar…
We propose a new approach to superrigidity phenomena and implement it for lattice representations and measurable cocycles with homeomorphisms of the circle as the target group. We are motivated by Ghys' theorem stating that any representation $\ro:Γ\to\Homeo_+(S^1)$ of an irreducible lattice $Γ$ in a semi-simple real Lie group $G$ of higher rank, either has a finite orbit or, up to a semi-conjugacy, extends to $G$ which acts through an epimorphism $G\to\PSL_{2}(\RR)$. Our approach, based on the study of abstract boundary theory and, specifically, on the notion of a generalized Weyl group, allo…
We prove cocycle and orbit equivalence superrigidity for lattices in SL(n,R) acting linearly on R^n, as well as acting projectively on certain flag manifolds, including the real projective space. The proof combines operator algebraic techniques with the property (T) in the sense of Zimmer for the action of SL(n,Z) on R^n, n \geq 4. We also show that the restriction of the associated orbit equivalence relation to a subset of finite Lebesgue measure, provides a II_1 equivalence relation with property (T) and yet fundamental group equal to R_+.
Let $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually c…
Let $G$ be either a profinite or a connected compact group, and $Γ, Λ$ be finitely generated dense subgroups. Assuming that the left translation action of $Γ$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $Γ\timesΛ$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $Λ$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W$^*$- superrigidity…
In this article, we will prove a full topological version of Popa's measurable cocycle superrigidity theorem for full shifts. More precisely, we prove that every Hölder continuous cocycle for the full shifts of every finitely generated group $G$ that has one end, undistorted elements and sub-exponential divergence function is cohomologous to a group homomorphism via a Hölder continuous transfer map if the target group is complete and admits a compatible bi-invariant metric. Using the ideas of Behrstock, Dru\c {t}u, Mosher, Mozes and Sapir, we show that the class of our acting groups is large i…
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a continuous perturbation of a constant cocycle is actually continuously conjugate back to the original constant cocycle modulo a cocycle that is continuous and "small". We give some applications to perturbat…
This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, techniques from partially hyperbolic dynamics, and the geometry and pseudo-Riemannian structure of the homogeneous space.
In the present paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov expoent for typical matrix cocyles. By using the results we obtain, we prove that in any non-uniquely ergodic minimal dynamical system, the Lyapunov-irregular points are typical for typical matrix cocyles.
We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence…
For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimension central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variatio…
Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Ávila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.
It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of nonsingular infinite direct products $T$ of transformations $T_n$, $n\in\Bbb N$, of finite type (IDPFT) is studied. It is shown that if $T_n$ is mildly mixing, $n\in\Bbb N$, the sequence of the Radon-Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and $T$ is conservative then the Maharam extension of $T$ is sharply weak mixing. This techniques provides a new approach to the nonsingular Gaussian transformations studi…
Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.
We show that several important normal subgroups $Γ$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $Γ\curvearrowright X$ is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preservi…
In this note we show existence of bounded, transitive cocycles over a transitive action of a finitely generated group, and bounded, ergodic cocycles over an ergodic, probability preserving action of $\Bbb Z^d$.
The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining $ξ$ of two shift-invariant measures $μ$ and $ν$, every point $x$ generic for $μ$ lifts to a pair $(x,y)$ generic for $ξ$ (such $y$ exists in the full symbolic space). Thi…
By using a cocycle generated by the step function $φ_{β, γ} = 1_{[0, β]} - 1_{[0, β]} (. + γ)$ over an irrational rotation $x \to x + α\mod 1$, we present examples which illustrate different aspects of the general theory of cylinder maps. In particular, we construct non ergodic cocycles with ergodic compact quotients, cocycles generating an extension $T_{α, φ}$ with a small centralizer. The constructions are related to diophantine properties of $α, β, γ$.
We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincaré sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman's Sub-additional Ergodic Theorem) that the set of `non-typical' points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any H$\ddot{o}$der continuous cocycles over hyperbolic systems, in this article we show that either all ergodic measures have same Maximal Lyapunov exponents or the set of Lyapunov `non-typical' points have full topological entropy and packing topological entropy. Moreover, we give an estimate of B…
Let T_αdenote the rotation T_αx=x+α(mod 1) by an irrational number αon the additive circle T=[0,1). Let β_1,..., β_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R^(d+1)) generated over T_αby the vectorial function Ψ_{d+1}(x):=(ϕ(x), ϕ(x+β_1),..., ϕ(x+β_d)), with ϕ(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that Ψ_{2} is regular for αwith bounded partial quotients. In the present paper we show that Ψ_{2} is regular for any irrational α. For higher dimensions, we give sufficient conditions f…
A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.
We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed. Smooth versions of some of the constructions are also given.
After relating the notion of $ω$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $ω(n) <n^{-ε}$, where $ε>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $ω$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over…
A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the Lyapunov exponents of a Banach space cocycle.
The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution.
We prove that, for semi-invertible linear cocycles, Oseledets subspaces associated to ergodic measures may be approximated by Oseledets subspaces associated to periodic points.
We prove that for semi-invertible and Hölder continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.
We prove that, for semi-invertible linear cocycles, Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.
We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $2\times 2$ matrix cocycles and our results apply to an open and dense subset of elliptic $\mathrm{SL}(2,\bR)$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov ex…
For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,μ)$ is W$^*$-superrigid in the following sense: if $L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimesΛ$, for an arbitrary free ergodic pmp action $Λ\curvearrowright (Y,ν)$, then the actions $\tilde B_n\curvearrowright X,Λ\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with a finite index subgroup …
It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_δ$ set…
We study the problem of lifting various mixing properties from a base automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where $\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$, $\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$ and $\tfs$ acts on the product space $(X\times Y,\cb\ot\cc,μ\otν)$ by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever $T$ is ergodic (mildly mixing, mixing) but $\tfs$ is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor $\ca\subset\cc$ of $\cs$ the cor…
We prove a nonuniformly hyperbolic version Livšic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, We also prove a $μ$-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system.
Given a linear cocycle over an ergodic homeomorphism on a compact metric space, we show that the existence of a uniform gap between the $p$-th and $(p+1)$-th Lyapunov exponent on a $C^0$-neighbourhood implies the existence of a dominated splitting of index $p$.
In 1968, V.I. Oseledets formulated the question of convergence in the Birkhoff theorem and the multiplicative ergodic theorem for measurable cocycles over flows under the condition of integrability for each individual t. A.M. Stepin and the author established (2016) the convergence along subsets of time of density 1. In this note, we show that moreover the convergence is fulfilled modulo time subsets of finite measure.
Let H be an infinite dimensional separable Hilbert space, X a compact Hausdorff space and f : X \rightarrow X a homeomorphism which preserves a Borel ergodic measure which is positive on non-empty open sets. We prove that the non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic f,H-skew products with non-trivial unstable bundles.
This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint Hölder continuity of the maximal Lyapunov exponent as a function of the cocycle and the transition kernel in the vicinity of any irreducible cocycle with simple maximal Lyapunov exponent. Our approach, via Furstenberg's formula, shows the Hölder continuous dependence on the data of the stationary measure of the projective cocycle and in particular provides a more computable Hölder exponent.
Let $Γ$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $Γ$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $Γ$.
