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Journal of Modern Dynamics

2010 , Volume 4 , Issue 3

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Linear cocycles over hyperbolic systems and criteria of conformality
Boris Kalinin and Victoria Sadovskaya
2010, 4(3): 419-441 doi: 10.3934/jmd.2010.4.419 +[Abstract](85) +[PDF](259.1KB)
Abstract:
In this paper, we study Hölder-continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant subbundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Hölder-continuous Riemannian metric.
The action of finite-state tree automorphisms on Bernoulli measures
Rostyslav Kravchenko
2010, 4(3): 443-451 doi: 10.3934/jmd.2010.4.443 +[Abstract](80) +[PDF](119.0KB)
Abstract:
We describe how a finite-state automorphism of a regular rooted tree changes the Bernoulli measure on the boundary of the tree. It turns out that a finite-state automorphism of polynomial growth, as defined by S. Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodym derivative. On the other hand, the image of the Bernoulli measure under the action of a strongly connected finite-state automorphism is singular to the measure itself.
The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis
Carlos Matheus and Jean-Christophe Yoccoz
2010, 4(3): 453-486 doi: 10.3934/jmd.2010.4.453 +[Abstract](67) +[PDF](655.0KB)
Abstract:
We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus $3$) and Forni and Matheus (in genus $4$). We show that, in both cases, the action on the nontrivial part of the homology is through finite groups. In particular, the action on some $4$-dimensional invariant subspace of the homology leaves invariant a root system of $D_4$ type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the nontrivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmüller disks of these two origamis are equal to zero.
Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups
Anatole Katok and Federico Rodriguez Hertz
2010, 4(3): 487-515 doi: 10.3934/jmd.2010.4.487 +[Abstract](92) +[PDF](331.2KB)
Abstract:
We prove absolute continuity of "high-entropy'' hyperbolic invariant measures for smooth actions of higher-rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds the existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.
Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds
Aaron W. Brown
2010, 4(3): 517-548 doi: 10.3934/jmd.2010.4.517 +[Abstract](86) +[PDF](357.1KB)
Abstract:
We prove a result motivated by Williams's classification of expanding attractors and the Franks--Newhouse Theorem on codimension-$1$ Anosov diffeomorphisms: If $\Lambda$ is a topologically mixing hyperbolic attractor such that $\dim\E^u$|$\Lambda$ = 1, then either $\Lambda$ is expanding or is homeomorphic to a compact abelian group (a toral solenoid). In the latter case, $f$|$\Lambda$ is conjugate to a group automorphism. As a corollary, we obtain a classification of all $2$-dimensional basic sets in $3$-manifolds. Furthermore, we classify all topologically mixing hyperbolic attractors in $3$-manifolds in terms of the classically studied examples, answering a question of Bonatti in [1].
Zygmund strong foliations in higher dimension
Yong Fang, Patrick Foulon and Boris Hasselblatt
2010, 4(3): 549-569 doi: 10.3934/jmd.2010.4.549 +[Abstract](92) +[PDF](243.5KB)
Abstract:
For a compact Riemannian manifold $M$, $k\ge2$ and a uniformly quasiconformal transversely symplectic $C^k$ Anosov flow $\varphi$:$\R\times M\to M$ we define the longitudinal KAM-cocycle and use it to prove a rigidity result: $E^u\oplus E^s$ is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that $E^u\oplus E^s$ is Lipschitz-continuous. Results proved elsewhere then imply that the flow is smoothly conjugate to an algebraic one.
Lipschitz continuous invariant forms for algebraic Anosov systems
Patrick Foulon and Boris Hasselblatt
2010, 4(3): 571-584 doi: 10.3934/jmd.2010.4.571 +[Abstract](109) +[PDF](169.0KB)
Abstract:
We prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz continuous invariant $1$-form. This has corollaries for rigidity of time-changes, and we give a particular application to geometric rigidity of quasiconformal Anosov flows.
   Several features of the reasoning are interesting; namely, the use of exterior calculus for Lipschitz continuous forms, the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.

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