ERA-MS
Prevalence of non-Lipschitz Anosov foliations
Boris Hasselblatt and Amie Wilkinson
keywords: stable foliation Holder structures conjugacy. Anosov splitting invariant foliations hyperbolic system holonomy Anosov system horospheric foliations
JMD
Preface
Dmitry Dolgopyat Giovanni Forni Rostislav Grigorchuk Boris Hasselblatt Anatole Katok Svetlana Katok Dmitry Kleinbock Raphaël Krikorian Jens Marklof
The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics.

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Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
keywords:
ERA-MS
Desingularization of surface maps
Erica Clay Boris Hasselblatt Enrique Pujals

We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.

keywords: Singularization hyperbolicity
DCDS
Pointwise hyperbolicity implies uniform hyperbolicity
Boris Hasselblatt Yakov Pesin Jörg Schmeling
We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.
keywords: nonuniform hyperbolicity Uniform hyperbolicity pointwise hyperbolicity.
DCDS
Critical regularity of invariant foliations
Boris Hasselblatt
We exhibit an open set of symplectic Anosov diffeomorphisms on which there are discrete "jumps" in the regularity of the unstable subbundle. It is either highly irregular almost everywhere ($C^\epsilon$ only on a negligible set) or better than $C^1$. In the latter case the Hölder exponent of the derivative is either about $\epsilon/2$ or almost 1.
keywords: invariant foliations. Critical regularity
DCDS
Topological entropy for nonuniformly continuous maps
Boris Hasselblatt Zbigniew Nitecki James Propp
The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map $f: X \rightarrow X$ from the case when $X$ is a compact metric space to the case when $X$ is allowed to be noncompact. These extensions all require the space $X$ to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When the map $f$ is uniformly continuous, it extends continuously to the completion, and the various notions of entropy reduce to the standard ones (applied to this extension). However, when uniform continuity is not assumed, these new quantities can differ. We consider extensions proposed by Bowen (maximizing over compact subsets and a definition of Hausdorff dimension type) and Friedland (using the compactification of the graph of $f$) as well as a straightforward extension of Bowen and Dinaburg's definition from the compact case, assuming that $X$ is totally bounded, but not necessarily compact. This last extension agrees with Friedland's, and both dominate the one proposed by Bowen (Theorem 6). Examples show how varying the metric outside its uniform class can vary both quantities. The natural extension of Adler--Konheim--McAndrew's original (metric-free) definition of topological entropy beyond compact spaces dominates these other notions, and is unfortunately infinite for a great number of noncompact examples.
keywords: topological entropy nonuniform continuity. totally bounded metric space compactification
JMD
Lipschitz continuous invariant forms for algebraic Anosov systems
Patrick Foulon Boris Hasselblatt
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.
keywords: Lipschitz regularity Anosov flow invariant forms smooth rigidity.
DCDS
Preface
Boris Hasselblatt
This issue of Discrete and Continuous Dynamical Systems is Anatole Katok at UC Berkeley dedicated to Anatole Katok and was conceived on the occasion of his 60th birthday. Anatole Katok was born in Washington, D.C. in 1944. In 1959 he placed second in the Moscow Mathematical Olympiad, and the year after entered Moscow State University, earning his mathematics doctorate in 1968 from Y. Sinai. After working in the department of mathematical methods at the Central Economics and Mathematics Institute for 10 years he emigrated with his family, moving via Vienna, Rome and Paris to the University ofMaryland. The position in the US allowed him to travel, attend and organize conferences, collaborate with other mathematicians and supervise students.
    From this time on, he organized more conferences, special years and other events than anybody else in the dynamics community. During his five years at Maryland Katok was instrumental in the development of their dynamical systems school, and after moving to first Caltech and then Penn State he founded a strong group in dynamical systems at each of these institutions. The schools at Maryland and Penn State have become leading world centers. He has always been active in mentoring younger generations. During his student years he devoted much energy to mathematics olympiads and circles, at Penn State he has been the driving force behind the Mathematics Advanced Study Semesters program for especially strong mathematics undergraduates, and he has supervised more than two dozen doctoral students.

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keywords:
JMD
Zygmund strong foliations in higher dimension
Yong Fang Patrick Foulon Boris Hasselblatt
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.
keywords: Zygmund regularity quasiconformal geometric rigidity smooth rigidity Anosov flow foliations. strong invariant subbundles
DCDS
Differentiability of the Hartman--Grobman linearization
Misha Guysinsky Boris Hasselblatt Victoria Rayskin
We show that the linearizing homeomorphism in the Hartman--Grobman Theorem is differentiable at the fixed point.
keywords: Hartman–Grobman Theorem linearizing homeomorphism fixed point.

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