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.
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.
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.
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
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In several contexts the defining invariant structures of a hyperbolic
dynamical system are smooth only in systems of algebraic origin, and we
prove new results of this smooth rigidity type for a class of flows.
For a transversely symplectic uniformly quasiconformal $C^2$ Anosov flow on
a compact Riemannian manifold we define the longitudinal KAM-cocycle
and use it to prove a rigidity result: The joint stable/unstable subbundle
is Zygmund-regular, and higher regularity implies vanishing of the
KAM-cocycle, which in turn implies that the subbundle is
Lipschitz-continuous and indeed that the flow is smoothly conjugate to an
algebraic one. To establish the latter, we prove results for algebraic
Anosov systems that imply smoothness and a special structure for any
Lipschitz-continuous invariant 1-form.
We obtain a pertinent geometric rigidity result: Uniformly quasiconformal
magnetic flows are geodesic flows of hyperbolic metrics.
Several features of the reasoning are interesting: The use of exterior
calculus for Lipschitz-continuous forms, that the arguments for geodesic
flows and infranilmanifoldautomorphisms are quite different, and the need
for mixing as opposed to ergodicity in the latter case.
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.
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
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.
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.
We show that the linearizing homeomorphism in the Hartman--Grobman
Theorem is differentiable at the fixed point.