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.

For more information please click the “Full Text” above.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
Fifty years of entropy in dynamics: 1958--2007
Anatole Katok
These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical” nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fifty-year period under discussion and on numerous occasions was fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre.

For the full preface, please click the "Full Text" link above.
keywords: Entropy.
Moscow dynamics seminars of the Nineteen seventies and the early career of Yasha Pesin
Anatole Katok
keywords: Mathematics in the Soviet Union. Moscow mathematical school non-uniform hyperbolicity partial hyperbolicity Pesin theory.
Hyperbolic measures and commuting maps in low dimension
Anatole Katok
We study invariant measures with non-vanishing Lyapunov characteristic exponents for commuting diffeomorphisms of compact manifolds. In particular we show that for $k=2,3$ no faithful $\mathbb{Z}^k$ real-analytic action on a $k$-dimensional manifold preserves a hyperbolic measure. In the smooth case similar statements hold for actions faithful on the support of the measure. Generalizations to higher dimension are proved under certain non-degeneracy conditions for the Lyapunov exponents.
keywords: compact manifolds commuting diffeomorphisms hyperbolic measure.
Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups
Anatole Katok Federico Rodriguez Hertz
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.
keywords: entropy nonuniform hyperbolicity measure rigidity synchronizing time change. Lyapunov metric
Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions
Danijela Damjanović Anatole Katok
We give a proof of cocycle rigidity in Hölder and smooth categories for Cartan actions on $SL(n, \mathbb R)$/$\Gamma$ and $SL(n, \mathbb C)$/$\Gamma$ for $n\ge 3$ and $\Gamma$ cocompact lattice, and for restrictions of those actions to subspaces which contain a two-dimensional plane in general position. This proof does not use harmonic analysis, it relies completely on the structure of stable and unstable foliations of the action. The key new ingredient is the use of the description of generating relations in the group $SL_n$.
keywords: rigidity Cocycles partial hyperbolicity. Weyl chamber flow
Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure
Roland Gunesch Anatole Katok
We describe in detail a construction of weakly mixing $C^\infty$ diffeomorphisms preserving a smooth measure and a measurable Riemannian metric as well as ${\mathbb} Z^k$ actions with similar properties. We construct those as a perturbation of elements of a nontrivial non-transitive circle action. Our construction works on all compact manifolds admitting a nontrivial circle action.
It is shown in the appendix that a Riemannian metric preserved by a weakly mixing diffeomorphism can not be square integrable.
keywords: Conjugation-­approximation method partition rearrangements almost isometries weakly mixing dioeeomorphisms measure­preserving exponential map.
Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
Anatole Katok Federico Rodriguez Hertz
Every $C^2$ action $\a$ of $\mathbb{Z}k$, $k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to the corresponding elements of an action $\ao$ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between $\a$ and $\a_0$. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for $\a$ for which the eigenvalues for $\a$ and $\a_0$ coincide. We describe some nontrivial examples of actions of this kind.
keywords: $\mathbb{Z}^k$ actions. nonuniform hyperbolicity measure rigidity
Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
Anatole Katok Federico Rodriguez Hertz
We prove that any real-analytic action of $SL(n,\Z),n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.
keywords: Global rigidity Zimmer program. real analytic
Local rigidity of homogeneous parabolic actions: I. A model case
Danijela Damjanovic Anatole Katok
We show a weak form of local differentiable rigidity for the rank $2$ abelian action of upper unipotents on $SL(2,R)$ $\times$ $SL(2,R)$ $/\Gamma$. Namely, for a $2$-parameter family of sufficiently small perturbations of the action, satisfying certain transversality conditions, there exists a parameter for which the perturbation is smoothly conjugate to the action up to an automorphism of the acting group. This weak form of rigidity for the parabolic action in question is optimal since the action lives in a family of dynamically different actions. The method of proof is based on a KAM-type iteration and we discuss in the paper several other potential applications of our approach.
keywords: Local rigidity KAM method. homogeneous actions

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