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JMD

Using the definition of dominated splitting, we introduce the notion of

*critical set*for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept.
DCDS

It is shown that stable accessibility property is $C^r$-dense among
partially hyperbolic diffeomorphisms with one-dimensional center
bundle, for $r \geq 2$, volume preserving or not. This establishes
a conjecture by Pugh and Shub for these systems.

JMD

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

JMD

We prove global rigidity results for some linear abelian actions on
tori. The type of actions we deal with includes in particular
maximal rank semisimple actions on $\mathbb T^N$.

JMD

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.

DCDS

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.

DCDS

We prove that a cohomology free flow on a manifold $M$ fibers over a diophantine translation on $\T^{\beta_1}$ where $\beta_1$ is the first Betti number of $M$.

JMD

This special issue presents some of the lecture notes of the courses held in the 2008 and 2011
Summer Institutes at the Mathematics Research and Conference Center of Polish Academy of
Sciences at Będlewo, Poland. The school was structured as daily courses with a double lecture each, in two parts of 45-50 minutes with a break in between.

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

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

keywords:

JMD

In [15] the authors proved the Pugh–Shub conjecture for partially
hyperbolic diffeomorphisms with 1-dimensional center,

*i.e.*, stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the ﬁrst examples of manifolds in which*all*conservative partially hyperbolic diffeomorphisms are ergodic.
JMD

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.

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