## Journals

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### Open Access Journals

DCDS

A Riemannian manifold is said to be uniformly secure if there is
a finite number $s$ such that all geodesics connecting an
arbitrary pair of points in the manifold can be blocked by $s$
point obstacles. We prove that the number of geodesics with length
$\leq T$ between every pair of points in a uniformly secure
manifold grows polynomially as $T \to \infty$. By results of
Gromov and Mañé, the fundamental group of such a manifold is
virtually nilpotent, and the topological entropy of its geodesic
flow is zero. Furthermore, if a uniformly secure manifold has no
conjugate points, then it is flat. This follows from the virtual
nilpotency of its fundamental group either via the theorems of
Croke-Schroeder and Burago-Ivanov, or by more recent work of
Lebedeva.

We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.

We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.

JMD

We establish stable ergodicity of diffeomorphisms with partially
hyperbolic attractors whose Lyapunov exponents along the central direction
are all negative with respect to invariant SRB-measures.

keywords:
Partial hyperbolicity
,
stable ergodicity
,
accessibility
,
Lyapunov exponents
,
SRB-measures.

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.

DCDS

This paper discusses relationships among the basic notions
that have been important in recent investigations of
the ergodicity of volume-preserving
partially hyperbolic diffeomorphisms. In particular we survey the possible definitions
of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.

DCDS

We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface.
We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.

keywords:
Lyapunov exponents
,
geodesic flow
,
rank 1 surfaces
,
Hausdorff dimension
,
shadowing.
,
entropy
,
pressure
,
multifractal formalism

## Year of publication

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