## Journals

- Advances in Mathematics of Communications
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DCDS

We study topological and ergodic properties of
some almost hyperbolic diffeomorphisms on two dimensional
manifolds. Under generic conditions, diffeomorphisms obtained from
Anosov by an isotopy pushing together the stable and unstable
manifolds to be tangent at a fixed point, are conjugate to
Anosov. For a finite codimension subset at the boundary of Anosov
there exist a SRB measure and an unique ergodic attractor.

DCDS

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary.
We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.

DCDS

For a sequence of subadditive potentials,
a method of choosing
state points with negative growth rates for an ergodic
dynamical system was given in [5]. This paper first
generalizes this result to the non-ergodic dynamics, and then
proves that under some mild additional hypothesis, one can choose
points with negative growth rates from a positive Lebesgue measure
set, even if the system does not preserve any measure that is
absolutely continuous with respect to Lebesgue measure.

JDG

We study the deterministic dynamics of networks ${\mathcal N}$ composed by $m$ non identical, mutually pulse-coupled cells. We assume
weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of $m$. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" $p \geq 1$, and that if the cells are mutually structurally identical or similar, then the synchronization is complete ($p= 1$) . Second, we prove that the amount of information $H$ that ${\mathcal N}$ generates or processes, equals $\log p$. Therefore, if ${\mathcal N}$ completely synchronizes, the information is null. Finally, we prove that ${\mathcal N}$ protects the cells from their risk of death.

keywords:
synchronization
,
impulsive ODE
,
Pulse-coupled networks
,
information.
,
cooperative evolutive game

DCDS

We prove that the
$C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks
near the generic unfolding of a quadratic homoclinic tangency of a
dissipative saddle, can be perturbed along an infinite dimensional
manifold of $C^3$ diffeomorphisms such that infinitely many sinks
persist simultaneously. On the other hand, if they are perturbed
along one-parameter families that unfold generically the quadratic
tangencies, then at most a finite number of those sinks have
continuation.

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