76 research outputs found
Dynamical Properties of Gaussian Thermostats
In this work we show that the set of Kupka-Smale Gaussian thermostats on a
compact manifold is generic. A Gaussian thermostat is Kupka-Smale if the closed
orbits are hyperbolic and the heteroclinic intersection are transversal.
We also show a dichotomy between robust transitivity and existence of
arbitrary number of attractors or repellers orbits. The main tools are the
concept of transitions adapted to the conformally symplectic context and a
perturbative theorem which is a version of the Franks lemma for Gaussian
thermostats.
Finally we provide some conditions in terms of geometrical invariants for an
invariant set of a Gaussian thermostat to have dominated splitting. From that
we conclude some dynamical properties for the surface case
Robust Transitivity in Hamiltonian Dynamics
A goal of this work is to study the dynamics in the complement of KAM tori
with focus on non-local robust transitivity. We introduce open sets
() of symplectic diffeomorphisms and Hamiltonian systems,
exhibiting "large" robustly transitive sets. We show that the
closure of such open sets contains a variety of systems, including so-called a
priori unstable integrable systems. In addition, the existence of ergodic
measures with large support is obtained for all those systems. A main
ingredient of the proof is a combination of studying minimal dynamics of
symplectic iterated function systems and a new tool in Hamiltonian dynamics
which we call symplectic blender.Comment: 52 pages, 3 figure
Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms
We prove that any diffeomorphism of a compact manifold can be approximated in
topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a
homoclinic tangency or a heterodimensional cycle) or by one which is
essentially hyperbolic (it has a finite number of transitive hyperbolic
attractors with open and dense basin of attraction)
The Evolutionary Robustness of Forgiveness and Cooperation
We study the evolutionary robustness of strategies in infinitely repeated
prisoners' dilemma games in which players make mistakes with a small
probability and are patient. The evolutionary process we consider is given by
the replicator dynamics. We show that there are strategies with a uniformly
large basin of attraction independently of the size of the population.
Moreover, we show that those strategies forgive defections and, assuming that
they are symmetric, they cooperate
Mildly dissipative diffeomorphisms of the disk with zero entropy
We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing
topological entropy which satisfy the mild dissipation property introduced in
[CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We
prove that these systems are either (generalized) Morse Smale or infinitely
renormalizable. In particular we prove for this class of diffeomorphisms a
conjecture of Tresser: any diffeomorphism in the interface between the sets of
systems with zero and positive entropy admits doubling cascades. This
generalizes for these surface dynamics a well known consequence of
Sharkovskii's theorem for interval maps
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