43 research outputs found
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