273 research outputs found
Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions
For finite-dimensional maps and periodic systems, Palmer rigorously proved
Smale horseshoe theorem using shadowing lemma in 1988. For infinite-dimensional
maps and periodic systems, such a proof was completed by Steinlein and Walther
in 1990, and Henry in 1994. For finite-dimensional autonomous systems, such a
proof was accomplished by Palmer in 1996. For infinite-dimensional autonomous
systems, the current article offers such a proof. First we prove an Inclination
Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize
graph transform and the concept of persistence of invariant manifold, to prove
the existence of a shadowing orbit
Arnold's Diffusion: from the a priori unstable to the a priori stable case
We expose some selected topics concerning the instability of the action
variables in a priori unstable Hamiltonian systems, and outline a new strategy
that may allow to apply these methods to a priori stable systems
Mixing-like properties for some generic and robust dynamics
We show that the set of Bernoulli measures of an isolated topologically
mixing homoclinic class of a generic diffeomorphism is a dense subset of the
set of invariant measures supported on the class. For this, we introduce the
large periods property and show that this is a robust property for these
classes. We also show that the whole manifold is a homoclinic class for an open
and dense subset of the set of robustly transitive diffeomorphisms far away
from homoclinic tangencies. In particular, using results from Abdenur and
Crovisier, we obtain that every diffeomorphism in this subset is robustly
topologically mixing
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