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### Open Access Journals

DCDS

We propose a general theorey of formally gradient differential
equations on unbounded one-dimensional domains, based on an
energy-flow inequality, and on the study of the induced
semiflow on the space of probability measures on the
phase space.
We prove that the $\omega$-limit set of each point contains
an equilibrium, and that the $\omega$-limit set of $\mu$-almost
every point in the phase space consists of equilibria,
where $\mu$ is any Borel probability measure invariant
for spatial translation.

keywords:
return times
,
Gradient semiflows
,
recurrent orbits
,
Lyapunov function
,
periodic orbits
,
invariant measures.

DCDS

We consider uniformly (DC) or periodically (AC) driven generalized
infinite elastic chains (a generalized Frenkel-Kontorova model) with
gradient dynamics. We first show that the union of supports of all
space-time invariant measures, denoted by $\mathcal{A}$, projects
injectively to a dynamical system on a 2-dimensional cylinder. We also prove
existence of space-time ergodic measures supported on a set of rotationaly
ordered configurations with an arbitrary (rational or irrational) rotation
number. This shows that the Aubry-Mather structure of ground states persists
if an arbitrary AC or DC force is applied. The set $\mathcal{A}$ attracts
almost surely (in probability) configurations with bounded spacing. In the
DC case, $\mathcal{A}$ consists entirely of equilibria and uniformly sliding
solutions. The key tool is a new weak Lyapunov function on the space of
translationally invariant probability measures on the state space, which
counts intersections.

DCDS

We generalize various notions of stability of invariant sets of dynamical
systems to invariant measures, by defining a topology on the set of
measures. The defined topology is similar, but not topologically equivalent
to weak* topology, and it also differs from topologies induced by the Riesz
Representation Theorem. It turns out that the constructed topology is a
solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.

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