87 research outputs found
Quasi-Invariant measures, escape rates and the effect of the hole
Let be a piecewise expanding interval map and be an abstract
perturbation of into an interval map with a hole. Given a number ,
, we compute an upper-bound on the size of a hole needed for the
existence of an absolutely continuous conditionally invariant measure (accim)
with escape rate not greater than . The two main ingredients of
our approach are Ulam's method and an abstract perturbation result of Keller
and Liverani.Comment: 15 page
Escape Rates Formulae and Metastability for Randomly perturbed maps
We provide escape rates formulae for piecewise expanding interval maps with
`random holes'. Then we obtain rigorous approximations of invariant densities
of randomly perturbed metabstable interval maps. We show that our escape rates
formulae can be used to approximate limits of invariant densities of randomly
perturbed metastable systems.Comment: Appeared in Nonlinearity, May 201
Metastability of Certain Intermittent Maps
We study an intermittent map which has exactly two ergodic invariant
densities. The densities are supported on two subintervals with a common
boundary point. Due to certain perturbations, leakage of mass through subsets,
called holes, of the initially invariant subintervals occurs and forces the
subsystems to merge into one system that has exactly one invariant density. We
prove that the invariant density of the perturbed system converges in the
-norm to a particular convex combination of the invariant densities of the
intermittent map. In particular, we show that the ratio of the weights in the
combination equals to the limit of the ratio of the measures of the holes.Comment: 19 pages, 2 figure
Linear response in the intermittent family: differentiation in a weighted -norm
We provide a general framework to study differentiability of SRB measures for
one dimensional non-uniformly expanding maps. Our technique is based on
inducing the non-uniformly expanding system to a uniformly expanding one, and
on showing how the linear response formula of the non-uniformly expanding
system is inherited from the linear response formula of the induced one. We
apply this general technique to interval maps with a neutral fixed point
(Pomeau-Manneville maps) to prove differentiability of the corresponding SRB
measure. Our work covers systems that admit a finite SRB measure and it also
covers systems that admit an infinite SRB measure. In particular, we obtain a
linear response formula for both finite and infinite SRB measures. To the best
of our knowledge, this is the first work that contains a linear response result
for infinite measure preserving systems.Comment: Final version. To appear in DCDS-
Variance continuity for Lorenz flows
The classical Lorenz flow, and any flow which is close to it in the
-topology, satisfies a Central Limit Theorem (CLT). We prove that the
variance in the CLT varies continuously.Comment: Final version. To appear in Annales Henri Poincar\'
Pseudo-Orbits, Stationary Measures and Metastability
We study random perturbations of multidimensional piecewise expanding maps.
We characterize absolutely continuous stationary measures (acsm) of randomly
perturbed dynamical systems in terms of pseudo-orbits linking the ergodic
components of absolutely invariant measures (acim) of the unperturbed system.
We focus on those components, called least-elements, which attract
pseudo-orbits. We show that each least element admits a neighbourhood which
supports exactly one ergodic acsm of the random system. We use this result to
identify random perturbations that exhibit a metastable behavior.Comment: To appear in Dynamical System
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