4,076 research outputs found
Superpolynomial and polynomial mixing for semiflows and flows
We give a review of results on superpolynomial decay of correlations, and
polynomial decay of correlations for nonuniformly expanding semiflows and
nonuniformly hyperbolic flows. A self-contained proof is given for semiflows.
Results for flows are stated without proof (the proofs are contained in
separate joint work with Balint and Butterley). Applications include
intermittent solenoidal flows, suspended Henon attractors, Lorenz attractors,
and various Lorentz gas models including the infinite horizon Lorentz gas.Comment: Final minor change
Decay of correlations for slowly mixing flows
We show that polynomial decay of correlations is prevalent for a class of
nonuniformly hyperbolic flows. These flows are the continuous time analogue of
a class of nonuniformly hyperbolic diffeomorphisms for which Young proved
polynomial decay of correlations. Roughly speaking, in situations where the
decay rate has previously been proved for diffeomorphisms, we
establish the decay rate for typical flows. Applications include
certain classes of semidispersing billiards, as well as dispersing billiards
with vanishing curvature. In addition, we obtain results for suspension flows
with unbounded roof functions. This includes the planar periodic Lorentz flow
with infinite horizon
Dynamics on unbounded domains; co-solutions and inheritance of stability
We consider the dynamics of semiflows of patterns on unbounded domains that
are equivariant under a noncompact group action. We exploit the unbounded
nature of the domain in a setting where there is a strong `global' norm and a
weak `local' norm. Relative equilibria whose group orbits are closed manifolds
for a compact group action need not be closed in a noncompact setting; the
closure of a group orbit of a solution can contain `co-solutions'.
The main result of the paper is to show that co-solutions inherit stability
in the sense that co-solutions of a Lyapunov stable pattern are also stable
(but in a weaker sense). This means that the existence of a single group orbit
of stable relative equilibria may force the existence of quite distinct group
orbits of relative equilibria, and these are also stable. This is in contrast
to the case for finite dimensional dynamical systems where group orbits of
relative equilibria are typically isolated
Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation
Over the last 10 years or so, advanced statistical properties, including
exponential decay of correlations, have been established for certain classes of
singular hyperbolic flows in three dimensions. The results apply in particular
to the classical Lorenz attractor. However, many of the proofs rely heavily on
the smoothness of the stable foliation for the flow.
In this paper, we show that many statistical properties hold for singular
hyperbolic flows with no smoothness assumption on the stable foliation. These
properties include existence of SRB measures, central limit theorems and
associated invariance principles, as well as results on mixing and rates of
mixing. The properties hold equally for singular hyperbolic flows in higher
dimensions provided the center-unstable subspaces are two-dimensional.Comment: Accepted version. To appear in Advances in Mat
Existence and smoothness of the stable foliation for sectional hyperbolic attractors
We prove the existence of a contracting invariant topological foliation in a
full neighborhood for partially hyperbolic attractors. Under certain bunching
conditions it can then be shown that this stable foliation is smooth.
Specialising to sectional hyperbolic attractors, we give a verifiable condition
for bunching. In particular, we show that the stable foliation for the
classical Lorenz equation (and nearby vector fields) is better than which
is crucial for recent results on exponential decay of correlations. In fact the
foliation is at least .Comment: Corrected estimate for smoothness of stable foliation. Clarification
of which results hold for general partially hyperbolic attractors. Some minor
typos fixed. Accepted for publication in Bull. London Math. So
A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion
We provide a systematic approach for deducing statistical limit laws via
martingale-coboundary decomposition, for nonuniformly hyperbolic systems with
slowly contracting and expanding directions. In particular, if the associated
return time function is square-integrable, then we obtain the central limit
theorem, the weak invariance principle, and an iterated version of the weak
invariance principle.Comment: Final versio
Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems
We prove an almost sure invariance principle that is valid for general
classes of nonuniformly expanding and nonuniformly hyperbolic dynamical
systems. Discrete time systems and flows are covered by this result. In
particular, the result applies to the planar periodic Lorentz flow with finite
horizon. Statistical limit laws such as the central limit theorem, the law of
the iterated logarithm, and their functional versions, are immediate
consequences.Comment: 21 pages, To appear in Communications in Mathematical Physic
Decay of correlations for nonuniformly expanding systems with general return times
We give a unified treatment of decay of correlations for nonuniformly
expanding systems with a good inducing scheme. In addition to being more
elementary than previous treatments, our results hold for general integrable
return time functions under fairly mild conditions on the inducing scheme
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