28 research outputs found
A Note on Operator Semigroups Associated to Chaotic Flows
The transfer operator associated to a flow (continuous time dynamical system)
is a one-parameter operator semigroup. We consider the operator-valued Laplace
transform of this one-parameter semigroup. Estimates on the Laplace transform
have be used in various settings in order to show the rate at which the flow
mixes. Here we consider the case of exponential mixing or rapid mixing (super
polynomial). We develop the operator theory framework amenable to this setting
and show that the same estimates may be used to produce results, in terms of
the operators, which go beyond the results for the rate of mixing. Such results
are useful for obtaining other statistical properties of the dynamical system
An Alternative Approach to Generalised BV and the Application to Expanding Interval Maps
We introduce a family of Banach spaces of measures, each containing the set
of measures with density of bounded variation. These spaces are suitable for
the study of weighted transfer operators of piecewise-smooth maps of the
interval where the weighting used in the transfer operator is not better than
piecewise H\"older continuous and the partition on which the map is continuous
may possess a countable number of elements. For such weighted transfer
operators we give upper bounds for both the spectral radius and for the
essential spectral radius
Parabolic Flows Renormalized by Partially Hyperbolic Maps
We consider parabolic flows on 3-dimensional manifolds which are renormalized
by circle extensions of Anosov diffeormorphisms. This class of flows includes
nilflows on the Heisenberg nilmanifold which are renormalized by partially
hyperbolic automorphisms. The transfer operators associated to the
renormalization maps, acting on anisotropic Sobolev spaces, are known to have
good spectral properties (this relies on ideas which have some resemblance to
representation theory but also apply to non-algebraic systems). The spectral
information is used to describe the deviation of ergodic averages and solutions
of the cohomological equation for the parabolic flow.Comment: Comments welcom
Open sets of Axiom A flows with Exponentially Mixing Attractors (with Erratum)
For any dimension we construct -open subsets of the space of
vector fields such that the flow associated to each vector field is
Axiom A and exhibits a non-trivial attractor which mixes exponentially with
respect to the unique SRB measure.Comment: 15 pages, no figures; content as published in journal; includes text
of the Erratum as an appendi
Anisotropic spaces and nilmanifold automorphisms
We introduce anisotropic Banach spaces on Heisenberg nilmanifolds and study
the resonance spectrum associated to partially hyperbolic automorphisms. In
this work we describe an alternative proof of the spectrum which takes
advantage of geometric-style anisotropic norms.Comment: Comments welcom
Discontinuities cause essential spectrum on surfaces
Two dimensional maps with discontinuities are considered. It is shown that,
in the presence of discontinuities, the essential spectrum of the transfer
operator is large whenever it acts on a Banach space with norm that is stronger
than or . Three classes of examples are introduced and studied.
In two dimensions there is complication due to the geometry of the
discontinuities, an issue not present in the one dimensional case and which is
explored in this work.Comment: Comments welcom