92 research outputs found
Consistent spectral approximation of Koopman operators using resolvent compactification
Koopman operators and transfer operators represent dynamical systems through
their induced linear action on vector spaces of observables, enabling the use
of operator-theoretic techniques to analyze nonlinear dynamics in state space.
The extraction of approximate Koopman or transfer operator eigenfunctions (and
the associated eigenvalues) from an unknown system is nontrivial, particularly
if the system has mixed or continuous spectrum. In this paper, we describe a
spectrally accurate approach to approximate the Koopman operator on for
measure-preserving, continuous-time systems via a ``compactification'' of the
resolvent of the generator. This approach employs kernel integral operators to
approximate the skew-adjoint Koopman generator by a family of skew-adjoint
operators with compact resolvent, whose spectral measures converge in a
suitable asymptotic limit, and whose eigenfunctions are approximately periodic.
Moreover, we develop a data-driven formulation of our approach, utilizing data
sampled on dynamical trajectories and associated dictionaries of kernel
eigenfunctions for operator approximation. The data-driven scheme is shown to
converge in the limit of large training data under natural assumptions on the
dynamical system and observation modality. We explore applications of this
technique to dynamical systems on tori with pure point spectra and the Lorenz
63 system as an example with mixing dynamics.Comment: 60 pages, 7 figure
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
Kernel Analog Forecasting: Multiscale Test Problems
Data-driven prediction is becoming increasingly widespread as the volume of
data available grows and as algorithmic development matches this growth. The
nature of the predictions made, and the manner in which they should be
interpreted, depends crucially on the extent to which the variables chosen for
prediction are Markovian, or approximately Markovian. Multiscale systems
provide a framework in which this issue can be analyzed. In this work kernel
analog forecasting methods are studied from the perspective of data generated
by multiscale dynamical systems. The problems chosen exhibit a variety of
different Markovian closures, using both averaging and homogenization;
furthermore, settings where scale-separation is not present and the predicted
variables are non-Markovian, are also considered. The studies provide guidance
for the interpretation of data-driven prediction methods when used in practice.Comment: 30 pages, 14 figures; clarified several ambiguous parts, added
references, and a comparison with Lorenz' original method (Sec. 4.5
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
HNN Extensions of von Neumann Algebras
Reduced HNN extensions of von Neumann algebras (as well as -algebras)
will be introduced, and their modular theory, factoriality and ultraproducts
will be discussed. In several concrete settings, detailed analysis on them will
be also carried out.Comment: Slightly revised versio
The ergodic theory of hyperbolic groups
These notes are a self-contained introduction to the use of dynamical and
probabilistic methods in the study of hyperbolic groups. Most of this material
is standard; however some of the proofs given are new, and some results are
proved in greater generality than have appeared in the literature. These notes
originated in a minicourse given at a workshop in Melbourne, July 11-15 2011.Comment: 37 pages, 5 figures; incorporates referee's comment
Reproducing kernel Hilbert space compactification of unitary evolution groups
A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W-tau on a reproducing kernel Hilbert space (RKHS). The operator W-tau is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W-tau generates a unitary evolution group {e(tW tau)}t epsilon R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W-tau as tau -> 0(+), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rossler system, are presented. (C) 2021 The Author(s). Published by Elsevier Inc.Peer reviewe
Renormalization analysis of catalytic Wright-Fisher diffusions
Recently, several authors have studied maps where a function, describing the
local diffusion matrix of a diffusion process with a linear drift towards an
attraction point, is mapped into the average of that function with respect to
the unique invariant measure of the diffusion process, as a function of the
attraction point. Such mappings arise in the analysis of infinite systems of
diffusions indexed by the hierarchical group, with a linear attractive
interaction between the components. In this context, the mappings are called
renormalization transformations. We consider such maps for catalytic
Wright-Fisher diffusions. These are diffusions on the unit square where the
first component (the catalyst) performs an autonomous Wright-Fisher diffusion,
while the second component (the reactant) performs a Wright-Fisher diffusion
with a rate depending on the first component through a catalyzing function. We
determine the limit of rescaled iterates of renormalization transformations
acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.Comment: 65 pages, 3 figure
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