237 research outputs found
Fourier transforms of Gibbs measures for the Gauss map
We investigate under which conditions a given invariant measure for the
dynamical system defined by the Gauss map is a Rajchman
measure with polynomially decaying Fourier transform We show that this
property holds for any Gibbs measure of Hausdorff dimension greater than
with a natural large deviation assumption on the Gibbs potential. In
particular, we obtain the result for the Hausdorff measure and all Gibbs
measures of dimension greater than on badly approximable numbers, which
extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result
implies that the Fourier-Stieltjes coefficients of the Minkowski's question
mark function decay to polynomially answering a question of Salem from
1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain
an equidistribution theorem for Gibbs measures, which extends in part a recent
result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear
and number theoretic nature of the Gauss map and large deviation theory for
Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more
elaborations, and included comments on Salem's problem. To appear in Math.
An
Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis
Data sets are often modeled as point clouds in , for large. It is
often assumed that the data has some interesting low-dimensional structure, for
example that of a -dimensional manifold , with much smaller than .
When is simply a linear subspace, one may exploit this assumption for
encoding efficiently the data by projecting onto a dictionary of vectors in
(for example found by SVD), at a cost for data points. When
is nonlinear, there are no "explicit" constructions of dictionaries that
achieve a similar efficiency: typically one uses either random dictionaries, or
dictionaries obtained by black-box optimization. In this paper we construct
data-dependent multi-scale dictionaries that aim at efficient encoding and
manipulating of the data. Their construction is fast, and so are the algorithms
that map data points to dictionary coefficients and vice versa. In addition,
data points are guaranteed to have a sparse representation in terms of the
dictionary. We think of dictionaries as the analogue of wavelets, but for
approximating point clouds rather than functions.Comment: Re-formatted using AMS styl
Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005
Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids
For points in real dimensions, we introduce a geometry for general digit
sets. We introduce a positional number system where the basis for our
representation is a fixed by matrix over \bz. Our starting point is a
given pair with the matrix assumed expansive, and
a chosen complete digit set, i.e., in bijective correspondence
with the points in \bz^d/A^T\bz^d. We give an explicit geometric
representation and encoding with infinite words in letters from .
We show that the attractor for an affine Iterated Function
System (IFS) based on is a set of fractions for our digital
representation of points in \br^d. Moreover our positional "number
representation" is spelled out in the form of an explicit IFS-encoding of a
compact solenoid \sa associated with the pair . The intricate
part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the
initial -IFS. Using these cycles we are able to write down
formulas for the two maps which do the encoding as well as the decoding in our
positional -representation.
We show how some wavelet representations can be realized on the solenoid, and
on symbolic spaces
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