702 research outputs found
An optimal transportation approach to the decay of correlations for non-uniformly expanding maps
We consider the transfer operators of non-uniformly expanding maps for
potentials of various regularity, and show that a specific property of
potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay
of the transfer operator of the same speed than entailed by the constant
potential. The method relies neither on Markov partitions nor on inducing, but
on functional analysis and duality, through the simplest principles of optimal
transportation. As an application, we notably show that for any map of the
circle which is expanding outside an arbitrarily flat neutral point, the set of
H{\"o}lder potentials exhibiting a spectral gap is dense in the uniform
topology. The method applies in a variety of situation, including
Pomeau-Manneville maps with regular enough potentials, or uniformly expanding
maps of low regularity with their natural potential; we also recover in a
united fashion variants of several previous results.Comment: v3: The published article (Ergodic Theory Dynam. Systems 40, 2020)
contained a significant error in Lemma 2.14, used in the core Theorem 4.1.
This is a consolidated version of the article, with the error corrected (and
a few other minor points improved along the way). In order to fix the error,
the assumption on coupling in 4.1 needs to be slightly modified (see also
Definition 2.12) and Lemma 2.14 (now numbered 2.15) has been adjusted.
Section 5.1 provides a criterion to ensure this new hypothesis in our cases
of interest, so that all other results are unaffected. I apologize to readers
of the previous version for this embarrassing mistake, and warmly thank
Manuel Stadlbauer for pointing out this error to m
On differentiable compactifications of the hyperbolic space
The group of direct isometries of the real n-dimensional hyperbolic space is
G=SOo(n,1). This isometric action admits many differentiable compactifications
into an action on the closed ball. We prove that all such compactifications are
topologically conjugate but not necessarily differentiably conjugate. We give
the classifications of real analytic and smooth compactifications.Comment: 11
On Lipschitz compactifications of trees
We study the Lipschitz structures on the geodesic compactification of a
regular tree, that are preserved by the automorphism group. They are shown to
be similar to the compactifications introduced by William Floyd, and a complete
description is given.Comment: 6 page
Approximation by finitely supported measures
Given a compactly supported probability measure on a Riemannian manifold, we
study the asymptotic speed at which it can be approximated (in Wasserstein
distance of any exponent p) by finitely supported measure. This question has
been studied under the names of ``quantization of distributions'' and, when
p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi
Tessellations.Comment: v2: the main result is extended to measures defined on a manifold.
v3: references added. 25 pp. To appear in ESAIM:COC
Effective high-temperature estimates for intermittent maps
Using quantitative perturbation theory for linear operators, we prove
spectral gap for transfer operators of various families of intermittent maps
with almost constant potentials ("high-temperature" regime). H\"older and
bounded p-variation potentials are treated, in each case under a suitable
assumption on the map, but the method should apply more generally. It is
notably proved that for any Pommeau-Manneville map, any potential with
Lispchitz constant less than 0.0014 has a transfer operator acting on Lip([0,
1]) with a spectral gap; and that for any 2-to-1 unimodal map, any potential
with total variation less than 0.0069 has a transfer operator acting on BV([0,
1]) with a spectral gap. We also prove under quite general hypotheses that the
classical definition of spectral gap coincides with the formally stronger one
used in (Giulietti et al. 2015), allowing all results there to be applied under
the high temperature bounds proved here: analyticity of pressure and
equilibrium states, central limit theorem, etc.Comment: v2: minor corrections and clarifications. To appear in ETDS; Ergodic
Theory and Dynamical Systems, Cambridge University Press (CUP), 201
A geometric study of Wasserstein spaces: Hadamard spaces
Optimal transport enables one to construct a metric on the set of
(sufficiently small at infinity) probability measures on any (not too wild)
metric space X, called its Wasserstein space W(X). In this paper we investigate
the geometry of W(X) when X is a Hadamard space, by which we mean that has
globally non-positive sectional curvature and is locally compact. Although it
is known that -except in the case of the line- W(X) is not non-positively
curved, our results show that W(X) have large-scale properties reminiscent of
that of X. In particular we define a geodesic boundary for W(X) that enables us
to prove a non-embeddablity result: if X has the visibility property, then the
Euclidean plane does not admit any isometric embedding in W(X).Comment: This second version contains only the first part of the preceeding
one. The visibility properties of W(X) and the isometric rigidity have been
split off to other articles after a referee's commen
The linear request problem
We propose a simple approach to a problem introduced by Galatolo and
Pollicott, consisting in perturbing a dynamical system in order for its
absolutely continuous invariant measure to change in a prescribed way. Instead
of using transfer operators, we observe that restricting to an infinitesimal
conjugacy already yields a solution. This allows us to work in any dimension
and dispense from any dynamical hypothesis. In particular, we don't need to
assume hyperbolicity to obtain a solution, although expansion moreover ensures
the existence of an infinite-dimensional space of solutions.Comment: v2: the approach has been further simplified, only basic differential
calculus is in fact needed instead of basic PD
The Cartan-Hadamard conjecture and The Little Prince
The generalized Cartan-Hadamard conjecture says that if is a domain
with fixed volume in a complete, simply connected Riemannian -manifold
with sectional curvature , then the boundary of
has the least possible boundary volume when is a round -ball with
constant curvature . The case and is an old result
of Weil. We give a unified proof of this conjecture in dimensions and
when , and a special case of the conjecture for \kappa
\textless{} 0 and a version for \kappa \textgreater{} 0. Our argument uses a
new interpretation, based on optical transport, optimal transport, and linear
programming, of Croke's proof for and . The generalization to
and is a new result. As Croke implicitly did, we relax the
curvature condition to a weaker candle condition
or .We also find counterexamples to a na\"ive
version of the Cartan-Hadamard conjecture: For every \varepsilon
\textgreater{} 0, there is a Riemannian 3-ball with
-pinched negative curvature, and with boundary volume bounded
by a function of and with arbitrarily large volume.We begin with
a pointwise isoperimetric problem called "the problem of the Little Prince."
Its proof becomes part of the more general method.Comment: v3: significant rewritting of some proofs, a mistake in the proof of
the ball counter-example has been correcte
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