67 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
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
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
Mixed sectional-Ricci curvature obstructions on tori
We establish new obstruction results to the existence of Riemannian metrics
on tori satisfying mixed bounds on both their sectional and Ricci curvatures.
More precisely, from Lohkamp's theorem, every torus of dimension at least three
admits Riemannian metrics with negative Ricci curvature. We show that the
sectional curvature of these metrics cannot be bounded from above by an
arbitrarily small positive constant. In particular, if the Ricci curvature of a
Riemannian torus is negative, bounded away from zero, then there exist some
planar directions in this torus where the sectional curvature is positive,
bounded away from zero. All constants are explicit and depend only on the
dimension of the torus
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
Optimal transport and dynamics of expanding circle maps acting on measures
Using optimal transport we study some dynamical properties of expanding
circle maps acting on measures by push-forward.
Using the definition of the tangent space to the space of measures introduced
by Gigli, their derivative at the unique absolutely continuous invariant
measure is computed. In particular it is shown that 1 is an eigenvalue of
infinite multiplicity, so that the invariant measure admits many deformations
into nearly invariant ones. As a consequence, we obtain counter-examples to an
infinitesimal version of Furstenberg's conjecture.
We also prove that this action has positive metric mean dimension with
respect to the Wasserstein metric.Comment: 35 pages; v4 includes a corrigendum (Lemma 4.2 statement and proofs
are corrected without influence on the main results) and an addendum
(application to an infinitesimal version of Furstenberg Conjecture, Theorem
1.7 and Corollary 1.8
Bad cycles and chaos in iterative Approval Voting
We consider synchronized iterative voting in the Approval Voting system. We
give examples with a Condorcet winner where voters apply simple, sincere,
consistent strategies but where cycles appear that can prevent the election of
the Condorcet winner, or that can even lead to the election of a ''consensual
loser'', rejected in all circumstances by a majority of voters. We conduct
numerical experiments to determine how rare such cycles are. It turns out that
when voters apply Laslier's Leader Rule they are quite uncommon, and we prove
that they cannot happen when voters' preferences are modeled by a
one-dimensional culture. However a slight variation of the Leader Rule
accounting for possible draws in voter's preferences witnesses much more bad
cycle, especially in a one-dimensional culture.Then we introduce a
continuous-space model in which we show that these cycles are stable under
perturbation. Last, we consider models of voters behavior featuring a
competition between strategic behavior and reluctance to vote for candidates
that are ranked low in their preferences. We show that in some cases, this
leads to chaotic behavior, with fractal attractors and positive entropy.Comment: v2: added a numerical study of rarity of bad cycles and equilibriums,
and a case of chaotic Continuous Polling Dynamics. Many other improvements
throughout the tex
Optimal transportation and stationary measures for Iterated Function Systems
In this article we show how ideas, methods and results from optimal
transportation can be used to study various aspects of the stationary
measuresof Iterated Function Systems equipped with a probability distribution.
We recover a classical existence and uniqueness result under a
contraction-on-average assumption, prove generalized moment bounds from which
tail estimates can be deduced, consider the convergence of the empirical
measure of an associated Markov chain, and prove in many cases the Lipschitz
continuity of the stationary measure when the system is perturbed, with as a
consequence a "linear response formula" at almost every parameter of the
perturbation.Comment: v3- small typos corrected. v2- many small modifications throughout,
added a bibliographical section, improved the exponential moment estimate for
the hyperbolic-parabolic example. Mathematical Proceedings, Cambridge
University Press (CUP), In pres
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