Permalinkhttps://doi.org/10.5802/ahl.24

Abstract

We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in $\mathrm{SL}(2,ℂ)$, as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on ${\mathbb{P}}^1\left(ℂ\right)$.

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Permalinkhttps://doi.org/10.5802/ahl.23

KeywordsStochastic Navier–Stokes system, large deviations principle, occupation measures, multiplicative ergodicity

Abstract

In this paper, we consider the 2D Navier–Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical systems, based on a Kifer-type criterion and a multiplicative ergodic theorem.

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Permalinkhttps://doi.org/10.5802/ahl.22

KeywordsIntegral $p$-adic Hodge theory, Harder–Narasimhan filtrations, Breuil–Kisin–Fargues modules

Abstract

We define and study Harder–Narasimhan filtrations on Breuil–Kisin–Fargues modules and related objects relevant to $p$-adic Hodge theory.

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Permalinkhttps://doi.org/10.5802/ahl.21

KeywordsChacon infinite measure preserving transformation, rank-one transformation, joinings.

Abstract

We construct an infinite measure preserving version of Chacon transformation, and prove that it has a property similar to Minimal Self-Joinings in finite measure: its Cartesian powers have as few invariant Radon measures as possible.

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Permalinkhttps://doi.org/10.5802/ahl.20

Keywordsrandom interlacements, range, transience, simple random walk, Doob’s $h$-transform

Abstract

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is “almost recurrent” in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a “large” set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset {ℝ}^2$ that does not “surround” the origin, we prove that a.s. there is an infinite number of $k$’s such that $kG\cap {\mathbb{Z}}^2$ is unvisited. These results suggest that the range of the conditioned walk has “fractal” behavior.

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Permalinkhttps://doi.org/10.5802/ahl.19

KeywordsSchrödinger equation, control/observability for PDE, semiclassical analysis

Abstract

The purpose of this note is to use the results and methods of [BBZ13] and [BZ12] to obtain control and observability by rough functions and sets on 2-tori, ${𝕋}^2={ℝ}^2/\mathbb{Z}\oplus \gamma \mathbb{Z}$. We show that for a non-trivial $W\in L^{\infty }\left({𝕋}^2\right)$, solutions to the Schrödinger equation, $(i{\partial }_t+\Delta )u=0$, satisfy ${\parallel u|}_{t=0}{\parallel }_{L^2\left({𝕋}^2\right)}\le K_T{\parallel Wu\parallel }_{L^2([0,T]\times {𝕋}^2)}$. In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions in $L^2\left({𝕋}^2\right)$ and controls in $L^4([0,T]\times {𝕋}^2)$. For continuous $W$ this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for ${𝕋}^2={ℝ}^2/{\left(2\pi \mathbb{Z}\right)}^2$ (the rational torus) and $T>\pi$ this can be deduced from the results of Jakobson [Jak97].

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Permalinkhttps://doi.org/10.5802/ahl.11

KeywordsPinning model, localization transition, free energy, correlated disorder, renewal, disorder relevance, Harris criterion, smoothing inequality, second moment.

Abstract

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha >0$, when the correlated sequence is given by another independent renewal set with loop exponent $\widehat{\alpha }>0$. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\widehat{\alpha }>2$ (summable correlations), disorder is irrelevant if $\alpha <1/2$ and relevant if $\alpha >1/2$, which extends the Harris criterion for independent disorder. The case $\widehat{\alpha }\in (1,2)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\widehat{\alpha }$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\widehat{\alpha }\in (0,1)$ is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

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Permalinkhttps://doi.org/10.5802/ahl.14

KeywordsMöbius group, Lobatchevsky space, hyperbolic space, infinite-dimensional space

Abstract

Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.

The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.

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Permalinkhttps://doi.org/10.5802/ahl.18

KeywordsCremona transformations, Jonquières transformations, length, dynamical length, linear systems, base-points

Abstract

The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonquières transformation. By the famous Noether–Castelnuovo theorem, every birational transformation $f$ is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written $\mathrm{lgth}\left(f\right)$. Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the linear system of $f$.

As an application of this computation, we give a few properties of the dynamical length of $f$ defined as the limit of the sequence $n\mapsto \mathrm{lgth}\left(f^n\right)/n$. It follows for example that an element of the Cremona group is distorted if and only if it is algebraic. The computation of the length may also be applied to the so called Wright complex associated with the Cremona group: This has been done recently by Lonjou. Moreover, we show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group (the length coming from its amalgamated structure). In another direction, we compute the lengths and dynamical lengths of all monomial transformations, and of some Halphen transformations. Finally, we show that the length is a lower semicontinuous map on the Cremona group endowed with its Zariski topology.

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Permalinkhttps://doi.org/10.5802/ahl.16

KeywordsRamsey theory, fixed point properties in topological dynamics

Abstract

The Kechris–Pestov–Todorcevic correspondence connects extreme amenability of topological groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts.

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Permalinkhttps://doi.org/10.5802/ahl.17

Keywordsrigorous computing, symbolic-numeric algorithms, D-finite functions, error bounds

Abstract

We describe a flexible symbolic-numeric algorithm for computing bounds on the tails of series solutions of linear differential equations with polynomial coefficients. Such bounds are useful in rigorous numerics, in particular in rigorous versions of the Taylor method of numerical integration of ODEs and related algorithms. The focus of this work is on obtaining tight bounds in practice at an acceptable computational cost, even for equations of high order with coefficients of large degree. Our algorithm fully covers the case of generalized series expansions at regular singular points. We provide a complete implementation in SageMath and use it to validate the method in practice.

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Permalinkhttps://doi.org/10.5802/ahl.13

KeywordsSquared Bessel processes, linear birth-death processes, intertwining, continuous and discrete scaling, spectral decomposition.

Abstract

By providing instances of approximation of linear diffusions by birth-death processes, Feller [Fel50] has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one.

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Permalinkhttps://doi.org/10.5802/ahl.12

Keywordspolynomial entropy, Brouwer homeomorphisms, wandering set

Abstract

Nous nous proposons d’étudier l’entropie polynomiale de la composante errante de n’importe quel système dynamique topologique inversible. Pour illustrer cette étude, nous calculerons l’entropie polynomiale de divers homéomorphismes de Brouwer, qui sont les homéomorphismes du plan sans point fixe et préservant l’orientation. En particulier, nous verrons que l’entropie polynomiale de tels homéomorphismes peut prendre n’importe quelle valeur supérieure ou égale à 2.

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Permalinkhttps://doi.org/10.5802/ahl.15

KeywordsHausdorff dimension, limsup sets, potentials

Abstract

Generalising a construction of Falconer, we consider classes of $G_{\delta }$-subsets of ${ℝ}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.

As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for $\alpha \ge 1$,

$${\dim }_{\mathrm{H}}\{y:|T_a^n\left(x\right)-y|<n^{-\alpha }\text{infinitely}\text{often}\}=\frac{1}{\alpha },$$

for almost every $x\in [1-a,1]$, where $T_a$ is a quadratic map with $a$ in a set of parameters described by Benedicks and Carleson.

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Permalinkhttps://doi.org/10.5802/ahl.10

KeywordsAbelian varieties, covering gonality, zero-cycles, Chow ring

Abstract

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have (covering) gonality at least $f\left(g\right)$, where $f\left(g\right)$ grows like $\log g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties $A$ of dimension $g$, e.g., if $g\ge 2k-1$, the set of divisors $D\in {\mathrm{Pic}}^0\left(A\right)$ such that $D^k=0$ in ${\mathrm{CH}}^k\left(A\right)$ is at most countable.

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