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Thierry de la Rue

Publications and source records attributed to Thierry de la Rue.

14 recordsLinked to original sources
arXiv2026

Frequency of patterns in smooth sequences over the alphabet \{1, 3\}

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1, 3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts, and propose some perspectives.

math.DS↗
arXiv2026

Frequencies of Patterns in Smooth Sequences Over the Alphabet $\{1,3\}$

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1,3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1,3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts.

cs.DM↗
arXiv2025

Confined Poisson extensions

This paper follows on from our previous work, where we introduced the notion of \emph{confined extensions}, and our purpose is to widen the context in which such extensions appear. We do so in the setup of Poisson suspensions: we take a $σ$-finite measure-preserving dynamical system $(X, μ, T)$ and a compact extension $(X \times G, μ\otimes m_G, T_ϕ)$, then we consider the corresponding Poisson extension $((X \times G)^*, (μ\otimes m_G)^*, (T_ϕ)_*) \overset{}{\to} (X^*, μ^*, T_*)$. Our results give two different conditions under which that extension is confined. Finally, to show that those conditions are not void, we give an example of a system $(X, μ, T)$ and a cocycle $ϕ$ so that the compact extension $(X \times G, μ\otimes m_G, T_ϕ)$ has an infinite ergodic index.

math.DS↗
arXiv2025

On orthogonality to uniquely ergodic systems

We solve Boshernitzan's problem of characterization (in terms of so called Furstenberg systems) of bounded sequences that are orthogonal to all uniquely ergodic systems. Some variations of Boshernitzan's problem involving characteristic classes are considered. As an application, we characterize sequences orthogonal to all uniquely ergodic systems whose (unique) invariant measure yields a discrete spectrum automorphism as those satisfying an averaged Chowla property.

math.DS↗
arXiv2024

Furstenberg systems of pretentious and MRT multiplicative functions

We prove structural results for measure preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwill, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

math.NT↗
arXiv2024

A central limit theorem for the variation of the sum of digits

We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $μ^{(r)}(d)$ as the density of integers $n\in \mathbb{N}$ for which the sum of digits increases by $d$ when we add $r$ to $n$. We give a probabilistic interpretation of $μ^{(r)}$ on the probability space given by the group of $b$-adic integers equipped with the normalized Haar measure. We split the base-$b$ expansion of the integer $r$ into so-called "blocks", and we consider the asymptotic behaviour of $μ^{(r)}$ as the number of blocks goes to infinity. We show that, up to renormalization, $μ^{(r)}$ converges to the standard normal law as the number of blocks of $r$ grows to infinity. We provide an estimate of the speed of convergence. The proof relies, in particular, on a $ϕ$-mixing process defined on the $b$-adic integers.

math.PR↗
arXiv2023

Joining properties of automorphisms disjoint with all ergodic systems

We study the class $Erg^\perp$ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $Erg^\perp,$ that is, each automorphism whose every joining with an element of $Erg^{\perp}$ yields a system which is again an element of $Erg^{\perp}$, must be an identity. Despite this fact, we show that $Erg^\perp$ is closed by taking Cartesian products. Finally, we prove that there are non-identity elements in $Erg^\perp$ whose self-joinings always yield elements in $Erg^\perp$. This shows that there are non-trivial characteristic classes included in $Erg^\perp$.

math.DS↗
arXiv2021

On arithmetic functions orthogonal to deterministic sequences

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic M\''obius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

math.DS↗
arXiv2021

Classification of Backward Filtrations and Factor Filtrations: Examples from Cellular Automata

We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik's theory, but we also study them from another point of view that takes into account the measurepreserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.

math.DS↗
arXiv2018

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},κ,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbolμ$ in all zero entropy systems, in particular, it makes $\boldsymbolμ$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

math.DS↗
arXiv2017

On typicality of translation flows which are disjoint with their inverse

In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_δ$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_δ$-condition.

math.DS↗
arXiv2007

How to clean a dirty floor: Probabilistic potential theory and the Dobrushin uniqueness theorem

Motivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let αbe a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" β_h = I_{1-h} + I_h αfor h: X \to [0,1] (here I_f denotes the diagonal matrix with entries f). We ask: For which "cleaning sequences" h_1, h_2, ... do we have c β_{h_1} ... β_{h_n} \to 0 for a suitable class of "dirt vectors" c? We show, under a modest condition on α, that this occurs whenever \sum_i h_i = \infty everywhere on X. More generally, we analyze the cleaning of subsets Λ\subseteq X and the final distribution of dirt on the complement of Λ. We show that when supp(h_i) \subseteq Λwith \sum_i h_i = \infty everywhere on Λ, the operators β_{h_1} ... β_{h_n} converge as n \to \infty to the "balayage operator" Π_Λ= \sum_{k=0}^\infty (I_Λα)^k I_{Λ^c). These results are obtained in two ways: by a fairly simple matrix formalism, and by a more powerful tree formalism that corresponds to working with formal power series in which the matrix elements of αare treated as noncommuting indeterminates.

math.PR↗
arXiv2005

Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on {0,...,l-1}, we show for a suitable class of functions g that these quantities, once properly renormalized, converge to (part of) the graph of a self-affine function. The latter only depends on the ergodic component of x, and is a deformation of the so-called Blancmange function. We also briefly describe the links with a series of works on Conway recursive $10,000 sequence.

math.PR↗