Product of two staircase rank one transformations that is not loosely Bernoulli
We construct two staircase rank one transformations whose Cartesian product is not loosely Bernoulli.
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arXiv subjects
Thierry De La Rue
Publications and source records attributed to Thierry De La Rue.
Ergodic Poisson Splittings
In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.
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Poisson suspensions and Sushis
In this paper, we prove that ergodic point processes with moments of all orders, driven by particular infinite measure preserving transformations, have to be a superposition of shifted Poisson processes. This rigidity result has a lot of implications in terms of joining and disjointness for the corresponding Poisson suspension. In particular, we prove that its ergodic self-joinings are Poisson joinings, which provides an analog, in the Poissonian context, of the GAG property for Gaussian dynamical systems.
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Nearly finite Chacon Transformation
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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Invariant measures for Cartesian powers of Chacon infinite transformation
We describe all boundedly finite measures which are invariant by Cartesian powers of an infinite measure preserving version of Chacon transformation. All such ergodic measures are products of so-called diagonal measures, which are measures generalizing in some way the measures supported on a graph. Unlike what happens in the finite-measure case, this class of diagonal measures is not reduced to measures supported on a graph arising from powers of the transformation: it also contains some weird invariant measures, whose marginals are singular with respect to the measure invariant by the transformation. We derive from these results that the infinite Chacon transformation has trivial centralizer, and has no nontrivial factor. At the end of the paper, we prove a result of independent interest, providing sufficient conditions for an infinite measure preserving dynamical system defined on a Cartesian product to decompose into a direct product of two dynamical systems.
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The Chowla and the Sarnak conjectures from ergodic theory point of view
We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences of numbers in {-1,0,1}, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.
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Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals
We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
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Standardness of monotonic Markov filtrations
We derive a practical standardness criterion for the filtration generated by a monotonic Markov process. This criterion is applied to show standardness of some adic filtrations.
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A dynamical point of view on the set of B-free integers
We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free integers. Moreover, we show that the distribution of patterns in small intervals also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.
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Weak limits of powers of Chacon's automorphism
We completely describe the weak closure of the powers of the Koopman operator associated to Chacon's classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of alpha-weak mixing for Chacon's automorphism.
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On spectral disjointness of powers for rank-one transformations and Möbius orthogonality
We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chacon's maps and Katok's map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of {U_T^k: k in Z}, of "sufficiently many" analytic functions of the operator U_T. Then we apply these disjointness results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Möbius function.
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Around King's Rank-One theorems: Flows and Z^n-actions
We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of rank-one flows. In the case of rank-one Z^n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors.
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Averaging along Uniform Random Integers
Motivated by giving a meaning to "The probability that a random integer has initial digit d", we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The URI-density, obtained by averaging along the elements of E, and the local URI-density, which we get by considering the k-th element of E and letting k go to infinity. We prove that the elements of E satisfy Benford's law, both in the sense of URI-density and in the sense of local URI-density. Moreover, if b_1 and b_2 are two multiplicatively independent integers, then the mantissae of a natural number in base b_1 and in base b_2 are independent. Connections of URI-density and local URI-density with other well-known notions of densities are established: Both are stronger than the natural density, and URI-density is equivalent to log-density. We also give a stochastic interpretation, in terms of URI-set, of the H_\infty-density.
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Dynamics of $λ$-continued fractions and $β$-shifts
For a real number $0<λ<2$, we introduce a transformation $T_λ$ naturally associated to expansion in $λ$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_λ$ provides an algorithm to expand any positive real number in $λ$-continued fraction. We prove the conjugacy between $T_λ$ and some $β$-shift, $β>1$. Some properties of the map $λ\mapstoβ(λ)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.
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Zero Krengel Entropy does not kill Poisson Entropy
We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann-Kakutani odometer), but whose associated Poisson suspension has positive entropy.
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Transposition game
We introduce a two-player game, in which each player extends a given sequence by picking a free element in a domain D of the real line. The aim of the players is to control the parity of the number of transpositions necessary to put the final sequence in order. We will see that the winner can be the last player, the second last player, the first player, the second player, the person who wants the parity to end up even or the person who wants the parity to end up odd. A special case of the game can be reduced to a game with nontrivial winning strategy, but describable in so simple a way that children can understand it and enjoy playing it.
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Growth rate for the expected value of a generalized random Fibonacci sequence
A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |λg_{n-1} +/- g_{n-2} |. When λ>=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When λ= λ_k = 2 cos(π/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-λ_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author.
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Almost-sure Growth Rate of Generalized Random Fibonacci sequences
We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = λF_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |λ\widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le 1$). Our main result is that, when $λ$ is of the form $λ_k = 2\cos (π/k)$ for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le 1$, and of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive and given by $$ \int_0^\infty \log x dν_{k, ρ} (x), $$ where $ρ$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ν_{k, ρ}$ is an explicit probability distribution on $\RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\le 1$ in the easier case $λ\ge 2$. Finally, we study the variations of the exponent as a function of $p$.
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