Shift-invariant Probability Measures Research Articles

Borel complexity of sets of normal numbers via generic points in subshifts with specification

We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base r r expansions, and their various generalisations: generalised Lüroth series expansions and β \beta -expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in [ 0 , 1 ) [0,1) . Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a Π 3 0 \boldsymbol {\Pi }^0_3 -complete set, meaning that it is a countable intersection of F σ F_\sigma -sets, but it is not possible to write it as a countable union of G δ G_\delta -sets). We also solve a problem of Sharkovsky–Sivak on the Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence the sets of normal numbers under consideration are Π 3 0 \boldsymbol {\Pi }^0_3 -complete.

Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift $\Omega_0$ made of all valid tilings using the set $\mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $\Omega_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112 that are equivalent to a set of 19 self-similar and aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel-Rao tilings, as we believe that $\Omega_0\setminus X_0$ is a null set for any shift-invariant probability measure on $\Omega_0$. The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles. The 2 other steps involve the addition of decorations to deal with fault lines and changing the base of the $\mathbb{Z}^2$-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles.

Central limit theorems with a rate of convergence for time-dependent intermittent maps

We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-norming CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the quenched CLT holds, assuming fiberwise centering. Finally, we show a multivariate CLT for intermittent quasi-static systems. Our approach is based on Stein’s method of normal approximation.

Stochastic dominance for shift-invariant measures

Let $X$ be the full shift on two symbols. The lexicographic order induces a partial order known as first-order stochastic dominance on the collection ${\mathcal{M}}_{X}$ of its shift-invariant probability measures. We present a study of the fine structure of this dominance order, denoted by $\prec$, and give criteria for establishing comparability or incomparability between measures in ${\mathcal{M}}_{X}$. The criteria also give an insight to the complicated combinatorics of orbits in the shift. As a by-product, we give a direct proof that Sturmian measures are totally ordered with respect to $\prec$.

Steganographic capacity for one-dimensional Markov cover

Abstract For shift-invariant probability measures on the set of infinite two-sided binary sequences (one-dimensional covers) we introduce the notion of capacity as a maximum portion of embedded into the cover uniformly distributed (purely random) binary sequence (message) that admits special correction of the cover restoring its distribution up to distribution of n-tuples (subwords of some fixed length n). “Special correction” is carried out using the proposed new algorithm that changes some of the cover’s symbols not occupied by embedded message. The features of the introduced capacity are examined for the Markov cover. In particular, we show how capacity may be significantly increased by weakening of the standard constraint that positions for message embedding have to be chosen by independent unfair coin tosses. Experimental results are presented for correction of real steganographic covers after LSB-embedding.

Limiting measures for addition modulo a prime number cellular automata

Linear cellular automata have many invariant measures in general. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure - the most natural one. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. In the two-state case, we characterize when iterates by addition modulo 2 cellular automata starting from a convex combination of strong mixing probability measures can converge. This also gives all invariant measures inside the class of those probability measures. We can obtain a similar result for iterates by addition modulo an odd prime number cellular automata starting from strong mixing probability measures.

Steganographic capacity for one-dimensional Markov cover} \runningtitle{Steganographic capacity for one-dimensional Markov cover} \author*[1]{Valeriy A. Voloshko} \runningauthor{V. A. Voloshko} \affil[1]{ Belarusian State University, e-mail: valeravoloshko@yandex.ru} \abstract{For shift-invariant probability measures on the set of infinite two-sided binary sequences (one-dimensional covers) we introduce the notion of capacity as a maximum portion of embedded into the cover uniformly distributed (purely random) binary sequence (message)

Для стационарных вероятностных мер на множестве двусторонних двоичных последовательностей-контейнеров вводится понятие емкости - максимальной доли вкрапления равномерно распределенной случайной последовательности, допускающей восстановление распределения контейнера с точностью до распределения вероятностей слов заданной длины. Восстановление распределения вероятностей при этом производится не затрагивающей вкрапления корректировкой информационных знаков контейнера. На примере марковского контейнера исследуются свойства введенной емкости. В частности, показывается, что последняя может быть существенно увеличена ослаблением стандартного требования независимости знаков встраивающего процесса. Представлены результаты компьютерных экспериментов на реальных данных.

On a measure with maximal entropy for a suspension flow over a countable alphabet Markov shift

Sufficient conditions on a shift invariant probability measure on the base of a suspension flow \(\{S_t\}\) over a countable alphabet Markov shift are stated under which the corresponding \(\{S_t\}\)-invariant measure is a measure with maximal entropy.

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

We show that for every metrizable Choquet simplex K and for every group G, which is infinite, countable, amenable and residually finite, there exists a Toeplitz G-subshift whose set of shift-invariant probability measures is affine homeomorphic to K. Furthermore, we get that for every integer d > 1 and every Toeplitz flow (X, T ), there exists a Toeplitz Z-subshift which is topologically orbit equivalent to (X, T ).

A Dynamical Point of View on the Set ofℬ-Free Integers

Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow to the more general context of |$\mathscr {B}$|-free integers, that is to say integers with no factor in a given family |$\mathscr {B}$| 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 |$\mathscr {B}$|-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 obtain the abundance of twin |$\mathscr {B}$|-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form |$[N,N+ \sqrt {N})$| also conforms to the same measure. When elements of |$\mathscr {B}$| are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

Sofic random processes

For a finite alphabet Σ, a subshift Y ⊂ Σℤ and an ergodic shift invariant probability measure with support Y, the future measures of the process (Y, ν) are the conditional measures of ν on the future, given the past. We introduce sofic processes as the processes that have finitely many future measures. We characterize the weighted Shannon graphs that canonically present sofic processes that are finitarily Markovian. With an example of Furstenberg as a starting point, we characterize the weighted Shannon graphs that canonically present sofic processes that are not finitarily Markovian. The semigroup measures of Kitchens and Tuncel yield sofic processes. We show that the class of sofic processes with semigroup measures is closed under measure preserving topological conjugacy.

Which beta-shifts have a largest invariant measure?

For a given beta-shift, the lexicographic order induces a partial order (known as first-order stochastic dominance) on the collection of its shift-invariant probability measures. We characterize those beta-shifts for which this partial order has a largest element. These beta-shifts are all of finite type, and their lexicographically largest point is a periodic sequence of a particular kind: it is Sturmian (that is, its shift-orbit is combinatorially equivalent to a rotation) with weight-per-symbol either an integer, or equal to p/(ap + 1) for some a, p ⩾ 1, or equal to A + p/(p + 1) for some p ⩾ 1 and A ⩾ 2. In these cases, the largest invariant measure is precisely the unique one supported by the shift-orbit of the lexicographically largest point in the beta-shift.

Maximizing measures of generic Hölder functions have zero entropy

We prove that for a generic real-valued Hölder continuous function f on a subshift of finite type, every shift-invariant probability measure that maximizes the integral of f must have zero entropy. An immediate corollary is that zero-temperature limits of equilibrium states of certain one-dimensional lattice systems generically have zero entropy. We prove an analogous statement for generic Lipschitz observations of expanding maps of the circle.

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way.

Invariant Cocycles Have Abelian Ranges

Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle.

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

If F is an exact symplectic map on the{\it d}-dimensional cylinder \(\mathbb{T}^d \times \mathbb{R}^d\), with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow \(\phi^*\) on the space of shift-invariant probability measures on the space of configurations \(({\mathbb{R}}^d)^{\mathbb{Z}}\). Stationary points of \(\phi^*\) are invariant measures of F, and the rotation vector and all spectral invariants are invariants of \(\phi^*\). Using \(\phi^*\) and the minimisation technique, we construct minimising measures with an arbitrary rotation vector \(\rho \in \mathbb{R}^d\), and with an additional assumption that F is strongly monotone, we show that the support of every minimising measure is a graph of a Lipschitz function. Using \(\phi^*\) and the relaxation technique, assuming a weak condition on \(\phi^*\) (satisfied e.g. in Hedlund's counter-example, and in the anti-integrable limit) we show existence of double-recurrent orbits of F (and F-ergodic measures) with an arbitrary rotation vector \(\rho \in \mathbb{R}^d\), and action arbitrarily close to the minimal action \(A(\rho)\).

Limit Sets of Cellular Automata Associated to Probability Measures

We introduce the concept of limit set associated to a cellular automaton (CA) and a shift invariant probability measure. This is a subshift whose forbidden blocks are exactly those, whose probabilities tend to zero as time tends to infinity. We compare this probabilistic concept of limit set with the concepts of attractors, both in topological and measure-theoretic sense. We also compare this notion with that of topological limit set in different dynamical situations.

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 −n where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1 (Ωμ) in the metric d is $$$$

Non-Markovian reversible Chapman-Kolmogorov measures on subshifts of finite type

We consider shift-invariant probability measures on subshift dynamical systems with a transition matrixA which satisfies the Chapman-Kolmogorov equation for some stochastic matrix Π compatible withA. We call them Chapman-Kolmogorov measures. A nonequilibrium entropy is associated to this class of dynamical systems. We show that ifA is irreducible and aperiodic, then there are Chapman-Kolmogorov measures distinct from the Markov chain associated with Π and its invariant row probability vectorq. If, moreover, (q, Π) is a reversible chain, then we construct reversible Chapman-Kolmogorov measures on the subshift which are distinct from (q, Π).