Shift-invariant Measures Research Articles

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.

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all \(\psi \)-mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.

Spectral metric spaces for Gibbs measures

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connesʼ corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-⁎-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.

Absolutely continuous random power series in reciprocals of Pisot numbers

We construct a continuum of shift invariant Borel probability measures on {−1,1}N such that corresponding random power series in reciprocals of Pisot numbers are absolutely continuous with respect to the Lebesgue measure.

Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology

We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet. The first turns out to be rather pathological, in that the countable space becomes neither first-countable nor sequential. Also, reversible automata form a closed set, while surjective ones are dense. The second topology, which is induced by a metric, is studied in more detail. Continuity of composition (under certain restrictions) and inversion, as well as closedness of the set of surjective automata, are proved, and some counterexamples are given. We then generalize this space, in the sense that every shift-invariant measure on the configuration space induces a pseudometric on cellular automata, and study the properties of these spaces. We also characterize the pseudometric spaces using the Besicovitch distance, and show a connection to the first (pathological) space.

Inverting the Furstenberg correspondence

Given a sequence of sets $A_n \subseteq \{0,\ldots,n-1\}$, the Furstenberg correspondence principle provides a shift-invariant measure on $2^N$ that encodes combinatorial information about infinitely many of the $A_n$'s. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large $n$ has to be to yield a sufficiently good approximation. We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on $2^N$, there is a computable element of $2^N$ that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse.

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.

Uncomputably Noisy Ergodic Limits

V'yugin has shown that there are a computable shift-invariant measure on Cantor space and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages A_n f. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given epsilon.

On the finite-dimensional marginals of shift-invariant measures

AbstractLet Σ be a finite alphabet, Ω=Σℤdequipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐnof ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different whend=1 and whend≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimensiond≥2 there are also extreme points which arise in other ways. We show that ℐn=ℐlocnwhend=1 , but in higher dimensions they differ fornlarge enough. We also show that while in dimension one ℐnare polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐnfor all large enoughn.

Equilibrium states for factor maps between subshifts

Let π : X → Y be a factor map, where ( X , σ X ) and ( Y , σ Y ) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a = ( a 1 , a 2 ) ∈ R 2 with a 1 > 0 and a 2 ⩾ 0 . Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes ∫ f d μ + a 1 h μ ( σ X ) + a 2 h μ ∘ π − 1 ( σ Y ) . In particular, taking f ≡ 0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a 1 h μ ( σ X ) + a 2 h μ ∘ π − 1 ( σ Y ) , which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dim H μ = dim H K .

Fisher Information and Semiclassical Treatments

We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.

Uniform Distributions on the Natural Numbers

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of ℕ for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.

Continuity of Information Transport in Surjective Cellular Automata

We introduce a local version of the Shannon entropy in order to describe information transport in spatially extended dynamical systems, and to explore to what extent information can be viewed as a local quantity. Using an appropriately defined information current, this quantity is shown to obey a local conservation law in the case of one-dimensional reversible cellular automata with arbitrary initial measures. The result is also shown to apply to one-dimensional surjective cellular automata in the case of shift-invariant measures. Bounds on the information flow are also shown.

Erdős measures, sofic measures, and Markov chains

We consider the random variable ζ = ξ1ρ+ξ2ρ2+…, where ξ1, ξ2, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξi = 0) = p0, P(ξi = 1) = p1, 0 < p0 < 1. Let β = 1/ρ be the golden number.The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η1ρ + η2ρ2 + … where the random variables ηk are {0, 1}-valued and ηkηk+1 = 0. The infinite random word η = η1η2 … ηn … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous.We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure.We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ1, ξ2, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles.

Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules

In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin.<br> First we show a representation result for classes of permutative cellular automata: those with associative type local rule are the product of a group cellular automaton with a translation map, and if they satisfy a scaling condition, they are the product of an affine cellular automaton (the alphabet is an Abelian group) with a translation map.<br> For cellular automata of this type with an Abelian factor group, and starting from a translation invariant probability measure with complete connections and summable decay, it is shown that the Cesàro mean of the iteration of this measure by the cellular automaton converges to the product of the uniform Bernoulli measure with a shift invariant measure.<br> Finally, the following characterization is shown for affine cellular automaton whose alphabet is a group of prime order: the uniform Bernoulli measure is the unique invariant probability measure which has positive entropy for the automaton, and is either ergodic for the shift or ergodic for the $\mathbb Z^2$-action induced by the shift and the automaton, together with a condition on the rational eigenvalues of the automaton.<br>

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.

Local Geometry of Fractals Given by Tangent Measure Distributions

Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ.

Measures that maximize weighted entropy for factor maps between subshifts of finite type

Let X, Y be topologically mixing subshifts of finite type and \pi : X \rightarrow Y a factor map. For each \alpha \geq 0, the weighted entropy function \phi_{\alpha} is defined by \phi_{\alpha} (\mu) = h (\mu) + \alpha h (\pi \mu) for each invariant measure \mu on X. To investigate whether for a given \alpha &gt; 0 there is a unique measure which achieves \sup_{\mu} \phi_{\alpha} (\mu), we use the concept of compensation functions which was first considered by Boyle and Tuncel and has been developed by Walters. We prove that if there is a certain kind (more general than summable variation) of compensation function, then for each \alpha \geq 0 the shift-invariant measure which maximizes the weighted entropy is unique. In particular, if the compensation function is locally constant, then the unique measure is Markov and mixing. We classify the 1-block codes from a 3-symbol subshift of finite type to a 2-symbol subshift in terms of what type of compensation function exists or does not exist, providing examples of factor maps which do and do not satisfy the hypothesis. Also we study general properties of compensation functions and the maximal weighted entropy map as a function of the weight.

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 \(A(\rho)\).

Shift invariant measures and simple spectrum

We consider some descriptive properties of supports of shift invariant measures on $ℂ^{ℤ}$ under the assumption that the closed linear span (in $L^{2}$) of the co-ordinate functions on $ℂ^{ℤ}$ is all of $L^{2}$.