Parry Measure Research Articles

Some measure rigidity and equidistribution results for β-maps

AbstractWe prove $\times a \times b$ measure rigidity for multiplicatively independent pairs when $a\in \mathbb {N}$ and $b>1$ is a ‘specified’ real number (the b-expansion of $1$ has a tail or bounded runs of $0$ s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $\times a$ invariant measure. The quantitative version together with the $\times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a-shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.

On the Perron root and eigenvectors associated with a subshift of finite type

In this paper, we describe the relationship between the Perron root and eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. As an application, we obtain the Perron eigenvectors for irreducible (0,1) matrices which are adjacency matrices for directed graphs. Moreover, we derive an alternate definition of the Parry measure in ergodic theory on an irreducible subshift of finite type.

Ahlfors regularity and fractal dimension of Smale spaces

AbstractWe prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen’s construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry’s measure, which is Ahlfors regular. By extending Bowen’s construction, we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff, box-counting and Assouad dimensions of a large class of Smale spaces.

On Normality in Shifts of Finite Type

In this paper we consider the notion of normality of sequences in shifts of finite type. A sequence is normal if the frequency of each block exists and is equal to the Parry measure of the block. We give a characterization of normality in terms of incompressibility by lossless transducers. The result was already known in the case of the full shift.

Topological conjugacy of topological Markov shifts and Ruelle algebras

We will characterize topological conjugation for two-sided topological Markov shifts (¯¯¯¯¯XA,¯¯¯σA) in terms of the associated asymptotic Ruelle C∗-algebra RA and its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical circle action. We will also show that the extended Ruelle algebra ˜RA, which is a unital and purely infinite version of RA, together with its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical torus action γA is a complete invariant for topological conjugacy of (¯¯¯¯¯XA,¯¯¯σA). The diagonal action of γA has a unique KMS-state on ˜RA, which is an extension of the Parry measure on ¯¯¯¯¯XA.

Equidistribution from fractal measures

We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.

A maximal entropy stochastic process for a timed automaton

Several ways of assigning probabilities to runs of timed automata (TA) have been proposed recently. When only the TA is given, a relevant question is to design a probability distribution which represents in the best possible way the runs of the TA. We give an answer to this question using a maximal entropy approach. We introduce our variant of a stochastic model, the stochastic process over runs, which permits to simulate random runs of any given length with a linear number of atomic operations. We adapt the notion of Shannon (continuous) entropy to such processes. Our main contribution is an explicit formula defining a process Y⁎ which maximizes the entropy. This ensures that, among the stochastic process over runs of a given TA, Y⁎ is the one that permits to sample runs of the TA in the most uniform way possible. Hence, our method could be used in a statistical model checking framework, providing a non-trivial yet natural way to generate runs in a quasi-uniform manner (described in the article). The formula defining Y⁎ is an adaptation of the so-called Shannon–Parry measure to the timed automata setting. We also show that Y⁎ enjoys well known properties in ergodic and information theory, namely, Y⁎ is ergodic and satisfies an asymptotic equipartition property.

Effective uniqueness of Parry measure and exceptional sets in ergodic theory

It is known that hyperbolic dynamical systems admit a unique invariant probability measure with maximal entropy. We prove an effective version of this statement and use it to estimate an upper bound for Hausdorff dimension of exceptional sets arising from dynamics.

NONSTATIONARY MIXING AND THE UNIQUE ERGODICITY OF ADIC TRANSFORMATIONS

We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

The Topological Pressure of Linear Cellular Automata

This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.

Almost Sure Stability of Discrete-Time Switched Linear Systems: A Topological Point of View

In this paper, we study the stability of discrete-time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem. A sufficient and necessary condition for $\mu_A$-almost sure stability is derived, where $\mu_A$ is the Parry measure of the topological Markov chain with a prescribed transition (0,1)-matrix A. The obtained $\mu_A$-almost sure stability is invariant under small perturbations of the system. The topological description of stable processes of switched linear systems in terms of Hausdorff dimension is given, and it is shown that our approach captures the maximal set of stable processes for linear switched systems. The obtained results cover the stochastic Markov jump linear systems, where the measure is the natural Markov measure defined by the transition probability matrix. Two examples are provided to illustrate the theoretical outcomes of the paper.

Parry measure and the topological entropy of chaotic repellers embedded within chaotic attractors

We study the topological entropy of chaotic repellers formed by those points in a given chaotic attractor that never visit some small forbidden hole-region in the phase space. The hole is a set of points in the phase space that have a sequence α=( α 0 α 1⋯ α l−1 ) as the first l letters in their itineraries. We point out that the difference between the topological entropies of the attractor and the embedded repeller is for most choices of α approximately equal to the Parry measure corresponding to α, μ P( α). When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than μ P( α). This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure μ P( α), and the topological entropies of the two chaotic sets.

Non-uniqueness of measures of maximal entropy for subshifts of finite type

AbstractIt is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

Kneading of Lorenz type for intervals and product spaces

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.