The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...

This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodic aspect, the formulae of measure-theoretic entropy, topological entropy and topological pressure are given in closed forms and Parry measure is demonstrated to be an equilibrium measure for some potential function. For topological aspect, an example is examined to show that the exhibition of snap-b...

We give an algorithm, based on the φ-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is to solve an inverse problem for the dynamical systems βx + α mod 1. The first part is an exposition of the φ-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the φ-expansion, necessary and sufficient conditions, w...

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 = (α0α1 . . . αl−1) as the first l letters in their itineraries. We point out that the difference between the topological entropies of the attractor an...

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of [PS05, PS07]. We show that under some natural assumptions on the inducing schemes – which hold for many known examples – any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the in...

Indeed, Margulis [lo] announced that for the geodesic flow on a d-dimensional compact manifold of curvature 1 the number of periodic orbits 7 with (minimal) period r(1) I x is asymptotic to ecd-‘jx/(d 1)x. This result bears a striking resemblance to the prime number theorem. Parry and Pollicott [12], following earlier work by Bowen [2, 41, generalized Margulis’ theorem to weakly mixing Axiom A ...

Let β > 1 be an algebraic number. The relations between the coefficient vector of its minimal polynomial and the digits of the Rényi β-expansion of unity are investigated in terms of the germ of curve associated with β, which is constructed from the Salem parametrization, and the Parry Upper function fβ(z). If β is a Parry number, the Parry Upper function fβ(z) is simply related to the dynamica...

Dr. Caleb Hillier Parry, 1755-1822, is remembered as a distinguished Bath physician, original describer of several syndromes, geologist and eminent agriculturalist, creator of the Parry Library, and close friend and collaborator of Edward Jenner. Scion of a landed family in Pembrokeshire, Parry began his education at a private school in Cirencester, where he first met his lifelong friend, Edwar...