Mark Pollicott

An analogue of the prime number theorem for closed orbits of Axiom A flows

For an Axiom A flow restricted to a basic set we extend the zeta function to an open set containing W(s) > h where h is the topological entropy. This enables us to give an asymptotic formula for the number of closed orbits by adapting the Wiener-Ikehara proof of the prime number theorem.

Meromorphic extensions of generalised zeta functions.

SummaryIn this paper we give a full description of the spectrum of the Ruelle-Perron-Frobenius operator acting on the Banach space of Holder continuous functions on a subshift of finite type (Theorem 1). These results are then used to extend the meromorphic domain of generalised zeta functions (Theorem 2). The most important application of these results is to the domain of the Smale zeta function for Axiom A flows (Theorem 3). In the course of this paper we settle questions raised by Ruelle and Sunada.

The Hausdorff dimension of -expansions with deleted digits

A process and apparatus for the gaseous reduction of sized iron ores wherein a steam-hydrocarbon fluid mixture, having a steam : carbon molar ratio ranging from 0.9:1 to 1.8:1, is catalytically reformed to produce a reducing gas mixture containing about 85 to 98 percent by volume carbon monoxide plus hydrogen, with a hydrogen:carbon monoxide volume ratio of at least 2:1. The reducing gas is transferred directly at elevated temperatures to a shaft furnace for reduction of ores therein, and the spent reducing gas is cleaned, cooled and dried, for use as fuel and/or cooling the reduced ores.

Escape rates for Gibbs measures

We study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of Hausdorff dimension of the survivor set.

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.

Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature

SummaryIn this article we develop a method of deriving asymptotic formulae for the orbital counting function for the action of certain discrete groups of isometries of simply connected negatively curved manifolds. We consider the particular case of normal subgroupsΓ ⊲Γ0 of a co-compact groupΓ0 for which the quotientΓ0/Γ ≌ ℤk. Even in the special case of manifolds ofconstant negative curvature, this leads to new results. In particular, we have asymptotic estimates for some groups which arenot geometrically finite.

Large deviations for intermittent maps

In this paper we study large deviation results for the Manneville-Pomeau map and related transformations to indifferent fixed points. In particular, we consider conditions under which the associated error term is polynomial or even exponential. For typical observables, polynomial estimates are optimal. However, under suitable conditions, the exponential error term arises from the compactness of the space of measures, despite the indifference of the fixed point.

Rates of mixing for potentials of summable variation

It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials. 0. Introduction In this paper we shall consider the rate of mixing of subshifts of finite type with respect to equilibrium states for potentials of summable variation. Let σ : XA → XA denote a transitive two-sided subshift of finite type and let g : XA → R be a function of summable variation. Thus there exists a unique equilibrium state μ for g [7]. Assume that f : XA → R is Hölder, then we will study the behaviour as N → +∞ of the correlation function ρ(N) := ∫ f ◦ σ .fdμ− (∫ fdμ )2 . It is a well known result that if the nth variation varn(g) tends to zero exponentially fast, then ρ(N) tends to zero exponentially fast [1] (i.e., if there exists 0 < β < 1 and C > 0 such that varn(g) ≤ Cβ, n ≥ 0, then there exists 0 < θ < 1 and D > 0 such that |ρ(N)| ≤ Dθ , N ≥ 0). In this paper we shall consider the rate at which ρ(N) tends to zero when varn(g) tends to zero at a sub-exponential rate. To help our exposition we shall concentrate on a number of particular cases. Our first main result is the following. Theorem 1. Let σ : XA → XA denote a transitive two-sided subshift of finite type. (1) Polynomially decay: If ∃r > 2, ∃C > 0 such that varn(g) ≤ C ( 1 nr ) , then |ρ(N)| = O ( 1 Nr−2− ) for any > 0. (2) Intermediate decay: If ∃0 < β < 1, ∃C > 0, ∃γ > 1 such that varn(g) ≤ C(β n) γ ), then ∃D > 0, ∃0 < θ < 1 such that |ρ(N)| = D ( θ N) γ− ) for any > 0. (3) Stretched exponential decay: If ∃0 < θ < 1, ∃C > 0 such that varn(g) ≤ C(θ 1/2 ), then ∃D > 0, ∃0 < β < 1 such that |ρ(N)| = D ( β 1/3 ) for any > 0. Received by the editors September 22, 1997. 1991 Mathematics Subject Classification. Primary 58Fxx. c ©1999 American Mathematical Society

The dimensions of some self-affine limit sets in the plane and hyperbolic sets

In this article we compute the Hausdorff dimension and box dimension (or capacity) of a dynamically constructed model similarity process in the plane with two distinct contraction coefficients. These examples are natural generalizations to the plane of the simple Markov map constructions for Cantor sets on the line. Some related problems have been studied by different authors; however, those results are directed toward generic results in quite general situations. This paper concentrates on computing explicit formulas in as many specific cases as possible. The techniques of previous authors and ours are correspondingly very different. In our calculations, delicate number-theoretic properties of the contraction coefficients arise. Finally, we utilize the results for the model problem to compute the dimensions of some affine horseshoes in ℝn, and we observe that the dimensions do not always coincide and their coincidence depends on delicate number-theoretic properties of the Lyapunov exponents.

Exponential mixing for the geodesic flow on hyperbolic three-manifolds

We give a short and direct proof of exponential mixing of geodesic flows on compact hyperbolic three-manifolds with respect to the Liouville measure. This complements earlier results of Collet-Epstein-Gallovotti, Moore, and Ratner for hyperbolic surfaces. Furthermore, since the analysis is even easier in three dimensions than in two dimensions (because of the absence of discrete series and the simplicity of the zonal spherical functions in this case), this apparently gives the simplest example of a flow with exponential mixing.