William Parry

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.

Compact abelian group extensions of discrete dynamical systems

SummaryThis paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramovs quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.

Synchronisation of canonical measures for hyperbolic attractors

Under suitable conditions it is shown how to change the velocity of aC2 AxiomA attractor so that the Sinai-Ruelle-Bowen measure coincides with the measure of maximal entropy. These measures are obtained as limits of certain closed orbital measures.

An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions

Following the classical procedure developed by Wiener and Ikehara for the proof of the prime number theorem we find an asymptotic formula for the number of closed orbits of a suspension of a shift of finite type when the suspended flow is topologically weak-mixing and when the suspending function is locally constant.

STABLE ERGODICITY OF SKEW EXTENSIONS BY COMPACT LIE GROUPS

Abstract We extend recent results of Adler, Kitchens & Shub, Parry, and Parry & Pollicott on the stable ergodicity and mixing of toral extensions to skew extensions by compact connected Lie groups. We show that Holder continuous extensions by a compact Lie group over a hyperbolic attractor are generically stably ergodic. If the group is compact semisimple, then Holder continuous extensions over hyperbolic sets are generically stably ergodic.

The Chebotarov theorem for Galois coverings of Axiom A flows

We consider G (Galois) coverings of Axiom A flows (restricted to basic sets) and prove an analogue of Chebotarevs theorem. The theorem provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class in G. An application answers a question raised by J. Plante. The basic method is then extended to compact group extensions and applied to frame bundle flows defined on manifolds of variable negative curvature.

Endomorphisms of a Lebesgue space

0. Introduction. Considerable progress has been made in the classification of measure preserving transformations during the last thirteen years, reaching a high point with the recent work of Ornstein [1]. Most of this theory has concentrated on invertible transformations (automorphisms) since it was here that the essential problems awaited solution. Viewed as two-sided shifts on symbol spaces, an isomorphism between invertible transformations amounts to a faithful coding between their respective infinite messages. A new problem appears, however, if a coding which does not anticipate the future is required. From this point of view such a coding establishes a correspondence between their associated one-sided shifts. This is one motivation for pursuing the classification of transformations which are not necessarily invertible (endomorphisms). No proofs will be given in this paper although it should be noted that one of the principal invariants mentioned here appears in [2]. We should also like to refer interested readers to the recent work of Versik [3], [4], to Rohlins paper [5] and the closing remarks of Rohlin in [6]. (The question raised in the last paragraph of [6] has a negative answer.)

The Livšic periodic point theorem for non-abelian cocycles

For hyperbolic systems and for Holder cocycles with values in a compact metric group, we extend Livsics periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.

Central limit asymptotics for shifts of finite type

We study the rate of convergence and asymptotic expansions in the central limit theorem for the class of Hölder continuous functions on a shift of finite type endowed with a stationary equilibrium state. It is shown that the rate of convergence in the theorem isO(n−1/2) and when the function defines a non-lattice distribution an asymptotic expansion to the order ofo(n−1/2) is given. Higher-order expansions can be obtained for a subclass of functions. We also make a remark on the central limit theorem for (closed) orbital measures.

SKEW PRODUCTS OF SHIFTS WITH A COMPACT LIE GROUP

We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew-products σ f where f : X → G is Holder and G is a compact Lie group. We show that generically σ f is weak-mixing and give constructive methods for achieving weak-mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σ f is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Holder continuous. This leads to conditions for weak-mixing or ergodicity in terms of a functional equation which involves Holder rather than measurable functions.