In this article we survey the mathematical research of the late William (Bill) Parry, FRS.

Let A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

We consider C2 Hénon-like families of diffeomorphisms of and study the boundary of the region of parameter values for which the non-wandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to the bifurcation parameter and even, to some extent, at the bifurcation parameter.

We construct conjugacies between linear and nonlinear non-uniform exponential contractions with discrete time. We also consider the general case of a non-autonomous dynamics defined by a sequence of maps. The results are obtained by considering both linear and nonlinear perturbations of the dynamics xm+1=Amxm defined by a sequence of linear operators Am. In the case of conjugacies between linear contractions we describe them explicitly. All the conjugacies are locally Hölder, and in fact are locally Lipschitz outside the origin. We also construct conjugacies between linear and nonlinear non-uniform exponential dichotomies, building on the arguments for contractions. All the results are obtained in Banach spaces.

We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

Every subgroup of the symmetric group defines a natural factor of the Cartesian power of a transformation. We calculate the set of values of the spectral multiplicity function of such factors (under certain conditions on the transformation) in terms of the number of orbits of diagonal actions of these subgroups. An analogous statement also holds on the unitary level for operators that preserve 1. In particular, we prove that for every positive integer n, there exists a transformation which is mixing of all orders and has a staircase multiplicity function of length n; that is, the essential values of the spectral multiplicity function are {1,2,…,n}.

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

In Bunimovich and Del Magno [Semi-focusing billiards: hyperbolicity. Comm. Math. Phys.262 (2006), 17–32], we proved that billiards in certain three-dimensional convex domains are hyperbolic. In this paper, we continue the study of these systems, and prove that they enjoy the Bernoulli property. This result answers affirmatively a long-standing question on the existence of ergodic billiards in convex domains in dimensions greater than two. Besides, it shows that the chaotic components of the first rigorously investigated three-dimensional billiards with mixed phase space (mushroom billiards), introduced in Bunimovich and Del Magno, are in fact Bernoulli.

We consider Ledrappier’s dynamical system, which was the first example of a -action which is 2-mixing but not 3-mixing. Our main result is that, excluding certain small ‘constructible’ sets, the system is mixing of every order.

A simple class of chaotic systems in a random environment is considered and their shadowing properties are studied. As an example of application, the fluctuation theorem is extended under the assumption of reversibility.

Let σ:ΣA→ΣA be a subshift of finite type, let be the set of all σ-invariant Borel probability measures on ΣA, and let be a Hölder continuous observable. There exists at least one σ-invariant measure μ which maximizes . The following question was asked by B. R. Hunt, E. Ott and G. Yuan: how quickly can the maximum of the integrals be approximated by averages along periodic orbits of period less than p? We give an example of a Hölder observable f for which this rate of approximation is slower than stretched-exponential in p.

For certain analytic data, we show that the eigenvalue sequence of the associated transfer operator ℒ is insensitive to the holomorphic function space on which ℒ acts. Explicit bounds on this eigenvalue sequence are established.

Let be the convex set of Borel probability measures on the circle invariant under the action of the transformation (mod 1). Its projection on the complex plane by the application is a compact convex subset of the unit disc, symmetric with respect to the x-axis, called the ‘fish’ by Bousch. Seeing the boundary of the upper half-fish as a function, we focus on its local regularity. We show that its multifractal spectrum is concentrated at , but that every pointwise regularity is realized in an uncountable dense set of points. The results rely on fine properties of Sturm measures.

In this paper, a group shift is an expansive action of on a compact metrizable zero-dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equal-entropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraically equivalent to these Bernoulli factors. A completely positive entropy group (even vector) shift need not be topologically conjugate to a Bernoulli shift, and the Pinsker factor of a vector shift need not split topologically.

Let G be a group and let be a subsemigroup. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of interaction groups which consists of an assignment of a positive operator Vg on A for each g in G, obeying a partial group law, and such that (Vg,Vg−1) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism crossed product constructions.

We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting nonlinearizable maps, a common hedgehog of Hausdorff dimension one, the minimum possible.

In the Michael Herman Memorial Volume, we stated and proved some rather simple invariant-manifold theorems, having many old and new applications. This paper presents the final state of the Lipschitzian part of the theory: the results are put into a more general framework and new properties are established. Both the hypotheses and the proofs of all our statements are quite simple. Smoothness will be treated as in Chaperon [Stable manifolds and the Perron–Irwin method. Ergod. Th. & Dynam. Sys.24 (2004), 1359–1394] in a forthcoming book.

Soit (M,g) une variété lorentzienne analytique réelle de dimension 3 compacte et connexe. Nous démontrons que l’existence d’une orbite ouverte (non vide) du pseudo-groupe des isométries locales implique que la métrique lorentzienne est localement homogène (i.e. le pseudo-groupe des isométries locales de g agit transitivement sur M).

In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction the flow in that direction is either completely periodic or minimal, and yet have minimal but non-uniquely ergodic directions.

We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence. Conditions for topological conjugacy to the closely related Toeplitz minimal set are also derived.