Arithmetic dynamics is discrete-time dynamics (function iteration) over discrete algebraic/arithmetical sets, such as algebraic number fields and rings, polynomial rings, finite fields, p-adic fields, algebraic curves, etc. This rapidly growing area of research lies at the interface between dynamics and algebra, particularly algebraic number theory. It is rich in history and motivations, and is a fertile ground for the development of algorithmic and computer-oriented theories.

Some constructs of algebraic dynamics (periodic orbits, attractors, Siegel discs), are clones of familiar dynamical objects. Others, like entropy or bifurcations, expose unexpected connections. The probabilistic phenomena are the most intriguing, being rooted in fluctuations of arithmetical origin. There one easily finds problems of considerable difficulty.


The strategic goal of my research in this area is to understand the arithmetical roots of chaotic behaviour over discrete sets, and to explore connections between the theory of dynamical systems and the theory of numbers. My perspective is that of a dynamicist.

As for dynamical systems, the basic models are linear. In my early work, I developed (with I Percival and M Bartuccelli) an arithmetical theory of strongly chaotic linear systems on the torus, applying classical ideal theory. Later (with D Arrowsmith), I have used p-adic arithmetic to characterize the process of linearization (Siegel discs) in simple discrete chaotic systems, elucidating some connections with the theory of formal groups.

The study of periodic orbits of polynomials and rational functions requires Galois theory, and gives rise to interesting dynamical zeta functions (with S Hatjispyros). I have exported the dynamical concept of bifurcation to dynamical systems over integral domains (with P Morton), by relating bifurcational states to zeros of certain discriminants. I have formulated some problems in cellular automata in the language of polynomials over finite fields.


Iintegrable vs. non-integrable behaviour is a main theme in dynamics. John Robert and I have explored various manifestations of this this dichotomy in algebraic mappings, when they are represented over finite fields. The distribution of orbit lengths turns out to be the relevant observable. (Its role is structurally similar to the distribution of energy level spacing in quantum dynamics.) We find that the presence of an integral of the motion causes a `quantization' of the cycle lengths, and we connected the resulting singular period distribution to problems in the theory of elliptic curves. The presence or absence of time-reversal symmetry also has a clear signature on a finite phase space. Based on strong experimental evidence, we conjectured the existence of a universal smooth (Gamma) distribution of cycle lengths for two-dimensional reversible algebraic mappings over finite fields. Subsequently, we have shown that this universal distribution is the limiting period distribution for the composition of two random involutions of a large, finite space, provided that the periods are suitably scaled, in accordance with the number of fixed points of the involutions. In recent work with Natascha Neumaerker and John Roberts we have observed the same period distribution is a zero-entropy system, conjectured to be mixing: the Casati-Prosen map.


Space discretization of smooth dynamical systems is an area rich in arithmetical phenomena. Work with D Nucinkis and D K Arrowsmith has demonstrated the existence of fractal features in the asymptotics of certain arithmetical functions obtained by discretizing a twist map.

In work with D Bosio and I Vladimirov, I have shown that the dynamics of round-off errors in discretized irrational rotations admits a natural representation as expanding dynamics over the p-adic integers. As a result, one obtains a central limit theorem for the propagation of round-off errors. Such dynamical systems seem ideally suited to generate pseudo-random numbers.

J Lowenstein and I have developed an arithmetical theory of invertible (hamiltonian) round-off errors for linear rational rotations, building on previous work with S Hatjispyros. Prominent structures found in this context are algebraic numbers, higher-dimensional embeddings and quasi-periodic tilings. The two-dimensional lattice which supports the dynamics is embedded uniformly in a torus of suitable dimension, where the dynamics is affine and discontinuous (see below). The complex ergodic-theoretic properties of these systems (found in many different contexts) here translate in anomalous transport properties of round-off errors.

Heather Reeve-Black and I have studied near-rational rotations in the round-off problem, with surprising findings. There is a non-smooth integrable Hamiltonian system, featuring a foliation by polygonal invariant curves, which represent the limit of vanishing discretisation of the space. We show that, for sufficiently small discretisation, a positive fraction of these invariant curves survives, leading to a discrete space version of the KAM scenario. The surviving curves are characterised in terms of congruences, and properties of the Gaussian integers.


Our investigations in the dynamics of round-off errors led us to the study of piecewise isometries, which has recently blossomed into an active area of research, fuelled by its geometrical appeal and important applications. In this class of systems the topology of the phase space is shaped by discontinuity rather than nonlinearity, resulting in strikingly complex behaviour from minimal ingredients. The irreducible complexity of these systems has delayed progress for over two decades. Their dynamics is shaped by diophantine properties, in ways that are still quite mysterious.

J Lowenstein and I have carried out a detailed analysis of the quadratic cases of the so-called `standard model' (a family of rotations on the two-dimensional torus). These are eight maps which are defined over quadratic number fields. In an early result (with K L Kouptsov) we established the renormalizability of these systems, and the vanishing of the measure of the closure of the discontinuity set. More recently, we have considered a one-parameter family of piecewise isometries containing one of the maps above as a special case. We have shown that the parameter values leading to renormalizability are precisely those that lie in the same quadratic field determined by the rotational part. In these works, computer-assisted proofs play a decisive role. We have also extended our study to a cubic case, featuring a multi-fractal renormalization structure. Another result that required computer-assistance was demonstrating a specific example that a positive measure of stable orbits can exist even in a region in which the so-called pseudo-hyperbolic points become dense.

G Poggiaspalla, J Lowenstein and I have applied some techniques developed for two-dimensional piecewise isometric systems to the study of interval-exchange transformations. Restricting these systems to algebraic number fields lead to very interesting maps on lattices, with strong arithmetical features.