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.
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.
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.