Finite Resolution Dynamics

We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Henon attractor is mixing at all resolutions coarser than 10^-5.Comment: 25 pages. Final version. To appear in Foundations of Computational Mathematic

Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps

We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.Comment: 9 page

Parameter exclusions in Henon-like systems

This survey is a presentation of the arguments in the proof that Henon-like maps f_a(x,y)=(1-a x^2,0) + R(a,x,y) with |R(a,x,y)|< b have a "strange attractor", with positive Lebesgue probability in the parameter "a", if the perturbation size "b" is small enough. We first sketch a "geometric model" of the strange attractor in this context, emphasising some of its key geometrical properties, and then focus on the construction and estimates required to show that this geometric model does indeed occur for many parameter values. Our ambitious aim is to provide an exposition at one and the same time intuitive, synthetic, and rigorous. We think of this text as an introduction and study guide to the original papers in which the results were first proved. We shall concentrate on describing in detail the overall structure of the argument and the way it breaks down into its (numerous) constituent sub-arguments, while referring the reader to the original sources for detailed technical arguments.Comment: 40 pages, 3 figure