101 research outputs found
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
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