Showing papers by "David Ruelle published in 1989"

Chaotic evolution and strange attractors

Abstract: This book, based on lectures given at the Accademia dei Lincei, is an accessible and leisurely account of systems that display a chaotic time evolution. This behaviour, though deterministic, has features more characteristic of stochastic systems. The analysis here is based on a statistical technique known as time series analysis and so avoids complex mathematics, yet provides a good understanding of the fundamentals. Professor Ruelle is one of the world's authorities on chaos and dynamical systems and his account here will be welcomed by scientists in physics, engineering, biology, chemistry and economics who encounter nonlinear systems in their research.

The thermodynamic formalism for expanding maps

Abstract: Letf:X↦X be an expanding map of a compact space (small distances are increased by a factor >1). A generating functionζ(z) is defined which countsf-periodic points with a weight. One can expressζ in terms of nonstandard “Fredholm determinants” of certain “transfer operators”, which can be studied by methods borrowed from statistical mechanics. In this paper we review the spectral properties of the transfer operators and the corresponding analytic properties ofζ(z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.

Resonances for intermittent systems

Abstract: There is increasing theoretical and numerical evidence that for many interesting dynamical systems the power spectrum of an observable A extends to a meromorphic function in the complex frequency plane. The position of the complex poles or 'resonances' is independent of the observable A which is monitored. The authors study the resonances for intermittent dynamical systems by using a probabilistic independence assumption about recurrence times. A close agreement between theory and numerical experiments is obtained.

An Extension of the Theory of Fredholm Determinants

Abstract: Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form e μ(dω) ℒω, where ℒω φ(x) = ϕω(x). φ(ψω x), , φ belongs to a space of Holder or C r functions, ϕω is Holder or C r , and ψω is a contraction or a C r contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps.

Differentiable Dynamical Systems

Abstract: Publisher Summary This chapter focuses on differentiable dynamical systems. Physics gives many examples of differentiable dynamical systems in infinite-dimensional Banach spaces. In fact, Banach spaces, also defined as finite-or infinite-dimensional, provide the natural framework for the study of differentiable maps. The time evolution of natural systems is often given by a differential equation. A hyperbolic fixed point or periodic orbit is also called a sink if it is attracting and a source if it is repelling. Otherwise, it is a saddle or of saddle type. The Grobman–Hartman theorem, for maps of flow, is satisfactory in providing a local topological model for dynamics. The study of fixed points and periodic orbits becomes more delicate when they are not hyperbolic. One is, however, forced to consider non-hyperbolic behavior in the theory of bifurcations.

Une extension de la théorie de Fredholm

Abstract: Analytic functions are introduced, which are analogous to the Fredholm determinant, but have finite convergence radius. These functions are associated with operators which are linear superpositions of operators of the form Φ→φ•(ΦοΨ), where Φ belongs to a space of Holder or smooth functions, φ is Holder or smooth, and attractor #7B-Ue On introduit des fonctions analytiques analogues au determinant de Fredholm, mais de rayon de convergence fini. Ces fonctions sont associees a des operateurs qui sont superpositions lineaires d'operateurs de la forme Φ→φ•(Φοψ) ou Φ appartient a un espace de fonctions holderiennes ou differentiables, φ est holderienne ou differentiable, et φ est une contraction ou une contraction differentiable