David Ruelle

Recurrence Plots of Dynamical Systems

A new graphical tool for measuring the time constancy of dynamical systems is presented and illustrated with typical examples.

Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems

We show that values of the correlation dimension estimated over a decade from the Grassberger-Procaccia algorithm cannot exceed the value 2 log10N if N is the number of points in the time series. When this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present. The estimation of Lyapunov exponents is also discussed.

Occurrence of strange AxiomA attractors near quasi periodic flows onTm,m≧3

It is shown that by a smallC2 (resp.C∞) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.

Ergodic theory of differentiable dynamical systems

Iff is a C1 + ɛ diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.

The Ergodic Theory of Axiom A Flows

Let M be a compact (Riemann) manifold and (f t ): M → M a differentiable flow. A closed (f t )-invariant set ∧ ⊂ M containing no fixed points is hyperbolic if the tangent bundle restricted to ∧ can be written as the Whitney sum of three (Tf t )-invariant continuous subbundles

Superstable interactions in classical statistical mechanics

{T_\Lambda }M = E + {E^s} + {E^u}

Repellers for real analytic maps

where E is the one-dimensional bundle tangent to the flow, and there are constants c λ>0 so that