The periodic structure of non-singular Morse-Smale flows.
The return sequence of the Bowen-Series map for punctured surfaces
For a non-compact hyperbolic surface M of finite area, we study a certain Poincaré section for the geodesic flow. The canonical, non-invertible factor of the first return map to this section is shown to be pointwise dual ergodic with return sequence (aₙ) given by aₙ = π/(4(Area(M) + 2π)) · n/(log n). We use this result to deduce that the section map itself is rationally ergodic, and that the geodesic flow associated to M is ergodic with respect to the Liouville measure. ...
The Riccati equation: pinching of forcing and solutions.
The simplest shadowing
Two different and easy proofs are presented that a hyperbolic linear homeomorphism of a Banach space admits the shadowing.
The size of scrambled sets: n-dimensional case
The structure of Lorenz attractors
The study of the chaotic behavior in retailer's demand model.
The Teichmüller geodesic flow and the geometry of the Hodge bundle
The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller...
The Teichmüller space of an Anosov diffeomorphis of T2.
The topological behaviour of a diffeomorphism near a fixed point.
The zeta functions of Ruelle and Selberg. I
Theory of adaptive adjustment.
There are Two Isotopic Morse-Smale Diffeomorphisms Which Cannot be Joined by Simple Arcs.
Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics
We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately...
Time-dependent billiards.
Time-dependent circular billiard.
Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere
This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via...
Topological entropy and variation for transitive maps
Topological invariance of the Collet–Eckmann property for S-unimodal maps
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
Topological Pressure for One-Dimensional Holomorphic Dynamical Systems
For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials φ the topological pressure is entirely determined by the values of φ on the repelling periodic points of f. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.