Symbolic dynamics and geodesic flows
Symbolic extensions for nonuniformly entropy expanding maps
A nonuniformly entropy expanding map is any ¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].
Symbolic extensions in intermediate smoothness on surfaces
We prove that maps with on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].
Symbolic extensions of smooth interval maps.
Symmetries and constants of the motion for dynamics in implicit form
Synchronization in ensembles of coupled maps with a major element.
Synchronization of nonautonomous dynamical systems.
Systèmes intégrables à -corps sur la droite
Szegő's first limit theorem in terms of a realization of a continuous-time time-varying systems
It is shown that the limit in an abstract version of Szegő's limit theorem can be expressed in terms of the antistable dynamics of the system. When the system dynamics are regular, it is shown that the limit equals the difference between the antistable Lyapunov exponents of the system and those of its inverse. In the general case, the elements of the dichotomy spectrum give lower and upper bounds.
Szemerédi theorem implies Furnsternberg theorem