Università Roma Tre

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.