We establish that Gaussian thermostats are geodesic flows of special metric connections. We give sufficient conditions for hyperbolicity of geodesic flows of metric connections in terms of their curvature and torsion.

We propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real Hénon family, we prove the existence of many regions of hyperbolic parameters in the parameter plane of the family.

We prove the hyperbolicity, ergodicity and thus the Bernoulli property of two hard balls in one of the following four polygons: the square, the equilateral triangle, the 45 − 45 − 90◦ triangle or the 30− 60− 90◦ triangle.

We brieey review the non-commutative generalisation, presented in 1], of the theory of hyperbolic dynamical systems; and then prove that hyperbolicity cannot be a paradigm for quantum chaos, except possibly in a certain asymptotic sense.

A diffeomorphism f : M → M on a compact manifold M is partially hyperbolic if there exists a continuous, nontrivial Df -invariant splitting TxM = E s x ⊕ E c x ⊕ E u x , x ∈ M of the tangent bundle such that the derivative is a contraction along E and an expansion along E, with uniform rates, and the behavior of Df along the center bundle E is in between its behaviors along E and E, again by a ...