Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$. A loop $h:S^1\to G$ is called strictly ergodic if for some irrational number the associated skew product map $T:S^1\times Y\to S^1\times Y$ defined by $T(t,y)=(t+\alpha ;h(t\left)y\right)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected...

Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.

We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

We present a very quick and easy proof of the classical Stepanov-Hopf ratio ergodic theorem, deriving it from Birkhoff's ergodic theorem by a simple inducing argument.