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Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y . A loop h : S 1 → G is called strictly ergodic if for some irrational number the associated skew product map T : S 1 × Y → S 1 × Y defined by T ( t , y ) = ( t + α ; h ( t ) y ) 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...

Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations

Franz Hofbauer (2009)

Commentationes Mathematicae Universitatis Carolinae

We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...

Hopf's ratio ergodic theorem by inducing

Roland Zweimüller (2004)

Colloquium Mathematicae

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.

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