Almost-invariant and essentially-invariant halfspaces

SEMIGROUP ACTIONS , WEAK ALMOST PERIODICITY, AND INVARIANT MEANS

Let S be a topological semigroup acting on a topological space X. We develop the theory of (weakly) almost periodic functions on X, with respect to S, and form the (weakly) almost periodic compactifications of X and S, with respect to each other. We then consider the notion of an action of Son a Banach space, and on its dual, and after defining S-invariant means for such a space, we give a...

Nonexistence of Invariant Rigid Structures and Invariant Almost Rigid Structures

We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the ”exotic” examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spaces. The actions on homogeneous spaces all preserve affine connections, wherea...

Fluctuation Induced Almost Invariant Sets

Statistically optimal almost-invariant sets

Chaotic dynamical systems are often transitive, although this transitivity is sometimes very weak. It is of interest to divide the phase space into large regions, between which there is relatively little communication of trajectories. We present fast, simple algorithms to find such divisions. The present work builds on the results of Froyland and Dellnitz [G. Froyland, M. Dellnitz, Detecting an...

Amenable Actions and Almost Invariant Sets

In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets.