A classification of orbits admitting a unique invariant measure

A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are S∞-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique S∞-invariant probability measure precisely when the structure is hig...

Countable Infinitary Theories Admitting an Invariant Measure

Let L be a countable language. We characterize, in terms of definable closure, those countable theories Σ of Lω1,ω(L) for which there exists an S∞-invariant probability measure on the collection of models of Σ with underlying set N. Restricting to Lω,ω(L), this answers an open question of Gaifman from 1964, via a translation between S∞-invariant measures and Gaifman’s symmetric measure-models w...

Ergodic Theory of Markov Chains Admitting an Infinite Invariant Measure.

NONINVERTIBLE TRANSFORMATIONS ADMITTING NO ABSOLUTELY CONTINUOUS ct-FINITE INVARIANT MEASURE

We study a family of H-to-1 conservative ergodic endomorphisms which we will show to admit no rj-finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as C°° endomorphisms of the 2-torus.

Topological spaces admitting a unique fractal structure

Each homeomorphism from the n-dimensional Sierpiński gasket into itself is a similarity map with respect to the usual metrization. Moreover, the topology of this space determines a kind of Haar measure and a canonical metric. We study spaces with similar properties. It turns out that in many cases, “fractal structure” is not a metric but a topological phenomenon.