The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for σ-finite invariant measures (Corollary 1). For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by ...

Finitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness measure is studied with respect group shifts vector space, orthogonal and symplectomorphisms equipped shift-invariant symplectic form. A considered locally finite, σ but it not countably additive. analog ergodic decomposition finitely additiv...

We study the statistical properties of the infinite horizon Lorentz gas after the introduction of small holes. Our basic approach is to prove the persistence of a spectral gap for the transfer operator associated with the billiard map in the presence of such holes. The new feature here is the interaction between the holes and the infinite horizon corridors, which causes previous approaches to f...

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits welldefined projections onto the quotient space of “corners” of finite size, must be finite. A similar result, Theorem 1, is also established for unitarily invariant measures on the space of all i...

We survey examples of dynamical systems on non–compact spaces which exhibit measure rigidity on the level of infinite invariant measures in one or more of the following ways: all locally finite ergodic invariant measures can be described; exactly one (up to scaling) admits a generalized law of large numbers; the generic points can be specified. The examples are horocycle flows on hyperbolic sur...

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which purely singular with respect to Lebesgue measure. Can such leave an infinite, $\sigma$-finite absolutely continuous measure? In this paper, using old criterion due Katznelson, we show the answer question no.

Durret t (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to it held starting in any semi-infinite initial state. We prove the same for the critical contact process. We also prove that the process sta...