Arkady Tempelman

Averaging sequences and modulated ergodic theorems for weakly almost periodic group representations

LetT be a weakly almost periodic (WAP) representation of a locally compact Σ-compact groupG by linear operators in a Banach spaceX, and letM = M(T) be its ergodic projection onto the space of fixed points (i.e.,Mx is the unique fixed point in the closed convex hull of the orbit ofx). A sequence of probabilities Μn is said toaverage T [weakly] if ∫T(t)x dΜn converges [weakly] toM(T)x for eachx ∃X. We callΜn [weakly]unitarily averaging if it averages [weakly] every unitary representation in a Hilbert space, and [weakly]WAPRaveraging if it averages [weakly] every WAP representation. We investigate some of the relationships of these notions, and connect them with properties of the regular representation (by translations) in the spaceWAP(G).

Ergodicity and Mixing

In the sequel X is a semigroup, T = {T x , x ∈ X} is a left dynamical system in a phase space (Ω, F, m), U will denote the isometric (right) representation of X in L 2 (Ω, F, m) associated with T. 3 is the σ—subalgebra of all T—invariant sets Λ ∈ F, and I B α is the subspace of all T—invariant functions f ∈ L B α , B being an arbitrary Banach space. If f ∈ L B α (Ω, F, m), M(f) = M(f∣T) is the mean of the function f with respect to T (see Subsect. 1.7.2). Let P be a probability measure on F E(f) = f Ω f dP, f ∈ L 1(Ω, F, m).

Mean Ergodic Theorems

Let X be a topological semigroup, B the σ-algebra of Borel sets in X, and {v n , n ∈ N} a net of Borel probability measures.

Pointwise Ergodic Theorems

We denote by \({\tilde F_B} = {\tilde F_B}(\Omega ,F,m)\) the space of all measurable B-valued functions with the seminorm

Maximal and Dominated Ergodic Theorems

|f|{P_b} = \mathop {\inf }\limits_{\alpha \geqslant 0} arctam[\alpha + m(\{ \omega :|f(\omega )|\} )];

Ergodic Theorems for Homogeneous Random Measures

convergence in \({\tilde F_R}\) is the same as convergence in m.

Ergodic Theorems for Group Actions

Let (X, B) be a measurable semigroup; B M (B) the Banach space of all signed measures of bounded variation on B with norm ‖v‖ = var v; P(B) the set of all probability measures on B; \(\tilde p\) the set of all probability measures v on X whose carriers c(v) are finite sets; and let F B be the subspace in Ф B consisting of the bounded measurable functions on X.

Specific characteristics and variational principle for homogeneous random fields

Let (X, B) be a measurable semigroup, N a linearly ordered set and {v n , n ∈ B} a net of probability measures on B.