Some properties of distal actions on locally compact groups

Abstract

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup K such that G/K is distal and the conjugacy action of G on K is ergodic; moreover, if G itself is (pointwise) distal then G is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.Mathematics Subject Classification: Primary: 37B05, 22D05 Secondary: 22E15, 22D45

The Point Value Maximization Problem for Positive Definite Functions Supported in a Given Subset of a Locally Compact Group

“…Note that (according to Ellis, who refers to Zippin, see the closing paragraph of [9]) distality was considered by Hilbert in an attempt to give a topological characterization of the concept of a rigid group of motions. The closely related notions of distal elements, contraction groups and equicontinuous elements provided tools for exploring the structures of locally compact topological groups, see, e.g., [3,5,6,17,23] and the references therein.…”

“…The closely related notions of distal elements, contraction groups and equicontinuous elements provided tools for exploring the structure of locally compact topological groups, see e.g. [3,5,6,17,23] and the references therein.…”

Distality of Certain Actions on -Adic Spheres

“…The notion of distality was introduced by Hilbert (cf. Moore [8]) and studied by many in different contexts (see Ellis [3], Furstenberg [4], Raja and Shah [10,11] and Shah [12], and references cited therein). Note that a homeomorphism T of a topological space is distal if and only if T n is so, for every n ∈ Z.…”

Contraction groups and passage to subgroups and quotients for endomorphisms of totally disconnected locally compact groups

“…In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup. (f) The compact open subgroups W ⊆ lev(α) with α(W ) = W form a basis of identity neigbourhoods in lev(α).If G is a Lie group over a totally disconnected local field (as in [3] and [18]) and α : G → G is an analytic endomorphism with small tidy subgroups, then con(α), lev(α) and con − (α) are Lie subgroups of G (in the strong sense of submanifolds) and the product map in (2) is an analytic diffeomorphism, see [9].By (e) in Theorem F, the automorphism α| lev(α) is distal, see [14]. Information concerning contractive automorphisms of locally compact groups can be found in [19] and [10]; contractive analytic automorphisms of Lie groups over a totally disconnected local field K are discussed in [21] (for K = Q p ) and [8].…”

“…By (e) in Theorem F, the automorphism α| lev(α) is distal, see [14]. Information concerning contractive automorphisms of locally compact groups can be found in [19] and [10]; contractive analytic automorphisms of Lie groups over a totally disconnected local field K are discussed in [21] (for K = Q p ) and [8].…”