Distal actions and shifted convolution property

Abstract

A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure μ on G, either sup x∈G μ n (Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that μ n x −n → ω K , the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.

Some properties of distal actions on locally compact groups

“…From above, as α-action on K/K i is also distal (cf. [31], Theorem 3.1), C K i (α) is closed for each i. By Lemma 4.4 (1), C(α) is also closed.…”

“…For each i, as L∩K i is α-invariant, if the α-action on L is distal, so is the corresponding action of α on L/L∩K i (cf. [31], Theorem 3.1), and hence…”

“…Let L be any α-invariant subgroup of G. As G/K is a Lie group from Theorem 2.4 of [15], we have that C L (α) ⊂ C LK (α) = C K (α)L = C(α)KL. Hence as K is normal, [31], Corollary 3.2).…”

“…But the Γ-action on K/C is distal and hence C 2 ⊂ C. Remark 2.2 1. As we can assume that Γ is a semigroup contained in Aut(K), let [Γ] be the subgroup generated by Γ in Aut(K), From Theorem 1 of [11], it is obvious that the Γ-action on K is distal if and only if the [Γ]-action on K is distal (see also the proof of Theorem 3.1 in [31]). By Theorem 2.1 of [4], the same statement holds for ergodic actions on a compact group.…”

Dynamics of Certain Distal Actions on Spheres

“…The notion of distality was introduced by Hilbert (cf. Ellis [7], Moore [12]) and studied by many in different contexts (see Abels [1,2], Furstenberg [9], Raja-Shah [14,15] and Shah [16], and references cited therein). Note that a homeomorphism T of a topological space is distal if and only if T n is so, for any n ∈ N.…”