Expansive dynamics on zero-dimensional groups

Expansive Homeomorphisms with the Shadowing Property on Zero Dimensional Spaces

Let X = {a}∪ {ai | i ∈ N} be a subspace of Euclidean space E such that limi→∞ ai = a and ai 6= aj for i 6= j. Then it is well known that the space X has no expansive homeomorphisms with the shadowing property. In this paper we show that the set of all expansive homeomorphisms with the shadowing property on the space Y is dense in the space H(Y ) of all homeomorphisms on Y , where Y = {a, b} ∪ {...

ALGEBRAIC Zd-ACTIONS ON ZERO-DIMENSIONAL COMPACT ABELIAN GROUPS

In 1967 Furstenberg proved that every infinite closed subset of T = R/Z simultaneously invariant under multiplication by 2 and by 3 is equal to T (cf. [8]), which motivated the still unresolved question whether this scarcity of invariant sets is paralleled by a corresponding scarcity of invariant probability measures: is Lebesgue measure the only nonatomic probability measure on T which is inva...

On the Structure of Some Deficiency Zero Groups

The Bernoulli Property for Expansive Z Actions on Compact Groups

We show that an expansive Z2 action on a compact abelian group is measurably isomorphic to a two–dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of the K property due to Lind, Schmidt and the author. As a corollary, we note that an expansive Z...

Expansive algebraic actions of countable abelian groups

This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic Zd-actions given by Schmidt using complex varieties. Also included is an application to a natural class of examples arising from unit subgroups of integral domains. 2000 Mathematics Subject Classification: 22D...