TY - JOUR
T1 - Rohlin properties for ℤd actions on the cantor set
AU - Hochman, Michael
PY - 2012
Y1 - 2012
N2 - We study the space H(d) of continuous ℤd-actions on the Cantor set, particularly questions related to density of isomorphism classes. For d = 1, Kechris and Rosendal showed that there is a residual conjugacy class. We show, in contrast, that for d ≥ 2 every conjugacy class in H(d) is meager, and that while there are actions with dense conjugacy class and the effective actions are dense, no effective action has dense conjugacy class. Thus, the action by the group homeomorphisms on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we show that in the spaces of transitive and minimal actions the effective action s are nowhere dense, and in particular there are minimal actions that are not approximable by minimal shifts of finite type.
AB - We study the space H(d) of continuous ℤd-actions on the Cantor set, particularly questions related to density of isomorphism classes. For d = 1, Kechris and Rosendal showed that there is a residual conjugacy class. We show, in contrast, that for d ≥ 2 every conjugacy class in H(d) is meager, and that while there are actions with dense conjugacy class and the effective actions are dense, no effective action has dense conjugacy class. Thus, the action by the group homeomorphisms on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we show that in the spaces of transitive and minimal actions the effective action s are nowhere dense, and in particular there are minimal actions that are not approximable by minimal shifts of finite type.
UR - http://www.scopus.com/inward/record.url?scp=82755163002&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2011-05319-5
DO - 10.1090/S0002-9947-2011-05319-5
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AN - SCOPUS:82755163002
SN - 0002-9947
VL - 364
SP - 1127
EP - 1143
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 3
ER -