TY - JOUR
T1 - Distal strongly ergodic actions
AU - Glasner, Eli
AU - Weiss, Benjamin
N1 - Publisher Copyright:
© 2022 European Mathematical Society.
PY - 2022
Y1 - 2022
N2 - Let η be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F3 on a compact metric space X, admitting an invariant probability measure μ, such that the resulting dynamical system .X;μ;F3/ is strongly ergodic and distal of rank η. In particular, this shows that there is an F3 system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.
AB - Let η be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F3 on a compact metric space X, admitting an invariant probability measure μ, such that the resulting dynamical system .X;μ;F3/ is strongly ergodic and distal of rank η. In particular, this shows that there is an F3 system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.
KW - Distal action
KW - spectral gap
KW - strong ergodicity
UR - http://www.scopus.com/inward/record.url?scp=85131399068&partnerID=8YFLogxK
U2 - 10.4171/GGD/650
DO - 10.4171/GGD/650
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AN - SCOPUS:85131399068
SN - 1661-7207
VL - 16
SP - 333
EP - 340
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
IS - 1
ER -