TY - JOUR
T1 - Some remarks about self-products and entropy
AU - Hochman, Michael
N1 - Publisher Copyright:
© The Author(s), 2024. Published by Cambridge University Press.
PY - 2024
Y1 - 2024
N2 - Let (X, β, μ, T) be a probability-preserving system with X compact and T a homeomorphism. We show that if every point in X × X is two-sided recurrent, then hμ (T)=0, resolving a problem of Benjamin Weiss, and that if hμ (T)=∞ then every full-measure set in X contains mean-asymptotic pairs (that is, the associated process is not tight), resolving a problem of Ornstein and Weiss.
AB - Let (X, β, μ, T) be a probability-preserving system with X compact and T a homeomorphism. We show that if every point in X × X is two-sided recurrent, then hμ (T)=0, resolving a problem of Benjamin Weiss, and that if hμ (T)=∞ then every full-measure set in X contains mean-asymptotic pairs (that is, the associated process is not tight), resolving a problem of Ornstein and Weiss.
KW - entropy
KW - mean asymptotic pairs
KW - tight processes
KW - topological recurrence
UR - http://www.scopus.com/inward/record.url?scp=85213852518&partnerID=8YFLogxK
U2 - 10.1017/etds.2024.77
DO - 10.1017/etds.2024.77
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AN - SCOPUS:85213852518
SN - 0143-3857
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
ER -