TY - JOUR
T1 - Neutralized Local Entropy and Dimension bounds for Invariant Measures
AU - Ben Ovadia, S.
AU - Rodriguez-Hertz, F.
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024/6/1
Y1 - 2024/6/1
N2 - We introduce a notion of a point-wise entropy of measures (i.e., local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.
AB - We introduce a notion of a point-wise entropy of measures (i.e., local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.
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U2 - 10.1093/imrn/rnae047
DO - 10.1093/imrn/rnae047
M3 - Article
AN - SCOPUS:85195786314
SN - 1073-7928
VL - 2024
SP - 9469
EP - 9481
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 11
ER -