TY - JOUR
T1 - Entropy of systolically extremal surfaces and asymptotic bounds
AU - Katz, Mikhail G.
AU - Sabourau, Stéphane
PY - 2005/8
Y1 - 2005/8
N2 - We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.
AB - We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.
UR - http://www.scopus.com/inward/record.url?scp=23444451555&partnerID=8YFLogxK
U2 - 10.1017/S0143385704001014
DO - 10.1017/S0143385704001014
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AN - SCOPUS:23444451555
SN - 0143-3857
VL - 25
SP - 1209
EP - 1220
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
IS - 4
ER -