Entropy of systolically extremal surfaces and asymptotic bounds

Details Entropy of systolically extremal surfaces and asymptotic bounds Entropy of systolically extremal surfaces and asymptotic bounds

2004-10-13

arxiv.org/abs/math/0410312v2

Mathematics

Differential Geometry

14 pages. Ergodic Theory and Dynamical Systems, to appear

Scientific paper

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. 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.

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