Related Fields, Volume Entropy
Asymptotic phenomena for the systole of surfaces of large genus have been shown to be related to interesting ergodic phenomena, and to properties of congruence subgroups of arithmetic groups.
Gromov's 1983 inequality for the homotopy systole implies, in particular, a uniform lower bound for the area of an aspherical surface in terms of its systole. Such a bound generalizes the inequalities of Loewner and Pu, albeit in a non-optimal fashion.
Gromov's seminal 1983 paper also contains asymptotic bounds relating the systole and the area, which improve the uniform bound (valid in all dimensions).
It was discovered recently (see paper by Katz and Sabourau below) that the volume entropy h, together with A. Katok's optimal inequality for h, is the "right" intermediary in a transparent proof of M. Gromov's asymptotic bound for the systolic ratio of surfaces of large genus.
The classical result of A. Katok states that every metric on a closed surface M with negative Euler characteristic satisfies an optimal inequality relating the entropy and the area.
It turns out that the minimal entropy of a closed surface can be related to its optimal systolic ratio. Namely, there is an upper bound for the entropy of a systolically extremal surface, in terms of its systole. By combining this upper bound with Katok's optimal lower bound in terms of the volume, one obtains a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, such an approach yields an improved multiplicative constant in Gromov's theorem.
As an application, this method implies that every metric on a surface of genus at least 20 satisfies Loewner's torus inequality. This improves the best earlier estimate of 50 which followed from an estimate of Gromov's.
Read more about this topic: Systolic Geometry
Famous quotes containing the words related, volume and/or entropy:
“Perhaps it is nothingness which is real and our dream which is non-existent, but then we feel think that these musical phrases, and the notions related to the dream, are nothing too. We will die, but our hostages are the divine captives who will follow our chance. And death with them is somewhat less bitter, less inglorious, perhaps less probable.”
—Marcel Proust (1871–1922)
“To be thoroughly conversant with a Man’s heart, is to take our final lesson in the iron-clasped volume of despair.”
—Edgar Allan Poe (1809–1845)
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)