Volume entropy

Among the various notions of entropy found in dynamical systems, differential geometry, and geometric group theory, an important role is played by the volume entropy.

Let $(M,g)$ be a closed surface with a Riemannian metric "g". Denote by $(\; ilde\{M\},\; ilde\{g\})$ the universal Riemannian cover of $(M,g)$. Choose a point $ilde\{x\}\_0in\; ilde\{M\}$.

The volume entropy (or asymptotic volume) $h(M,\; g)$ of a surface $(M,g)$ is defined by setting

:$h(M,g)\; =\; lim\_\{R\; ightarrow\; +\; infty\}\; frac\{log\; left(\; mathrm\{vol\}\_\{\; ilde\{g\; B(\; ilde\{x\}\_0,R)\; ight)\}\{R\},$

where $mathrm\{vol\}\_\{\; ilde\{g\; B(\; ilde\{x\}\_0,R)$ is the volume (area) of the ball of radius $R$ centered at $ilde\{x\}\_0in\; ilde\{M\}$.

Since $M$ is compact, the limit above exists, and does not depend on the point $ilde\{x\}\_0\; in\; ilde\{M\; \}$. This asymptotic invariant describes the exponential growth rate of the volume in the universal cover.

Application in differential geometry of surfaces

Katok's entropy inequality was recently exploited to obtain a tight asymptotic bound for the systolic ratio of surfaces of large genus, see systoles of surfaces.

References

*Katok, A.: Entropy and closed geodesics, Ergo. Th. Dynam. Sys. 2 (1983), 339--365.

*Katok, A.; Hasselblatt, B.: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and L. Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.

* Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Ergo. Th. Dynam. Sys. 25 (2005), 1209-1220.