Volume entropy

Last updated April 15, 2019

The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples.

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Compact space Topological notions of all points being "close"

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X|_p, Y|_p) is a smooth function. The family g_p of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

Definition

Let (M, g) be a compact Riemannian manifold, with universal cover ${\tilde {M}}.$ Choose a point ${\tilde {x}}_{0}\in {\tilde {M}}$ .

The volume entropy (or asymptotic volume growth) $h=h(M,g)$ is defined as the limit

h(M,g)=\lim _{R\rightarrow +\infty }{\frac {\log \left(\operatorname {vol} B(R)\right)}{R}},

where B(R) is the ball of radius R in ${\tilde {M}}$ centered at ${\tilde {x}}_{0}$ and vol is the Riemannian volume in the universal cover with the natural Riemannian metric.

A. Manning proved that the limit exists and does not depend on the choice of the base point. This asymptotic invariant describes the exponential growth rate of the volume of balls in the universal cover as a function of the radius.

Properties

Volume entropy h is always bounded above by the topological entropy h_top of the geodesic flow on M. Moreover, if M has nonpositive sectional curvature then h = h_top. These results are due to Manning.
More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed Riemannian manifold without conjugate points (Freire and Mañé).
Locally symmetric spaces minimize entropy when the volume is prescribed. This is a corollary of a very general result due to Besson, Courtois, and Gallot (which also implies Mostow rigidity and its various generalizations due to Corlette, Siu, and Thurston):

In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic from the north-pole fail to be length-minimizing if it passes through the south-pole.

William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

Let X and Y be compact oriented connected n-dimensional smooth manifolds and f: Y→X a continuous map of non-zero degree. If g₀ is a negatively curved locally symmetric Riemannian metric on X and g is any Riemannian metric on Y then

h^{n}(Y,g)\operatorname {vol} (Y,g)\geq |\operatorname {deg} (f)|h^{n}(X,g_{0})\operatorname {vol} (X,g_{0}),

and for n≥ 3, the equality occurs if and only if (Y,g) is locally symmetric of the same type as (X,g₀) and f is homotopic to a homothetic covering (Y,g) → (X,g₀).

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.

Systolic geometry minimum length of a noncontractible closed curve in a metric space

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.

In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949. Given a closed surface, its systole, denoted sys, is defined to the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys². The systolic ratio SR is the reciprocal quantity sys²/area. See also Introduction to systolic geometry.

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References

Besson, G., Courtois, G., Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. (French) [Entropy and rigidity of locally symmetric spaces with strictly negative curvature] Geom. Funct. Anal. 5 (1995), no. 5, 731–799
Katok, A.: Entropy and closed geodesics, Erg. Th. Dyn. 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. Erg. Th. Dyn. Sys. 25 (2005), 1209-1220
Manning, A.: Topological entropy for geodesic flows. Ann. of Math. (2) 110 (1979), no. 3, 567–573

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