On the Relations Among Various Entropy Characteristics of Dynamical Systems

Positive topological entropy of Reeb flows on spherizations

“…In this special case, Theorem 1 has been proved by Dinaburg [9], Gromov [21], Paternain [35,36], and Paternain-Petean [38] in their study of geodesic flows. More generally, the sets Σ q are all convex if and only if Σ is the level set of a Finsler metric.…”

Integrable geodesic flows with positive topological entropy

“…Indeed, it was proved by Dinaburg, that if the fundamental group of a manifold M n has an exponential growth, then the topological entropy of the geodesic flow of any Riemannian metric on M n is positive [2]. The latter corollary also follows from Proposition 3: it is known that the topological entropy equals the supremum of the measure entropies taken over all ergodic invariant Borel measures.…”

“…If this quantity vanishes, then π 1 (M n ) has a subexponential growth [2] and, if in addition M n is a C ∞ simply-connected manifold, then Y is rationallyelliptic (this follows from results of Gromov and Yomdin) [6,7]. Integrability implies vanishing of the topological entropy under some additional conditions which were established in [6,7,10] and restrict not only the singular set but also the behaviour of the flow on this set.…”

The volume entropy of a surface decreases along the Ricci flow

“…The volume of a ball of radius R in the universal cover ( M,g) of a Riemannian manifold (M, g) grows exponentially at a rate h(g) := lim R→∞ R −1 log vol(B(x, R, g)) ≥ 0 called the volume entropy of (M, g) [25]; see also [8,28]. The geodesic flow on the unit sphere bundle of a fixed Riemannian manifold (M, g) of negative sectional curvature has the Anosov or hyperbolic property and is a major example of a structurally stable flow, see [2] and [17, §17.6].…”

“…This estimate (which is a special case of Corollary II.1 in [9]), together with (4), (5), (9), implies that −K dµ τ > −2πχ(M) as required for (8). Question 2.…”