×

Topological entropy and Thurston’s norm of atoroidal surface bundles over the circle. (English) Zbl 0647.57006

Let M be a compact orientable 3-manifold which is atoroidal, in the sense that it does not contain any non-trivial embedded torus. W. P. Thurston investigated which cohomology classes in H 1(M;\({\mathbb{R}})\) can be represented by non-singular closed differential forms. In particular, he proved that the subspace of such classes can be divided into finitely many cones, each expressed in function of a certain norm on H 1(M;\({\mathbb{R}})\) [W. P. Thurston, Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)]. To each such non-singular class \(\alpha\), we can associate a number h(\(\alpha)\) defined as follows: The function \(\alpha\to h(\alpha)\) is continuous homogeneous of degree 1; if the cohomology class of \(\alpha\) in H 1(M;\({\mathbb{R}})\) is a primitive class of H 1(M;\({\mathbb{Z}})\), so that the foliation defined by \(\alpha\) is a fibration over the circle, h(\(\alpha)\) is the maximum of the inverses of the topological entropies of all diffeomorphisms of the fiber which are isotopic to the monodromy of the fibration. D. Fried constructed this map h, and proved that it is concave on each of the Thurston cells [D. Fried, Comment. Math. Helv. 57, 237-259 (1982; Zbl 0503.58026)]. The author proves that h is actually strictly concave on these cells. His techniques are based on an approach developed by D. Long and U. Oertel [Hyperbolic surface bundles over the circle, preprint].
Reviewer: F.Bonahon

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R30 Foliations in differential topology; geometric theory
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37A99 Ergodic theory
58A12 de Rham theory in global analysis
PDFBibTeX XMLCite