The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center
Let \(M\) be a compact manifold and \(f: M\to M\) a diffeomorphism. Denote by \(f_{*,k}\) the corresponding automorphism of the \(k\)-th homology group. Then the spectral radius \(\mathrm{sp}(f_*)\) is defined to be the maximum of the spectral radii of the \(f_{*,k}\). Further, denote by \(h(f)\) the topological entropy of \(f\). The authors show that under specific hyperbolicity conditions on \(f\) the inequality \(h(f)\geq \mathrm{sp}(f_*)\) holds, thus proving a special case of Shub's conjecture of \textit{M. Shub} [Bull. Am. Math. Soc. 80, 27--41 (1974; Zbl 0305.58014)].
scientific article; zbMATH DE number 5641741
scientific article