Periodic Points for Sphere Maps Preserving Monopole Foliations

Abstract

Let S 2 be a two-dimensional sphere. We consider two types of its foliations with one singularity and maps f : S 2 → S 2 preserving these foliations, more and less regular. We prove that in both cases f has at least | deg(f)| fixed points, where deg(f) is a topological degree of f. In particular, the lower growth rate of the number of fixed points of the iterations of f is at least log | deg(f)|. This confirms the Shub's conjecture in these classes of maps.

The Topological Entropy Conjecture

“…Lately, for results about random entropy expansiveness and dominated splittings, see [38], and for results about the relations of topological entropy and Lefschetz numbers, see [39][40][41]. Furthermore, for a variational principle for subadditive preimage topological pressure for continuous bundle random dynamical systems, see [42].…”