# Shub's conjecture for smooth longitudinal maps of
*S*^{m}

## Grzegorz Graff, Michal Misiurewicz and Piotr Nowak-Przygodzki

### Abstract

Let *f* be a smooth map of the *m*-dimensional
sphere *S*^{m} to itself, preserving the longitudinal
foliation. We estimate from below the number of fixed points of the
iterates of *f*, reduce the Shub's conjecture for longitudinal
maps to a lower dimensional classical version, and prove the
conjecture in case *m*=2 and in a weak form for *m*=3.