Monotone quotients of surface diffeomorphisms

Abstract

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha }$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy.

14 pages, 4 figures