Symplectomorphisms and discrete braid invariants

TL;DR

This paper develops a Morse-type forcing theory for periodic points of symplectomorphisms of the 2-disc, using braid invariants and Conley index theory to relate periodic points to mapping classes.

Contribution

It introduces a novel approach combining symplectomorphism decompositions with braid invariants and Conley index theory to predict periodic points.

Findings

Decomposition of symplectomorphisms into positive twists.

Application of Conley index theory to braid classes.

Establishment of a Morse type forcing theory for periodic points.

Abstract

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the Conley index theory of discrete braid classes as introduced in [Ghrist et al., C. R. Acad. Sci. Paris S\'er. I Math., 331(11), 2000, Invent. Math., 152(2), 2003] in order to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.