Conservative surface homeomorphisms with finitely many periodic points

Generic Equidistribution of Periodic Orbits for Area-Preserving Surface Maps

“…Theorem B follows from a short argument combining Theorem A and a theorem of Oxtoby-Ulam [OU41]. Theorem A follows from Theorem C, which may be of independent interest, and the results in [LC06,LC22].…”

“…The resulting time-one maps have finitely many periodic points, all of which are fixed, and are isotopic to the identity relative to their fixed point set. The current state-of-the-art, to our knowledge, is due to Le Calvez [LC22]. He shows, barring some edge cases when g ď 1, that any area-preserving homeomorphism of a closed surface is either periodic, has periodic points of unbounded minimal period, or after passing to an iterate only has finitely many fixed points and is isotopic to the identity relative to the fixed point set.…”

“…Assume that Σ is closed and has genus ě 2. It is known [LC22] that either φ has periodic points of unbounded minimal period or it has periodic Nielsen-Thurston class. Therefore, if φ has periodic Nielsen-Thurston class and some iterate φ q P Homeo 0 pΣ, ωq is rational, Theorem A implies φ is either periodic or has periodic points of unbounded minimal period.…”