Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1608.03249

On Periodic Points of Symplectomorphisms on Surfaces

Marta Batoréo
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct a symplectic flow on a surface of genus g greater than one with exactly 2g-2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly non-degenerate) symplectomorphism of a surface (with genus g greater than one) isotopic to the identity has infinitely many periodic points if there exists a fixed point with non-zero mean index. From this result, we obtain two corollaries, namely that such a symplectomorphism with an elliptic fixed point or with strictly more than 2g-2 fixed points has infinitely many periodic points provided that the flux of the isotopy is "irrational".

Explore related subjects

math.SGmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marta Batoréo. 2016-08-10. On Periodic Points of Symplectomorphisms on Surfaces. https://doi.org/10.2140/pjm.2018.294.19

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality

This paper relates the minimal Harder--Narasimhan slope to the surjectivity of real-linear Cauchy--Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

math.SG↗
arXiv2026

Welschinger invariants and the Conway polynomial

Welschinger showed that counts of connected holomorphic disks with Lagrangian boundary in symplectic 6-manifolds, meeting at least one boundary constraint, can be made invariant by correcting them with counts of disconnected disks weighted by certain "self-linking" numbers. We show his invariant is the lowest order term in an all-genus curve count where curves are weighted by the Conway polynomials of their boundaries. This in turn is a specialization of the skein-valued curve count, but can be defined without the 4-chain and vector field used in that setup.

math.SG↗
arXiv2026

KAM splittings and equidistributed periodic orbits for stable hypersurfaces

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

math.SG↗