arXiv subjects

Marta Batoréo

Publications and source records attributed to Marta Batoréo.

6 recordsLinked to original sources

arXiv2026

Remarks on periodic points, flux, and the mean index in symplectic dynamics

It is shown that a symplectic isotopy with non-degenerate end-point on a closed symplectically Calabi--Yau manifold has simple periodic points of arbitrarily large period under a simple non-vanishing condition on the Morse--Novikov homology associated with its flux. The proof combines a sharp estimate for the Conley--Zehnder index under the assumption that the mean index is an even integer with a classical asymptotic iteration argument in the non-Hamiltonian Floer-theoretic setting. We also apply this estimate to the symplectic dynamics of the standard symplectic $4$-torus.

math.SG↗

arXiv2024

The number of periodic points of surface symplectic diffeomorphisms

We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus $g$; it implies that a symplectic diffeomorphism isotopic to the identity with more than $2g-2$ fixed points, counted homologically, has infinitely many periodic points. Furthermore, we present examples of symplectic diffeomorphisms with a prescribed number of periodic points. In particular, we construct symplectic flows on surfaces possessing only one fixed point and no other periodic orbits.

math.SG↗

arXiv2016

On Periodic Points of Symplectomorphisms on Surfaces

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".

math.SG↗

arXiv2015

On non-contractible hyperbolic periodic orbits and periodic points of symplectomorphisms

We prove the existence of infinitely many periodic points of symplectomorphisms isotopic to the identity if they admit at least one (non-contractible) hyperbolic periodic orbit and satisfy some condition on its flux. The obtained periodic points correspond to periodic orbits whose free homotopy classes are formed by iterations of the hyperbolic periodic orbit. Our result is proved for a certain class of closed symplectic manifolds and the main tool we use is a variation of Floer theory for non-contractible periodic orbits and symplectomorphisms, the Floer-Novikov theory. For a certain class of symplectic manifolds, the theorem generalizes the main results proved for Hamiltonian diffeomorphisms in GG14 and for symplectomorphisms and contractible orbits in Ba15.

math.DS↗

arXiv2014

On hyperbolic points and periodic orbits of symplectomorphisms

We prove the existence of infinitely many periodic orbits of symplectomorphisms isotopic to the identity if they admit at least one hyperbolic periodic orbit and satisfy some condition on the flux. Our result is proved for a certain class of closed symplectic manifolds and the main tool we use is a variation of Floer theory for symplectomorphisms, the Floer-Novikov theory. The proof relies on an important result on hyperbolic orbits, namely, that a \emph{Floer-Novikov} trajectory which converges to an iteration $k$ of the hyperbolic orbit and crosses its fixed neighborhood has energy bounded bellow by a strictly positive constant independent of $k$. The main theorem follows from this feature of hyperbolic orbits and certain properties of quantum homology on the class of symplectic manifolds we work with.

math.SG↗

arXiv2011

On the rigidity of the coisotropic Maslov index on certain rational symplectic manifolds

We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a result on the rigidity of the Maslov index for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.

math.SG↗