Existence of minimal flows on nonorientable surfaces

José Ginés Espín Buendía; Daniel Peralta-salas; Gabriel Soler López

doi:10.3934/dcds.2017178

Article Contents

2017, Volume 37, Issue 8: 4191-4211. Doi: 10.3934/dcds.2017178

This issue Previous Article Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points Next Article Positive ground state solutions for a quasilinear elliptic equation with critical exponent

Existence of minimal flows on nonorientable surfaces

1.
Departamento de Matemáticas, Universidad de Murcia (Campus de Espinardo), 30100 Espinardo-Murcia, Spain
2.
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
3.
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Paseo Alfonso XIII, 52, 30203 Cartagena, Spain

^* Corresponding author: josegines.espin@um.es
^* Corresponding author: josegines.espin@um.es

Received: July 31, 2016

Revised: February 28, 2017

Published: May 2017

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Abstract

Surfaces admitting flows all whose orbits are dense are called minimal. Minimal orientable surfaces were characterized by J.C. Benière in 1998, leaving open the nonorientable case. This paper fills this gap providing a characterization of minimal nonorientable surfaces of finite genus. We also construct an example of a minimal nonorientable surface with infinite genus and conjecture that any nonorientable surface without combinatorial boundary is minimal.

Keywords:

Flow,
surface,
minimality,
transitivity,
interval and circle exchange transformations,
suspension.

Mathematics Subject Classification: 37C70, 37C10, 37E0.

Citation:

Full Text(HTML)

Figure 1. Construction of $M_T$ by means of a $(6, 3) $-i.e.t. with $\pi = (-3, 4, -5, 6, 1, -2) $. The circle $C$ is nonorientable. The arrows on the images of the $m_i$ mark if they are flipped by $T$

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Figure 2. A standard saddle point (left) and a $6$-saddle point (right)

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