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August 2017, 37(8): 4191-4211. doi: 10.3934/dcds.2017178

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

Received  August 2016 Revised  March 2017 Published  April 2017

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.

Citation: José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
References:
[1]

C. Angosto Hernández and G. Soler López, Minimality and the Rauzy-Veech algorithm for interval exchange transformations with flips, Dyn. Syst., 28 (2013), 539-550. doi: 10.1080/14689367.2013.824950.

[2]

S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, Providence, 1996.

[3]

P. Arnoux, échanges d'intervalles et flots sur les surfaces, Monograph. Enseign. Math., 29 (1981), 5-38.

[4]

J. C. Benière, Feuilletage Minimaux Sur Les Surfaces non Compactes, Ph.D thesis, Université de Lyon, 1998.

[5]

R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562. doi: 10.1090/S0002-9939-1969-0247028-5.

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[7]

C. Gutiérre, Smoothing continuous flows on two manifolds and recurrences, Ergod. Th. & Dynam. Sys., 6 (1986), 17-44. doi: 10.1017/S0143385700003278.

[8]

C. Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc., 241 (1978), 311-320. doi: 10.1090/S0002-9947-1978-0492303-2.

[9]

C. Gutiérrez, Smooth nonorientable nontrivial recurrence on two-manifolds, J. Differential Equations, 29 (1978), 388-395. doi: 10.1016/0022-0396(78)90048-7.

[10]

C. GutiérrezG. Hector and A. López, Interval exchange transformations and foliations on infinite genus 2-manifolds, Ergod. Th. & Dynam. Sys., 24 (2004), 1097-1180. doi: 10.1017/S0143385704000069.

[11]

C. GutiérrezS. LloydV. MedvedevB. Pires and E. Zhuzhoma, Transitive circle exchange transformations with flips, Discrete Contin. Dynam. Systems, 26 (2010), 251-263. doi: 10.3934/dcds.2010.26.251.

[12]

V. Jiménez López and G. Soler López, Transitive flows on manifolds, Rev. Mat. Iberoamericana, 20 (2004), 107-130. doi: 10.4171/RMI/382.

[13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[14]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[15]

K. Kuratowski, Topology. Vol. I, Academic Press, New York, 1966.

[16]

G. Levitt, Pantalons et feuilletages des surfaces, Topology, 21 (1982), 9-33. doi: 10.1016/0040-9383(82)90039-8.

[17]

A. Linero and G. Soler López, Minimal interval exchange transformations with flips, To appear in Ergodic Theory Dynam. Systems., (). doi: 10.1017/etds.2017.5.

[18]

I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269. doi: 10.1090/S0002-9947-1963-0143186-0.

[19]

R. A. Smith and S. Thomas, Some examples of transitive smooth flows on differentiable manifolds, J. London Math. Soc., 37 (1988), 552-568. doi: 10.1112/jlms/s2-37.3.552.

[20]

G. Soler López, Transitive and minimal flows and interval exchange transformations, in Advances in discrete dynamics (eds. J. S. Cánovas), Nova Science Publishers, (2013), 163-191.

[21]

G. Soler López, ω-límites de Sistemas Dinámicos Continuos, Master thesis, Universidad de Murcia, 2011.

[22]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

show all references

References:
[1]

C. Angosto Hernández and G. Soler López, Minimality and the Rauzy-Veech algorithm for interval exchange transformations with flips, Dyn. Syst., 28 (2013), 539-550. doi: 10.1080/14689367.2013.824950.

[2]

S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, Providence, 1996.

[3]

P. Arnoux, échanges d'intervalles et flots sur les surfaces, Monograph. Enseign. Math., 29 (1981), 5-38.

[4]

J. C. Benière, Feuilletage Minimaux Sur Les Surfaces non Compactes, Ph.D thesis, Université de Lyon, 1998.

[5]

R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562. doi: 10.1090/S0002-9939-1969-0247028-5.

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[7]

C. Gutiérre, Smoothing continuous flows on two manifolds and recurrences, Ergod. Th. & Dynam. Sys., 6 (1986), 17-44. doi: 10.1017/S0143385700003278.

[8]

C. Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc., 241 (1978), 311-320. doi: 10.1090/S0002-9947-1978-0492303-2.

[9]

C. Gutiérrez, Smooth nonorientable nontrivial recurrence on two-manifolds, J. Differential Equations, 29 (1978), 388-395. doi: 10.1016/0022-0396(78)90048-7.

[10]

C. GutiérrezG. Hector and A. López, Interval exchange transformations and foliations on infinite genus 2-manifolds, Ergod. Th. & Dynam. Sys., 24 (2004), 1097-1180. doi: 10.1017/S0143385704000069.

[11]

C. GutiérrezS. LloydV. MedvedevB. Pires and E. Zhuzhoma, Transitive circle exchange transformations with flips, Discrete Contin. Dynam. Systems, 26 (2010), 251-263. doi: 10.3934/dcds.2010.26.251.

[12]

V. Jiménez López and G. Soler López, Transitive flows on manifolds, Rev. Mat. Iberoamericana, 20 (2004), 107-130. doi: 10.4171/RMI/382.

[13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[14]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[15]

K. Kuratowski, Topology. Vol. I, Academic Press, New York, 1966.

[16]

G. Levitt, Pantalons et feuilletages des surfaces, Topology, 21 (1982), 9-33. doi: 10.1016/0040-9383(82)90039-8.

[17]

A. Linero and G. Soler López, Minimal interval exchange transformations with flips, To appear in Ergodic Theory Dynam. Systems., (). doi: 10.1017/etds.2017.5.

[18]

I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc., 106 (1963), 259-269. doi: 10.1090/S0002-9947-1963-0143186-0.

[19]

R. A. Smith and S. Thomas, Some examples of transitive smooth flows on differentiable manifolds, J. London Math. Soc., 37 (1988), 552-568. doi: 10.1112/jlms/s2-37.3.552.

[20]

G. Soler López, Transitive and minimal flows and interval exchange transformations, in Advances in discrete dynamics (eds. J. S. Cánovas), Nova Science Publishers, (2013), 163-191.

[21]

G. Soler López, ω-límites de Sistemas Dinámicos Continuos, Master thesis, Universidad de Murcia, 2011.

[22]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

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