• Previous Article
    Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points
  • DCDS Home
  • This Issue
  • Next Article
    Positive ground state solutions for a quasilinear elliptic equation with critical exponent
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. Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

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

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

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[21]

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

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[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. Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

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

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

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[21]

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

[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. Google Scholar

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)
[1]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010

[2]

Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251

[3]

Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123

[4]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[5]

Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006

[6]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

[7]

Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53

[8]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35

[9]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003

[10]

Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271

[11]

Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139

[12]

Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903

[13]

Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379

[14]

Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099

[15]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289

[16]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4619-4635. doi: 10.3934/dcds.2016001

[17]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431

[18]

Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867

[19]

Fernando Alcalde Cuesta, Françoise Dal'Bo, Matilde Martínez, Alberto Verjovsky. Corrigendum to "Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology". Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4585-4586. doi: 10.3934/dcds.2017196

[20]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (14)
  • HTML views (1)
  • Cited by (0)

[Back to Top]