September  2018, 38(9): 4243-4257. doi: 10.3934/dcds.2018185

Dynamics of induced homeomorphisms of one-dimensional solenoids

Centro de Investigación en Matemáticas, A.C., Jalisco S/N, Col. Valenciana CP: 36023, Guanajuato, Gto, México

Received  April 2017 Revised  October 2017 Published  June 2018

Fund Project: This paper is part of the author's doctoral dissertation. Research was supported by CONACyT scholarship for Doctoral Students

We study the displacement function of homeomorphisms isotopic to the identity of the universal one-dimensional solenoid and we get a characterization of the lifting property for an open and dense subgroup of the isotopy component of the identity. The dynamics of an element in this subgroup is also described using rotation theory.

Citation: Francisco J. López-Hernández. Dynamics of induced homeomorphisms of one-dimensional solenoids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4243-4257. doi: 10.3934/dcds.2018185
References:
[1]

J. Aliste-Prieto, Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line, Ergodic Theory Dynam. Systems, 30 (2010), 565-594. doi: 10.1017/S0143385709000145. Google Scholar

[2]

J. Aliste-Prieto and T. Jager, Almost periodic structures and the semiconjugacy problem, Journal of Differential Equations, 252 (2012), 4988-5001. doi: 10.1016/j.jde.2012.01.030. Google Scholar

[3]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. Google Scholar

[4]

M. Cruz-López and A. Verjorvsky, Poincaré theory for compact abelian one-dimensional solenoidal groups, arXiv: 1308.1853v2 [math.DS].Google Scholar

[5]

J. Franks, Realizing rotation vector for torus homeomorphisms, Transactions of the American Mathematical Society, 311 (1989), 107-115. doi: 10.1090/S0002-9947-1989-0958891-1. Google Scholar

[6]

É. Ghys, Groups acting on the circle, L'Enseignement Mathématique, 47 (2001), 329-407. Google Scholar

[7]

T. Jäger, Linearization of conservative toral homeomorphisms, Invent. Math., 176 (2009), 601-616. doi: 10.1007/s00222-008-0171-5. Google Scholar

[8]

T. Jäger and A. Koropecki, Poincaré theory for decomposable cofrontiers, Ann. Henri Poincaré, 18 (2017), 85-112. doi: 10.1007/s00023-016-0523-4. Google Scholar

[9]

J. Kwapisz, Poincaré rotation number for maps of the real line with almost periodic displacement, Nonlinearity, 13 (2000), 1841-1854. doi: 10.1088/0951-7715/13/5/320. Google Scholar

[10]

J. Kwapisz, Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua, Fundamenta Matematicae, 168 (2001), 251-278. doi: 10.4064/fm168-3-3. Google Scholar

[11]

F. J. López-Hernández, The displacement function of solenoidal homeomorphisms, M. Sc. Thesis CIMAT, A. C. 2013.Google Scholar

[12]

M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc., 40 (1989), 490-506. doi: 10.1112/jlms/s2-40.3.490. Google Scholar

[13]

H. Poincaré, Memoire sur les courbes définis par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422. Google Scholar

[14]

M. Pollicott, Rotation sets for homeomorphism and homology, Transactions of the American Mathematical Society, 331 (1992), 881-894. doi: 10.1090/S0002-9947-1992-1094554-2. Google Scholar

[15]

J. T. RogersJr. and J. L. Tollefson, Homeomorphisms homotopic to induced homeomorphisms of weak solenoidal spaces, Colloq. Math., 25 (1972), 81-87. doi: 10.4064/cm-25-1-81-87. Google Scholar

show all references

References:
[1]

J. Aliste-Prieto, Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line, Ergodic Theory Dynam. Systems, 30 (2010), 565-594. doi: 10.1017/S0143385709000145. Google Scholar

[2]

J. Aliste-Prieto and T. Jager, Almost periodic structures and the semiconjugacy problem, Journal of Differential Equations, 252 (2012), 4988-5001. doi: 10.1016/j.jde.2012.01.030. Google Scholar

[3]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N. Y., 1947. Google Scholar

[4]

M. Cruz-López and A. Verjorvsky, Poincaré theory for compact abelian one-dimensional solenoidal groups, arXiv: 1308.1853v2 [math.DS].Google Scholar

[5]

J. Franks, Realizing rotation vector for torus homeomorphisms, Transactions of the American Mathematical Society, 311 (1989), 107-115. doi: 10.1090/S0002-9947-1989-0958891-1. Google Scholar

[6]

É. Ghys, Groups acting on the circle, L'Enseignement Mathématique, 47 (2001), 329-407. Google Scholar

[7]

T. Jäger, Linearization of conservative toral homeomorphisms, Invent. Math., 176 (2009), 601-616. doi: 10.1007/s00222-008-0171-5. Google Scholar

[8]

T. Jäger and A. Koropecki, Poincaré theory for decomposable cofrontiers, Ann. Henri Poincaré, 18 (2017), 85-112. doi: 10.1007/s00023-016-0523-4. Google Scholar

[9]

J. Kwapisz, Poincaré rotation number for maps of the real line with almost periodic displacement, Nonlinearity, 13 (2000), 1841-1854. doi: 10.1088/0951-7715/13/5/320. Google Scholar

[10]

J. Kwapisz, Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua, Fundamenta Matematicae, 168 (2001), 251-278. doi: 10.4064/fm168-3-3. Google Scholar

[11]

F. J. López-Hernández, The displacement function of solenoidal homeomorphisms, M. Sc. Thesis CIMAT, A. C. 2013.Google Scholar

[12]

M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc., 40 (1989), 490-506. doi: 10.1112/jlms/s2-40.3.490. Google Scholar

[13]

H. Poincaré, Memoire sur les courbes définis par une équation différentielle, Journal de Mathématiques, 7 (1881), 375-422. Google Scholar

[14]

M. Pollicott, Rotation sets for homeomorphism and homology, Transactions of the American Mathematical Society, 331 (1992), 881-894. doi: 10.1090/S0002-9947-1992-1094554-2. Google Scholar

[15]

J. T. RogersJr. and J. L. Tollefson, Homeomorphisms homotopic to induced homeomorphisms of weak solenoidal spaces, Colloq. Math., 25 (1972), 81-87. doi: 10.4064/cm-25-1-81-87. Google Scholar

[1]

Deissy M. S. Castelblanco. Restrictions on rotation sets for commuting torus homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5257-5266. doi: 10.3934/dcds.2016030

[2]

Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399

[3]

Ronald de Man. On composants of solenoids. Electronic Research Announcements, 1995, 1: 87-90.

[4]

Matthieu Hillairet, Ayman Moussa, Franck Sueur. On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow. Kinetic & Related Models, 2019, 12 (4) : 681-701. doi: 10.3934/krm.2019026

[5]

Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443

[6]

Manfred Einsiedler and Elon Lindenstrauss. Rigidity properties of \zd-actions on tori and solenoids. Electronic Research Announcements, 2003, 9: 99-110.

[7]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213

[8]

Światosław R. Gal, Jarek Kędra. On distortion in groups of homeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 609-622. doi: 10.3934/jmd.2011.5.609

[9]

Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291

[10]

Salvador Addas-Zanata, Fábio A. Tal. Homeomorphisms of the annulus with a transitive lift II. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 651-668. doi: 10.3934/dcds.2011.31.651

[11]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711

[12]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593

[13]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169

[14]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007

[15]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391

[16]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[17]

Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511

[18]

Rafael Ortega. Trivial dynamics for a class of analytic homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 651-659. doi: 10.3934/dcdsb.2008.10.651

[19]

Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103

[20]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (132)
  • HTML views (110)
  • Cited by (0)

Other articles
by authors

[Back to Top]