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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

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