Restrictions on rotation sets for commuting torus homeomorphisms

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October 2016, 36(10): 5257-5266. doi: 10.3934/dcds.2016030

Restrictions on rotation sets for commuting torus homeomorphisms

Deissy M. S. Castelblanco ^1,

1.	Rua do Matão 1010, IME-USP, São Paulo, SP, Brazil

Received October 2015 Revised February 2016 Published July 2016

Abstract
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Let $K_1,\: K_2\subset \mathbb{R}^2$ be two convex, compact sets. We would like to know if there are commuting torus homeomorphisms $f$ and $h$ homotopic to the identity, with lifts $\tilde f$ and $\tilde h$ such that $K_1$ and $K_2$ are their rotation sets respectively. In this work, we prove some cases where it cannot happen, assuming some restrictions on rotation sets.

Keywords: commuting homeomorphisms., Torus homeomorphisms, rotation set.

Mathematics Subject Classification: Primary: 37E30, 37E45, 37B0.

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

References:

[1]	M. Benayon, Sobre Grupos Abelianos Irrotacionais de Homeomorfismos do Toro, Ph.D thesis, Universidade Federal Fluminense, 2013.
[2]	P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint, arXiv:1503.09127v1.
[3]	P. Dávalos, On annular maps of the torus and sublinear diffusion, preprint, arXiv:1311.0046.
[4]	J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-115. doi: 10.1090/S0002-9947-1989-0958891-1.
[5]	J. Franks and M. Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249. doi: 10.1090/S0002-9939-1990-1021217-5.
[6]	A. Kocksard and A. Koropecki, Free curves and periodic points for torus homeomorphisms, Ergod. Th. Dynam. Sys., 28 (2008), 1895-1915. doi: 10.1017/S0143385707001083.
[7]	A. Koropecki and F. Tal, Strictly toral dynamics, Invent math., 196 (2014), 339-381. doi: 10.1007/s00222-013-0470-3.
[8]	J. Llibre and R. S. Mackay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergod. Th. Dynam. Sys., 11 (1991), 115-128. doi: 10.1017/S0143385700006040.
[9]	M. Misiurewics 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.
[10]	K. Parkhe, Commuting homeomorphisms with non-commuting lifts, preprint, arXiv:1409.6422v2.

show all references

References:

[1]	M. Benayon, Sobre Grupos Abelianos Irrotacionais de Homeomorfismos do Toro, Ph.D thesis, Universidade Federal Fluminense, 2013.
[2]	P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint, arXiv:1503.09127v1.
[3]	P. Dávalos, On annular maps of the torus and sublinear diffusion, preprint, arXiv:1311.0046.
[4]	J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-115. doi: 10.1090/S0002-9947-1989-0958891-1.
[5]	J. Franks and M. Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc., 109 (1990), 243-249. doi: 10.1090/S0002-9939-1990-1021217-5.
[6]	A. Kocksard and A. Koropecki, Free curves and periodic points for torus homeomorphisms, Ergod. Th. Dynam. Sys., 28 (2008), 1895-1915. doi: 10.1017/S0143385707001083.
[7]	A. Koropecki and F. Tal, Strictly toral dynamics, Invent math., 196 (2014), 339-381. doi: 10.1007/s00222-013-0470-3.
[8]	J. Llibre and R. S. Mackay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergod. Th. Dynam. Sys., 11 (1991), 115-128. doi: 10.1017/S0143385700006040.
[9]	M. Misiurewics 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.
[10]	K. Parkhe, Commuting homeomorphisms with non-commuting lifts, preprint, arXiv:1409.6422v2.

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