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

Restrictions on rotation sets for commuting torus homeomorphisms

1. 

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

Received  October 2015 Revised  February 2016 Published  July 2016

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.
Citation: 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
References:
[1]

M. Benayon, Sobre Grupos Abelianos Irrotacionais de Homeomorfismos do Toro,, Ph.D thesis, (2013). Google Scholar

[2]

P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint,, , (). Google Scholar

[3]

P. Dávalos, On annular maps of the torus and sublinear diffusion, preprint,, , (). Google Scholar

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J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107. doi: 10.1090/S0002-9947-1989-0958891-1. Google Scholar

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A. Kocksard and A. Koropecki, Free curves and periodic points for torus homeomorphisms,, Ergod. Th. Dynam. Sys., 28 (2008), 1895. doi: 10.1017/S0143385707001083. Google Scholar

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A. Koropecki and F. Tal, Strictly toral dynamics,, Invent math., 196 (2014), 339. doi: 10.1007/s00222-013-0470-3. Google Scholar

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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. doi: 10.1017/S0143385700006040. Google Scholar

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

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K. Parkhe, Commuting homeomorphisms with non-commuting lifts, preprint,, , (). Google Scholar

show all references

References:
[1]

M. Benayon, Sobre Grupos Abelianos Irrotacionais de Homeomorfismos do Toro,, Ph.D thesis, (2013). Google Scholar

[2]

P. Le Calvez and F. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint,, , (). Google Scholar

[3]

P. Dávalos, On annular maps of the torus and sublinear diffusion, preprint,, , (). Google Scholar

[4]

J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107. doi: 10.1090/S0002-9947-1989-0958891-1. Google Scholar

[5]

J. Franks and M. Misiurewicz, Rotation sets of toral flows,, Proc. Amer. Math. Soc., 109 (1990), 243. doi: 10.1090/S0002-9939-1990-1021217-5. Google Scholar

[6]

A. Kocksard and A. Koropecki, Free curves and periodic points for torus homeomorphisms,, Ergod. Th. Dynam. Sys., 28 (2008), 1895. doi: 10.1017/S0143385707001083. Google Scholar

[7]

A. Koropecki and F. Tal, Strictly toral dynamics,, Invent math., 196 (2014), 339. doi: 10.1007/s00222-013-0470-3. Google Scholar

[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. doi: 10.1017/S0143385700006040. Google Scholar

[9]

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

[10]

K. Parkhe, Commuting homeomorphisms with non-commuting lifts, preprint,, , (). Google Scholar

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