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September  2011, 31(3): 651-668. doi: 10.3934/dcds.2011.31.651

Homeomorphisms of the annulus with a transitive lift II

Salvador Addas-Zanata 1, and  Fábio A. Tal 1,

1. 

Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

Received  June 2010 Revised  September 2010 Published  August 2011

Let $f$ be a homeomorphism of the closed annulus $A$ that preserves the orientation, the boundary components and that has a lift $\tilde f$ to the infinite strip $\tilde A$ which is transitive. We show that, if the rotation number of $\tilde f$ restricted to both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $\Gamma\subset\tilde A$ such that $\Gamma\subset]-\infty,0]\times[0,1]$, $\Gamma$ is unbounded, the projection of $\Gamma$ to $A$ is dense, $\Gamma-(1,0)\subset\Gamma$ and $\tilde{f}(\Gamma)\subset \Gamma.$ Also, if $p_1$ is the projection on the first coordinate of $\tilde A$, then there exists $d>0$ such that, for any $\tilde z\in\Gamma,$ $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$
Keywords: Transitivity, connected subsets of the plane., rotation set.
Mathematics Subject Classification: 37E30, 37E4.
Citation: Salvador Addas-Zanata, Fábio A. Tal. Homeomorphisms of the annulus with a transitive lift II. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 651-668. doi: 10.3934/dcds.2011.31.651
References:
[1]

S. Addas-Zanata and F. A. Tal, Homeomorphisms of the annulus with a transitive lift, to appear in Math. Z., 2010. Google Scholar

[2]

S. Alpern and V. Prasad, Typical recurrence for lifts of mean rotation zero annulus homeomorphisms, Bull. London Math. Soc., 23 (1991), 477-481. doi: 10.1112/blms/23.5.477.  Google Scholar

[3]

S. Alpern and V. Prasad, Typical transitivity for lifts of rotationless annulus or torus homeomorphisms, Bull. London Math. Soc., 27 (1995), 79-81. doi: 10.1112/blms/27.1.79.  Google Scholar

[4]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.  Google Scholar

show all references

References:
[1]

S. Addas-Zanata and F. A. Tal, Homeomorphisms of the annulus with a transitive lift, to appear in Math. Z., 2010. Google Scholar

[2]

S. Alpern and V. Prasad, Typical recurrence for lifts of mean rotation zero annulus homeomorphisms, Bull. London Math. Soc., 23 (1991), 477-481. doi: 10.1112/blms/23.5.477.  Google Scholar

[3]

S. Alpern and V. Prasad, Typical transitivity for lifts of rotationless annulus or torus homeomorphisms, Bull. London Math. Soc., 27 (1995), 79-81. doi: 10.1112/blms/27.1.79.  Google Scholar

[4]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.  Google Scholar

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