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Fabio Armando Tal

Publications and source records attributed to Fabio Armando Tal.

22 records · Page 2Linked to original sources
arXiv2012

Dynamics of homeomorphisms of the torus homotopic to Dehn twists

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In other words, if we consider the lift $\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

math.DS↗
arXiv2012

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

We construct a $C^\infty$ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is $\{(0,0)\}$, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.

math.DS↗
arXiv2008

Homeomorphisms of the annulus with a transitive lift

Let $f$ be a homeomorphism of the closed annulus $A$ that preserves orientation, 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 both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $Γ\subset\tilde A$ such that $Γ\subset]-\infty,0]\times[0,1]$, $Γ$ is unlimited, the projection of $Γ$ to $A$ is dense, $Γ-(1,0)\subsetΓ$ and $\tilde{f}(Γ)\subset Γ.$ Also, if $p_1$ is the projection in the first coordinate in $\tilde A$, then there exists $d>0$ such that, for any $\tilde z\inΓ,$ $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.

math.DS↗