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Search: MSC category 37B20 ( Notions of recurrence )

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1. CMB 2012 (vol 57 pp. 240)

Bernardes, Nilson C.
Addendum to ``Limit Sets of Typical Homeomorphisms''
Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f : X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$ (respectively to the alpha limit set $\alpha(f,x)$) is topologically conjugate to the universal odometer.

Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets
Categories:37B20, 54H20, 28C15, 54C35, 54E52

2. CMB 2011 (vol 55 pp. 225)

Bernardes, Nilson C.
Limit Sets of Typical Homeomorphisms
Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is non-sensitive at each point of $\omega(f,x)$, and the function $a \to \omega(f,a)$ is continuous at $x$.

Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets
Categories:37B20, 54H20, 28C15, 54C35, 54E52

3. CMB 2011 (vol 54 pp. 311)

Marzougui, Habib
Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows
We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder
Categories:37B20, 37E35

4. CMB 2004 (vol 47 pp. 332)

Charette, Virginie; Goldman, William M.; Jones, Catherine A.
Recurrent Geodesics in Flat Lorentz $3$-Manifolds
Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely classified when $\Gamma$ is cyclic. This implies that for any pair of periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$.

Keywords:geometric structures on low-dimensional manifolds, notions of recurrence
Categories:57M50, 37B20

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