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Stable sets of planar homeomorphisms with translation pseudo-arcs

The author have been supported by MINECO, MTM2015-63612-P

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  • For every $ n ∈ {\mathbb N}$ we construct orientation preserving planar homeomorphisms $ g_n$ such that $ Fix(g_n)=\{0\}$, the fixed point index of $ g_n$ at $ 0$, $ i_{{\mathbb R}^2}(g_n,0)$, is equal to $ -n$ and the stable (respectively unstable) sets of $ g_n$ at $ 0$ decompose into exactly $ n+1$ connected branches $ \{U_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$ (resp.$ \{U_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$) such that:

    a) $ S_i \cap S_j= \{0\} = U_i \cap U_j$ for any $ i, j ∈ \{1,2, \dotsn+1\}$ with $ i\ne j$.

    b) $ S_i \cap U_j= \{0\}$ for any $ i, j ∈ \{1,2, \dots n+1\}$.

    c) For every $ j ∈ \{1,2, \dots n+1\}$, $ S_j \setminus\{0\}$ and $ U_j \setminus \{0\}$ admit translation pseudo-arcs. This means that there exist pseudo-arcs $ K_j\subset S_j $ and points $ p_{j\star} , g_n(p_{j\star}) ∈ K_j$, such that $ g_n(K_j)\cap K_j=\{ g_n(p_{j\star} )\} $ and

    $S_j\setminus \{ 0\}=\bigcup\limits_{m=-∞}^{∞} g_n^m (K_j)$

    and analogously for $ U_j$.

    We also study the closure of the class of above homeomorphisms in the (complete) metric space of planar orientation preserving homeomorphisms.

    Mathematics Subject Classification: Primary: 54H25, 54H20.


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  • Figure 2.  Periodic crooked chain of period 5.

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