December  2019, 12(8): 2379-2390. doi: 10.3934/dcdss.2019149

Stable sets of planar homeomorphisms with translation pseudo-arcs

Departamento de Geometría y Topología, Universidad Complutense de Madrid, Spain

Received  November 2016 Revised  June 2017 Published  January 2019

Fund Project: The author have been supported by MINECO, MTM2015-63612-P.

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
$ \{S_{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.
Citation: Francisco R. Ruiz del Portal. Stable sets of planar homeomorphisms with translation pseudo-arcs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2379-2390. doi: 10.3934/dcdss.2019149
References:
[1]

S. Baldwin and E. E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two dimensions, Proc. Amer. Math. Soc., 109 (1990), 823-828.  doi: 10.2307/2048225.  Google Scholar

[2]

R. H. Bing, Concerning hereditarily indecomposable compacta, Pacific J. Math., 1 (1951), 43-51.  doi: 10.2140/pjm.1951.1.43.  Google Scholar

[3]

K. Borsuk, Theory of Shape, Monografie Matematyczne 59, PWN, Warsaw, 1975.  Google Scholar

[4]

L. E. Brouwer, Beweis des ebenen Translationssatzes, Math. Annalen, 72 (1912), 37-54.  doi: 10.1007/BF01456888.  Google Scholar

[5]

M. Brown, A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.   Google Scholar

[6]

M. Brown, Homeomorphisms of two-dimensional manifolds, Houston Math. J., 11 (1985), 455-469.   Google Scholar

[7]

R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foreman Co. Glenview Illinois, London, 1971.  Google Scholar

[8]

J. Campos and R. Ortega, Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems, Journal Diff. Equations, 138 (1997), 157-170.  doi: 10.1006/jdeq.1997.3265.  Google Scholar

[9]

C. O. Christenson and W. L. Voxman, Aspects of Topology, BCS Associates, Moscow, Idaho, 1998.  Google Scholar

[10]

E. N. Dancer and R. Ortega, The index or Lyapunov stable fixed points, Journal Dynamics and Diff. Equations, 6 (1994), 631-637.  doi: 10.1007/BF02218851.  Google Scholar

[11]

A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology, 4 (1965), 1-8.  doi: 10.1016/0040-9383(65)90044-3.  Google Scholar

[12]

M. Handel, A pathological area preserving $ C^{∞}$ diffeomorphism of the plane, Proc. Amer. Math. Soc., 86 (1982), 163-168.  doi: 10.2307/2044419.  Google Scholar

[13]

F. Le Roux, Homomorphismes de surfaces - Thor$ \grave{e}$mes de la fleur de Leau-Fatou et de la variété stable, Astrisque, 292 (2004), Vi+210pp.  Google Scholar

[14]

R. D. Nussbaum, The Fixed Point Index and Some Applications, Sminaire de Mathmatiques suprieures, Les Presses de L'Universit de Montral, 1985.  Google Scholar

[15]

F. R. Ruiz del Portal and J. M. Salazar, Fixed point index of iterations of local homeomorphisms of the plane: A Conley-index approach, Topology, 41 (2002), 1199-1212.  doi: 10.1016/S0040-9383(01)00035-0.  Google Scholar

[16]

F. R. Ruiz del Portal and J. M. Salazar, A stable/unstable manifold theorem for local homeomorphisms of the plane, Ergodic Th. and Dynamical Systems, 25 (2005), 301-317.  doi: 10.1017/S0143385704000367.  Google Scholar

[17]

F. R. Ruiz del Portal and J. M. Salazar, A Poincar formula for the fixed point indices of the iterations of arbitrary planar homeomorphisms, Fixed Point Theory Appl., (2010), ID233069, 31pp. doi: 10.1155/2010/323069.  Google Scholar

show all references

References:
[1]

S. Baldwin and E. E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two dimensions, Proc. Amer. Math. Soc., 109 (1990), 823-828.  doi: 10.2307/2048225.  Google Scholar

[2]

R. H. Bing, Concerning hereditarily indecomposable compacta, Pacific J. Math., 1 (1951), 43-51.  doi: 10.2140/pjm.1951.1.43.  Google Scholar

[3]

K. Borsuk, Theory of Shape, Monografie Matematyczne 59, PWN, Warsaw, 1975.  Google Scholar

[4]

L. E. Brouwer, Beweis des ebenen Translationssatzes, Math. Annalen, 72 (1912), 37-54.  doi: 10.1007/BF01456888.  Google Scholar

[5]

M. Brown, A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.   Google Scholar

[6]

M. Brown, Homeomorphisms of two-dimensional manifolds, Houston Math. J., 11 (1985), 455-469.   Google Scholar

[7]

R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foreman Co. Glenview Illinois, London, 1971.  Google Scholar

[8]

J. Campos and R. Ortega, Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems, Journal Diff. Equations, 138 (1997), 157-170.  doi: 10.1006/jdeq.1997.3265.  Google Scholar

[9]

C. O. Christenson and W. L. Voxman, Aspects of Topology, BCS Associates, Moscow, Idaho, 1998.  Google Scholar

[10]

E. N. Dancer and R. Ortega, The index or Lyapunov stable fixed points, Journal Dynamics and Diff. Equations, 6 (1994), 631-637.  doi: 10.1007/BF02218851.  Google Scholar

[11]

A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology, 4 (1965), 1-8.  doi: 10.1016/0040-9383(65)90044-3.  Google Scholar

[12]

M. Handel, A pathological area preserving $ C^{∞}$ diffeomorphism of the plane, Proc. Amer. Math. Soc., 86 (1982), 163-168.  doi: 10.2307/2044419.  Google Scholar

[13]

F. Le Roux, Homomorphismes de surfaces - Thor$ \grave{e}$mes de la fleur de Leau-Fatou et de la variété stable, Astrisque, 292 (2004), Vi+210pp.  Google Scholar

[14]

R. D. Nussbaum, The Fixed Point Index and Some Applications, Sminaire de Mathmatiques suprieures, Les Presses de L'Universit de Montral, 1985.  Google Scholar

[15]

F. R. Ruiz del Portal and J. M. Salazar, Fixed point index of iterations of local homeomorphisms of the plane: A Conley-index approach, Topology, 41 (2002), 1199-1212.  doi: 10.1016/S0040-9383(01)00035-0.  Google Scholar

[16]

F. R. Ruiz del Portal and J. M. Salazar, A stable/unstable manifold theorem for local homeomorphisms of the plane, Ergodic Th. and Dynamical Systems, 25 (2005), 301-317.  doi: 10.1017/S0143385704000367.  Google Scholar

[17]

F. R. Ruiz del Portal and J. M. Salazar, A Poincar formula for the fixed point indices of the iterations of arbitrary planar homeomorphisms, Fixed Point Theory Appl., (2010), ID233069, 31pp. doi: 10.1155/2010/323069.  Google Scholar

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