Stable sets of planar homeomorphisms with translation pseudo-arcs

Francisco R. Ruiz del Portal

doi:10.3934/dcdss.2019149

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

Stable sets of planar homeomorphisms with translation pseudo-arcs

Francisco R. Ruiz del Portal

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$

$ 0$

$ i_{{\mathbb R}^2}(g_n,0)$

, is equal to

$ -n$

and the stable (respectively unstable) sets of

$ g_n$

$ 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:

$ S_i \cap S_j= \{0\} = U_i \cap U_j$

for any

$ i, j ∈ \{1,2, \dotsn+1\}$

with

$ i\ne j$

$ 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.

Keywords: Planar homeomorphisms, stable/unstable sets, fixed point index, pseudo-arcs, translation arc.

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

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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