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On a semigroup problem
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 |
For every
| $ n ∈ {\mathbb N}$ |
| $ g_n$ |
| $ Fix(g_n)=\{0\}$ |
| $ g_n$ |
| $ 0$ |
| $ i_{{\mathbb R}^2}(g_n,0)$ |
| $ -n$ |
| $ g_n$ |
| $ 0$ |
| $ n+1$ |
| $ \{S_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$ |
| $ \{U_{j}\}_{j ∈ \{1,2, \dots, n+1\}}$ |
a)
| $ S_i \cap S_j= \{0\} = U_i \cap U_j$ |
| $ i, j ∈ \{1,2, \dotsn+1\}$ |
| $ i\ne j$ |
b)
| $ S_i \cap U_j= \{0\}$ |
| $ i, j ∈ \{1,2, \dots n+1\}$ |
c) For every
| $ j ∈ \{1,2, \dots n+1\}$ |
| $ S_j \setminus\{0\}$ |
| $ U_j \setminus \{0\}$ |
| $ K_j\subset S_j $ |
| $ p_{j\star} , g_n(p_{j\star}) ∈ K_j$ |
| $ g_n(K_j)\cap K_j=\{ g_n(p_{j\star} )\} $ |
| $S_j\setminus \{ 0\}=\bigcup\limits_{m=-∞}^{∞} g_n^m (K_j)$ |
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 and 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. |
| [2] |
R. H. Bing,
Concerning hereditarily indecomposable compacta, Pacific J. Math., 1 (1951), 43-51.
doi: 10.2140/pjm.1951.1.43. |
| [3] |
K. Borsuk, Theory of Shape, Monografie Matematyczne 59, PWN, Warsaw, 1975. |
| [4] |
L. E. Brouwer,
Beweis des ebenen Translationssatzes, Math. Annalen, 72 (1912), 37-54.
doi: 10.1007/BF01456888. |
| [5] |
M. Brown,
A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.
|
| [6] |
M. Brown,
Homeomorphisms of two-dimensional manifolds, Houston Math. J., 11 (1985), 455-469.
|
| [7] |
R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foreman Co. Glenview Illinois, London, 1971. |
| [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. |
| [9] |
C. O. Christenson and W. L. Voxman, Aspects of Topology, BCS Associates, Moscow, Idaho, 1998. |
| [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. |
| [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. |
| [12] |
M. Handel,
A pathological area preserving $ C^{∞}$ diffeomorphism of the plane, Proc. Amer. Math. Soc., 86 (1982), 163-168.
doi: 10.2307/2044419. |
| [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. |
| [14] |
R. D. Nussbaum, The Fixed Point Index and Some Applications, Sminaire de Mathmatiques suprieures, Les Presses de L'Universit de Montral, 1985. |
| [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. |
| [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. |
| [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. |
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. |
| [2] |
R. H. Bing,
Concerning hereditarily indecomposable compacta, Pacific J. Math., 1 (1951), 43-51.
doi: 10.2140/pjm.1951.1.43. |
| [3] |
K. Borsuk, Theory of Shape, Monografie Matematyczne 59, PWN, Warsaw, 1975. |
| [4] |
L. E. Brouwer,
Beweis des ebenen Translationssatzes, Math. Annalen, 72 (1912), 37-54.
doi: 10.1007/BF01456888. |
| [5] |
M. Brown,
A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.
|
| [6] |
M. Brown,
Homeomorphisms of two-dimensional manifolds, Houston Math. J., 11 (1985), 455-469.
|
| [7] |
R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foreman Co. Glenview Illinois, London, 1971. |
| [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. |
| [9] |
C. O. Christenson and W. L. Voxman, Aspects of Topology, BCS Associates, Moscow, Idaho, 1998. |
| [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. |
| [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. |
| [12] |
M. Handel,
A pathological area preserving $ C^{∞}$ diffeomorphism of the plane, Proc. Amer. Math. Soc., 86 (1982), 163-168.
doi: 10.2307/2044419. |
| [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. |
| [14] |
R. D. Nussbaum, The Fixed Point Index and Some Applications, Sminaire de Mathmatiques suprieures, Les Presses de L'Universit de Montral, 1985. |
| [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. |
| [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. |
| [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. |
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