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, 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.

Figure 2.  Periodic crooked chain of period 5.
[1]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[2]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979

[3]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015

[4]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391

[5]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911

[6]

Deissy M. S. Castelblanco. Restrictions on rotation sets for commuting torus homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5257-5266. doi: 10.3934/dcds.2016030

[7]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451

[8]

Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797

[9]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[10]

Jiamin Zhu, Emmanuel Trélat, Max Cerf. Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1347-1388. doi: 10.3934/dcdsb.2016.21.1347

[11]

Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403

[12]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

[13]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[14]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381

[15]

E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207

[16]

V. Carmona, E. Freire, E. Ponce, F. Torres. The continuous matching of two stable linear systems can be unstable. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 689-703. doi: 10.3934/dcds.2006.16.689

[17]

Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555

[18]

Thierry Gallay, Guido Schneider, Hannes Uecker. Stable transport of information near essentially unstable localized structures. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 349-390. doi: 10.3934/dcdsb.2004.4.349

[19]

Kanat Abdukhalikov, Sihem Mesnager. Explicit constructions of bent functions from pseudo-planar functions. Advances in Mathematics of Communications, 2017, 11 (2) : 293-299. doi: 10.3934/amc.2017021

[20]

Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (12)
  • HTML views (410)
  • Cited by (0)

Other articles
by authors

[Back to Top]