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Porosity and hypercyclic operators

Author(s): Frédéric Bayart
Journal: Proc. Amer. Math. Soc. 133 (2005), 3309-3316.
MSC (2000): Primary 47A16, 28A05
Posted: May 9, 2005
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Abstract: We study if the set of hypercyclic vectors of a hypercyclic operator is the complement of a $\sigma$-porous set. This leads to interesting results for both points of view: a limitation of the size of hypercyclic vectors, and new examples of first category sets which are not $\sigma$-porous.

References:

[Bi]
G.D. BIRKHOFF, Démonstration d'un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris 189 (1929), 473-475.

[Bou]
P.S. BOURDON, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), 845-847. MR 1148021 (93i:47002)

[Gre]
K-G. GROSSE-ERDMANN, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345-381. MR 1685272 (2000c:47001)

[Hu]
P.R. HURST, A model for invertible composition operators on $H^2$, Proc. Amer. Math. Soc. 124 (1996) 1847-1856.MR 1307532 (96h:47037)

[MMPZ]
M. E. MERA; M. MORAN; D. PREISS; L. ZAJICEK, Porosity, $\sigma$-porosity and measures, Nonlinearity 16 (2003), no. 1, 247-255.MR 1950786 (2003m:28003)

[NRW]
E. NORDGREN, P. ROSENTHAL, F. S. WINTROBE,
Invertible composition operators on $H^{p}$, J. Funct. Anal., 73 (1987), 324 - 344. MR 0899654 (89c:47044)

[Re]
C. READ, The invariant subspace problem for a class of Banach spaces., Israel J. Math. 63 (1988), no. 1, 1-40. MR 0959046 (90b:47013)

[Rol]
S. ROLEWICZ, On orbits of elements, Studia Math. 32 (1969) 17-22. MR 0241956 (39:3292)

[Sal]
H. SALAS,
Hypercyclic weighted shifts, Trans. AMS, 347 (1995), pp 993 - 1004. MR 1249890 (95e:47042)

[Sh]
J. H . SHAPIRO,
Composition Operator and Classical function theory, Springer-Verlag, New-York, 1991. MR 1237406 (94k:47049)

[Za1]
L. ZAJICEK, Porosity and $\sigma$-porosity, Real Anal. Exchange 13 (1987/88), no. 2, 314-350.MR 0943561 (89e:26009)

[Za2]
L. ZAJICEK, Small non-$\sigma$-porous sets in topologically complete metric spaces, Colloq. Math. 77 (1998), no. 2, 293-304.MR 1628994 (2000b:28001)

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Additional Information:

Frédéric Bayart
Affiliation: Laboratoire Bordelais d'Analyse et de Géométrie, UMR 5467, Université Bordeaux 1, 351 Cours de la Libération, F-33405 Talence cedex, France
Email: bayart@math.u-bordeaux.fr

DOI: 10.1090/S0002-9939-05-07842-1
PII: S 0002-9939(05)07842-1
Keywords: Porous sets, hypercyclic operators
Received by editor(s): January 27, 2004
Received by editor(s) in revised form: June 17, 2004
Posted: May 9, 2005
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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