Geodesic
Local dimensions for Poincaré recurrences
Afraimovich, Valentin1 ; Chazottes, Jean-René1 ; Saussol, Benoît2
1 IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico
2 Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Electronic research announcements of the American Mathematical Society, Tome 06 (2000), pp. 64-74 Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

Pointwise dimensions and spectra for measures associated with Poincaré recurrences are calculated for arbitrary weakly specified subshifts with positive entropy and for the corresponding special flows. It is proved that the Poincaré recurrence for a “typical” cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Precise formulas for dimensions of measures associated with Poincaré recurrences are derived, which are comparable to Young’s formula for the Hausdorff dimension of measures and Abramov’s formula for the entropy of special flows.
DOI : 10.1090/S1079-6762-00-00082-2

Afraimovich, Valentin 1 ; Chazottes, Jean-René 1 ; Saussol, Benoît 2

1 IICO-UASLP, A. Obregon 64, San Luis Potosi SLP, 78210 Mexico
2 Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal 
@article{10_1090_S1079_6762_00_00082_2,
     author = {Afraimovich, Valentin and Chazottes, Jean-Ren\'e and Saussol, Beno{\^\i}t},
     title = {Local dimensions for {Poincar\'e} recurrences},
     journal = {Electronic research announcements of the American Mathematical Society},
     pages = {64--74},
     year = {2000},
     volume = {06},
     doi = {10.1090/S1079-6762-00-00082-2},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-00-00082-2/}
}
TY  - JOUR
AU  - Afraimovich, Valentin
AU  - Chazottes, Jean-René
AU  - Saussol, Benoît
TI  - Local dimensions for Poincaré recurrences
JO  - Electronic research announcements of the American Mathematical Society
PY  - 2000
SP  - 64
EP  - 74
VL  - 06
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-00-00082-2/
DO  - 10.1090/S1079-6762-00-00082-2
ID  - 10_1090_S1079_6762_00_00082_2
ER  - 
%0 Journal Article
%A Afraimovich, Valentin
%A Chazottes, Jean-René
%A Saussol, Benoît
%T Local dimensions for Poincaré recurrences
%J Electronic research announcements of the American Mathematical Society
%D 2000
%P 64-74
%V 06
%U http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-00-00082-2/
%R 10.1090/S1079-6762-00-00082-2
%F 10_1090_S1079_6762_00_00082_2
Afraimovich, Valentin; Chazottes, Jean-René; Saussol, Benoît. Local dimensions for Poincaré recurrences. Electronic research announcements of the American Mathematical Society, Tome 06 (2000), pp. 64-74. doi: 10.1090/S1079-6762-00-00082-2

[1] Afraimovich, V. Pesin’s dimension for Poincaré recurrences Chaos 1997 12 20

[2] Bowen, Rufus, Walters, Peter Expansive one-parameter flows J. Differential Equations 1972 180 193

[3] Katok, Anatole, Hasselblatt, Boris Introduction to the modern theory of dynamical systems 1995

[4] Ornstein, Donald Samuel, Weiss, Benjamin Entropy and data compression schemes IEEE Trans. Inform. Theory 1993 78 83

[5] Penné, Vincent, Saussol, Benoît, Vaienti, Sandro Dimensions for recurrence times: topological and dynamical properties Discrete Contin. Dynam. Systems 1999 783 798

[6] Pesin, Yakov B. Dimension theory in dynamical systems 1997

[7] Young, Lai Sang Dimension, entropy and Lyapunov exponents Ergodic Theory Dynam. Systems 1982 109 124

Cité par Sources :