|
Poincaré recurrence for observations
Author(s):
Jérôme
Rousseau;
Benoît
Saussol
Journal:
Trans. Amer. Math. Soc.
362
(2010),
5845-5859.
MSC (2010):
Primary 37C45, 37B20;
Secondary 37A25, 37DXX, 37M25
Posted:
June 10, 2010
MathSciNet review:
2661498
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system we study Poincaré recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relative to the push-forward are equal.
References:
-
- 1.
- Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. MR 1793194 (2001k:37035)
- 2.
- L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys. 219 (2001), no. 2, 443-463. MR 1833809 (2002c:37035)
- 3.
- Sean M. Bates and Carlos G. Moreira, De nouvelles perspectives sur le théorème de Morse-Sard, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 1, 13-17. MR 1805620 (2002b:58010)
- 4.
- Michael D. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617-631. MR 1231839 (94k:28028)
- 5.
- N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete Contin. Dynam. Systems 5 (1999), no. 2, 425-448. MR 1665752 (99k:58134)
- 6.
- M. Maurice Dodson and Simon Kristensen, Hausdorff dimension and Diophantine approximation, Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 305-347. MR 2112110 (2005m:11139)
- 7.
- Dmitry Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157-205. MR 1919377 (2003m:37037)
- 8.
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
- 9.
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons Ltd., Chichester, 1997. MR 1449135 (99f:28013)
- 10.
- Makoto Ohtsuka, Area formula, Bull. Inst. Math. Acad. Sinica 6 (1978), no. 2, part 2, 599-636. MR 528671 (80a:28003)
- 11.
- William Ott and James A. Yorke, Learning about reality from observation, SIAM J. Appl. Dyn. Syst. 2 (2003), no. 3, 297-322 (electronic). MR 2031277 (2004m:37041)
- 12.
- Benoît Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 259-267. MR 2191396 (2006k:37005)
- 13.
- Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585-650. MR 1637655 (99h:58140)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2010):
37C45, 37B20,
37A25, 37DXX, 37M25
Retrieve articles in all Journals with
MSC (2010):
37C45, 37B20,
37A25, 37DXX, 37M25
Additional Information:
Jérôme
Rousseau
Affiliation:
Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques UMR CNRS 6205, 6 avenue Victor le Gorgeu, CS93837, F-29238 Brest Cedex 3 France
Email:
jerome.rousseau@univ-brest.fr
Benoît
Saussol
Affiliation:
Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques UMR CNRS 6205, 6 avenue Victor le Gorgeu, CS93837, F-29238 Brest Cedex 3 France
Email:
benoit.saussol@univ-brest.fr
DOI:
10.1090/S0002-9947-2010-05078-0
PII:
S 0002-9947(2010)05078-0
Keywords:
Poincaré recurrence,
dimension theory,
Received by editor(s):
July 7, 2008
Posted:
June 10, 2010
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
