## POINCARÉ RECURRENCE: OLD AND NEW

Citations: | 2 - 0 self |

### BibTeX

@MISC{Barreira_poincarérecurrence:,

author = {Luis Barreira},

title = {POINCARÉ RECURRENCE: OLD AND NEW},

year = {}

}

### OpenURL

### Abstract

The classical theorem of Poincaré on recurrence only gives information of qualitative nature. On the other hand it is clearly a matter of intrinsic difficulty and not of lack of interest that less is known concerning the quantitative behavior of recurrence. Here we discuss recent developments that include the almost everywhere

### Citations

69 |
Entropy and data compression schemes
- Ornstein, Weiss
- 1993
(Show Context)
Citation Context ...to the stable and unstable directions, as if they were independent. 5. Relation to entropy Ornstein and Weiss obtained related results in the special case of symbolic dynamics. Namely, they showed in =-=[9]-=- that if σ + : Σ + → Σ + is a one-sided subshift and µ + is ans8 ergodic σ + -invariant probability measure on Σ + , then for µ + -almost every (i1i2 · · · ) ∈ Σ + , log inf{n > 0 : (in+1 · · · in+k) ... |

63 |
Sur le probleme des trois corps et les equations de la dynamique
- Poincare
(Show Context)
Citation Context ...ds as well as the almost product structure of hyperbolic measures. The notion of nontrivial recurrence goes back to Poincaré in his study of the three-body problem. He proved in his celebrated memoir =-=[11]-=- of 1890 that whenever a dynamical system preserves volume almost all trajectories return arbitrarily close to their initial position and that they do this an infinite number of times. More precisely,... |

25 |
Dimension and product structure of hyperbolic measures
- Pesin, Schmeling
- 1999
(Show Context)
Citation Context ...ry point x ∈ X. log τr(x) log µ(B(x, r)) lim = lim r→0 − log r r→0 log r (5)sThe existence for almost every point of the limit in the right-hand side of (5) is due to Barreira, Pesin and Schmeling in =-=[2]-=-. We remark that besides showing that the inequalities in (2) are in fact identities almost everywhere under the hypotheses of Theorem 3.1, the identity (5) also says that the lower and upper recurren... |

25 |
Quantitative recurrence results
- Boshernitzan
- 1993
(Show Context)
Citation Context ...easure m, for which d m (x) = dm(x) = 1 for every x. In particular R(x) < d m (x) for every x in the circle (and thus the first inequality in (2) is strict everywhere). Boshernitzan proved earlier in =-=[7]-=- that if the α-dimensional Hausdorff measure mα is σ-finite on X (that is, if X can be written as a countable union of sets Xi for i = 1, 2, . . . such that mα(Xi) < ∞ for all i), and T preserves a fi... |

22 | On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture
- Pesin, Weiss
- 1996
(Show Context)
Citation Context ...e a stronger statement than that in (3), and the first inequality in (2) may be sharper than that in Theorem 2.2. This possibility indeed occurs, as the following example illustrates. Example 2.2. In =-=[10]-=-, Pesin and Weiss presented an example of a Hölder homeomorphism T : X → X on a closed subset X of [0, 1], preserving a probability measure µ such that: there exist disjoint sets A1, A2 ⊂ [0, 1] with ... |

11 |
The metric entropy of diffeomorphisms Part II: Relations between entropy,exponents and dimension
- Ledrappier, Young
- 1985
(Show Context)
Citation Context ... X, where µ s x and µux measurable partitions ξs and ξu defined by the local stable and unstable manifolds. The existence for almost every point of the limits in (6) is due to Ledrappier and Young in =-=[8]-=-. Barreira, Pesin and Schmeling showed in [2] that measures supported on hyperbolic sets possess an almost product structure (the statement is also valid in the much more general case of hyperbolic me... |

10 |
Hausdoiff dimension of measure via Poincare recurrence
- Barreira, Saussol
- 2001
(Show Context)
Citation Context ...lim sup r→0 log τr(x) − log r . These quantities measure the rate with which the orbit of x returns to an arbitrarily small neighborhood of this point. The following result of Barreira and Saussol in =-=[3]-=- provides upper bounds for the lower and upper recurrence rates in terms of the lower and upper pointwise dimensions of µ at the point x. These are defined respectively by d µ (x) = lim inf r→0 log µ(... |

6 | 2002] “Hyperbolicity and recurrence in dynamical systems: A survey of recent results
- Barreira
(Show Context)
Citation Context ...eorem 2.2. This possibility indeed occurs, as the following example illustrates. Example 2.2. In [10], Pesin and Weiss presented an example of a Hölder homeomorphism T : X → X on a closed subset X of =-=[0, 1]-=-, preserving a probability measure µ such that: there exist disjoint sets A1, A2 ⊂ [0, 1] with positive µ-measure, and there exist positive constants c1 and c2 with c1 �= c2 such that d µ (x) = dµ(x) ... |

6 |
Product structure of Poincaré recurrence, Ergodic Theory Dynam
- Barreira, Saussol
(Show Context)
Citation Context ..., and thus (in view of (4)) for which the left-hand side of (5) attains its maximal possible value almost everywhere. A related result in the case of repellers was obtained by Barreira and Saussol in =-=[4]-=-. Let T : M → M be a differentiable map of a smooth manifold. Recall that a compact T - invariant set X ⊂ M is called a repeller of T if there exist constants c > 0 and β > 1 such that �dxT n v� ≥ cβ ... |

6 | Measures of maximal dimension for hyperbolic diffeomorphisms, preprint
- Barreira, Wolf
(Show Context)
Citation Context ... with a compact hyperbolic set X, if µ is an ergodic equilibrium measure of a Hölder continuous function, then for µ-almost every point x ∈ X. log τr(x) log µ(B(x, r)) lim = lim r→0 − log r r→0 log r =-=(5)-=-sThe existence for almost every point of the limit in the right-hand side of (5) is due to Barreira, Pesin and Schmeling in [2]. We remark that besides showing that the inequalities in (2) are in fact... |

6 |
Pointwise dimension and ergodic decompositions, preprint
- Barreira, Wolf
(Show Context)
Citation Context ... dimension of a measure µ on X. This is defined by dimH µ = inf{dimH Z : µ(Z) = µ(X)}, where dimH Z denotes the Hausdorff dimension of the set Z ⊂ X. It can be shown (see for example Proposition 3 in =-=[6]-=-) that dimH µ = ess sup{d µ (x) : x ∈ X}. (4) Boshernitzan’s results in [7] can be reformulated in the following manner (see [3] for details). Theorem 2.2. If T preserves a finite measure µ on X, then... |

2 | Dimension theory of smooth dynamical systems
- Schmeling
- 2001
(Show Context)
Citation Context ...deas with the study of hyperbolic measures by Barreira, Pesin and Schmeling in [2] and results and ideas of Saussol, Troubetzkoy and Vaienti in [12] and of Schmeling and Troubetzkoy in [14] (see also =-=[13]-=-). In view of work of Barreira and Wolf in [5], in the case of surface diffeomorphisms it always exists an ergodic measure of “maximal recurrence”, i.e., a measure at which the supremum sup µ dimH µ (... |

2 | Scaling properties of hyperbolic measures
- Schmeling, Troubetzkoy
- 1998
(Show Context)
Citation Context ... combines new ideas with the study of hyperbolic measures by Barreira, Pesin and Schmeling in [2] and results and ideas of Saussol, Troubetzkoy and Vaienti in [12] and of Schmeling and Troubetzkoy in =-=[14]-=- (see also [13]). In view of work of Barreira and Wolf in [5], in the case of surface diffeomorphisms it always exists an ergodic measure of “maximal recurrence”, i.e., a measure at which the supremum... |