Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective

By C. Bonatti. 2004

484 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/6791ad4c39d1caaf7de9adca512587454cf30a17

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Citations in the corpus

Some characterizations of domination (2008, background): “Let V be a continuous vector bundle over a compact Hausdorff space X , and of finite dimension d. Denote by Vx the fibre over x ∈ X .”
Stable ergodicity for partially hyperbolic attractors with negative central exponents (2007, background): “Therefore, the class of all limit measures for sequences of type (2) coincides with the class of all u-measures, while the class of limit measures for sequences of type (1) may be smaller.”
New criteria for ergodicity and nonuniform hyperbolicity (2009, background): “It is interesting to note that blenders were originally introduced by Bonatti and Díaz in [14] to obtain robust transitivity.”
Partial hyperbolicity far from homoclinic bifurcations (2008, background): “iformly contracted by f and the bundle Eu is uniformly contracted by f−1. The partially hyperbolic structure extends to any invariant compact set contained in a small neighbourhood of K. (We refer to [BDV] for a survey on the properties satisﬁed by partially hyperbolic sets and dominated splittings.”
Classification of partially hyperbolic diffeomorphisms in 3‐manifolds with solvable fundamental group (2013, background): “Partially hyperbolic diffeomorphisms play a central role in both the study of stable ergodicity and robust transitivity (see [2, 32]).”
Exponential estimates for plurisubharmonic functions and stochastic dynamics (2008, background): “ i = lim n→∞ 1 √ n h ψ′(x)−ψ′(gn(x)) i = 0 in distribution. So, ψdoes not satisﬁes the CLT (sometimes, one says that it satisﬁes the CLT for σ= 0). The CLT can be deduced from some strong mixing, see [3, 19, 24, 34]. In the following result, E(ψ|Fn) denotes the expectation of ψwith respect to Fn, th”
Anosov representations and dominated splittings (2016, background): “otion of hyperbolicity, as introduced by Anosov and Smale [Sma] plays a central role. Early on, it was noted that weaker forms of hyperbolicity (partial, nonuniform, etc.) should also be studied: see [BDV] for a detailed account. One of these is the the notion of dominated splittings, which can be t”
Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems (2011, methodology): “A follow-up paper [10] of JB and TF will analyze a set of robustly transitive diffeomorphisms on T4, based on examples of Bonatti and Viana [5], which have the weakest possible form of hyperbolicity for robustly transitive diffeomorphisms: a dominated splitting [3, 12] (see §2).”
Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence (2009, background): “We point out that the robustness of this example provides an open set of flows which are not Morse–Smale, nor hyperbolic, and which are also non-structurally stable (see [10, 32]).”
On the regularization of conservative maps (2008, unknown): “But ∑ p∈P volF (s(W )) ∩ B(p, 1) = volF (s(W ))∩B = volW ∩B = ∑ p∈P volW∩B(p, 1), so we must have volF (s(W ))∩ B(p, 1) = volW ∩B(p, 1) also for p = p2n−k , and property (2) follows.”