Some characterizations of domination

By Jairo Bochi, Nicolas Gourmelon. 2008

116 citations recorded by the source corpus.

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets Σ in $${{\rm GL}(d, \mathbb R)}$$ with the property that any cocycle with values in Σ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a two-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/50bf51751793b3b36224a0d50c58b4de82abf8b2

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Anosov representations and dominated splittings (2016, methodology): “owing theorem asserts that the existence of a dominated splitting can be detected in terms of singular values, and also describes the invariant subbundles in these terms: Theorem 2.2 (Bochi{Gourmelon [BG]). Let T Z or R. Let t tutPT be a linear ow on a vector bundle E, bering over a continuous ow t˚”
Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles (2008, methodology): “2Added in proof: After this paper was completed, Bochi and Gourmelon [3] obtained generalizations of Proposition 2.”
Ledrappier-Young formula and exact dimensionality of self-affine measures (2015, background): “stinct Lyapunov exponents and the case of matrices satisfying the totally dominated splitting condition with quasi self-aﬃne measures. In the latter case, because of the result of Bochi and Gourmelon [7], we can deﬁne a H¨older continuous function from the symbolic space to the space of sequences of”
Conformality for a robust class of non-conformal attractors (2019, methodology): “The goal of this section is to extend the results of Bochi-Gurmelon [4] and Bochi-PotrieS.”
ERGODIC OPTIMIZATION OF BIRKHOFF AVERAGES AND LYAPUNOV EXPONENTS (2017, background): “As shown in Bochi and Gourmelon [2009], i 2 f1; : : : ; d 1g is a index of domination if and only if there is an exponential gap between i -th and (i+1)-th singular values; more precisely: there are constants 1 > 1 andn1 1 such that si (F (x))/si+1(F (x))”
On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems (2016, background): “Several of them also either assume positivity of the matrix entries or assume domination, a generalised notion of positivity which for invertible 2 × 2 matrices is equivalent to the hypothesis that the matrices strictly preserve a finite collection of cones (for more information see [6]).”