Adapted metrics for dominated splittings

By Nicolas Gourmelon. Ergodic Theory and Dynamical Systems, 2007

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ERGODIC OPTIMIZATION OF BIRKHOFF AVERAGES AND LYAPUNOV EXPONENTS (2017, background): “1, we reobtain a result of Gourmelon [31], which says that it is always possible to find an adapted Riemannian norm for which dominations are seen in the first iterate (i.”
Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms (2010, background): “To simplify the presentation, one can change theRiemannianmetric (see [24]) and find κ ∈ (0, 1) such that for each point x ∈ H(p f0), and each unitary vectors u ∈ Ecs x and v ∈ Ecu x , one has ‖D f0.”
Entropy-expansiveness for partially hyperbolic diffeomorphisms (2010, background): “ equivalent to the original one and λ ∈ (0,1) such that (3) Yn j=0 kDf|Ecs,i(fj(x))k 0 ·kDf −1|Ecu,i+1(fj(x))k 0 < λ n for all n ≥ 0, all x ∈ Λ, and all i ∈ {0,...,k}. Proof. Fix i ∈ {0,...,k}. By [G] there exist an adapted Riemannian metric equivalent to the original one and λi ∈ (0,1) such that”
Some characterizations of domination (2008, background): “It is also said that E dominates F. It is always possible to find an adapted metric, that is, an inner product such that C = 1, see [ 10 ].1 Some properties about dominated”