Lyapunov Exponents of Linear Cocycles: Continuity via Large by Pedro Duarte, Silvius Klein

By Pedro Duarte, Silvius Klein

The goal of this monograph is to give a common approach to proving continuity of Lyapunov exponents of linear cocycles. the tactic makes use of an inductive method according to a common, geometric model of the Avalanche precept. the most assumption required by means of this technique is the supply of applicable huge deviation sort estimates for amounts with regards to the iterates of the bottom and fiber dynamics linked to the linear cocycle. We identify such estimates for varied versions of random and quasi-periodic cocycles. Our procedure has its origins in a paper of M. Goldstein and W. Schlag. Our current paintings expands upon their strategy in either intensity and breadth. We finish this monograph with a listing of similar open difficulties, a few of that may be handled utilizing the same approach.

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These results refer to Schrödinger cocycles, as defined in Sect. 2, and continuity is understood relative to the energy parameter and/or the frequency. In [10], the authors prove joint continuity in the energy E and the frequency ω, at all points (E, ω) with ω irrational. In [14], for the one frequency case, assuming a strong Diophantine condition on the frequency, the authors prove a sharp fiber LDT estimate and establish the AP for SL(2, R) matrices. Based on these ingredients, they develop an inductive procedure that leads to Hölder continuity of the (top) Lyapunov exponent as a function of the energy E, under the assumption of a positive lower bound on the Lyapunov exponent.

The kth exterior power of V , denoted by ∧k V , is the linear span of all simple k vectors of V . Elements of ∧k V are called k-vectors. An easy consequence of this formal definition is that v1 ∧ · · · ∧ vk = 0 if and only if v1 , . . , vk are linearly dependent. Another simple consequence is that given two bases {v1 , . . , vk } and {w1 , . . , wk } of the same k-dimensional linear subspace of V , if for some real matrix A = (aij ) we have wi = kj=1 aij vj for all i = 1, . . , k, then w1 ∧ · · · ∧ wk = (det A) v1 ∧ · · · ∧ vk .

V(g∗ ), we have gw = α g v∗ + gu with gu ⊥ v∗ and Choosing √a unit vector v √ 2 gu ≤ √1 − α s2 (g) = 1 − α 2 σ g . We define the number 0 ≤ κ ≤ σ so that gu = 1 − α 2 κ g . Hence α2 g 2 ≤ α2 g 2 + gu + gu 2 = g 2 = gw 2 , and also gw 2 = α2 g 2 = g α2 ⊕ κ 2 ≤ g 2 2 2 α 2 + (1 − α 2 )κ 2 α2 ⊕ σ 2 , which proves (a). Item (b) follows from g v ∧ gu v∗ ∧ gu g v ∧ gw = = gv gw g gw gw √ 2 σ 1−α g σ δ(w, ˆ v(g)) gu ≤ = . = gw α g α δ ϕg (w), ˆ v(g∗ ) = Next proposition relates the expansion rift ρ(g, g ) with the angle α(g, g ) and the upper angle β(g, g ).

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