Large coupling asymptotics for the entropy of quasi-periodic operators
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https://doi.org/10.1007/s11425-019-1662-8
Large coupling asymptotics for the entropy of quasi-periodic operators Lingrui Ge1 & Jiangong You2,∗ 1Department 2Chern
of Mathematics, University of California at Irvine, Irvine, CA 92612, USA; Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China Email: [email protected], [email protected] Received September 26, 2019; accepted January 21, 2020
Abstract
In this paper, we give an asymptotic estimate for the entropy, i.e., the sum of all positive Lya-
punov exponents, of the quasi-periodic finite-range operator with a large trigonometric polynomial potential and Diophantine frequency. Keywords MSC(2010)
large coupling asymptotics, entropy, quasi-periodic operator, quantitative almost reducibility 81Q10, 47B39
Citation: Ge L R, You J G. Large coupling asymptotics for the entropy of quasi-periodic operators. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1662-8
1
Introduction
For given λ ∈ R+ , x ∈ T, θ, α ∈ Tν and the real trigonometric polynomials W (x) =
d ∑
Wk e2πikx
and V (θ) =
∑
Vk e2πi⟨k,θ⟩ ,
|k|6s
k=−d
we consider the one-dimensional quasi-periodic finite-range operator on ℓ2 (Z) defined as (LW,λV,α,θ u)n =
d ∑
Wk un−k + λV (θ + nα)un ,
n ∈ Z.
(1.1)
k=−d
Usually, θ is called the initial phase, α is called the frequency, λ is called the coupling constant and V (θ) is called the potential. The eigenvalue equations LW,λV,α,θ u = Eu * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
Ge L R et al.
2
Sci China Math
λV of (1.1) are equivalent to the 2d-dimensional cocycles (α, SE,W ) ∈ C ω (Tν , GL(2d, C))1) , where
λV SE,W (θ)
1 = Wd
−Wd−1 · · · −W1 E − W0 − λV (θ) −W−1 · · · −W−d+1 −W−d
Wd
..
.
.
Wd Let λV (SE,W )n (θ)
Let
=
{ λV λV λV SE,W (θ + (n − 1)α) · · · SE,W (θ + α)SE,W (θ), λV )−n (θ ((SE,W
+ nα))
1 n→∞ n
λV Ld (α, SE,W ) = lim
−1
n > 0, n 6 −1.
,
∫ Tν
λV ln ∥Λd (SE,W )n (θ)∥dθ,
λV where Λd denotes the d-th exterior product. Ld (α, SE,W ) is in fact the sum of all non-negative Lyapunov λV 2) λV λV exponents of (α, SE,W ). We call Ld (α, SE,W ) the entropy of the operator (1.1). In the case d = 1, SE,W λV λV is actually a 2 × 2 matrix and L1 (α, SE,W ) is in fact the non-negative Lyapunov exponent of (α, SE,W ). λV In this paper, we are interested in the asymptotic estimate of Ld (α, SE,W ) when the coupling constant λ tends to the infinity. λV For fixed E, α, W, V , capturing the behavior of Ld (α, SE,W ) for large coupling constant λ has been richly studied. Notice that (1.1) reduces to the famous almost Mathieu operator if ν = 1 and V, W are 2λ cos equal to 2 cos 2π(·). Its eigenvalue equation is equivalent to (α, SE,2 cos ). The pioneer work was due to Herman [18] who proved that 2λ cos L1 (α, SE,2 cos ) > ln λ
for λ > 13) . Later, Sorets and Spencer [23] proved that λV L1 (α, SE,2 cos ) >
1 ln λ 2
for any real analytic potential V , i.e
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