The Lyapunov spectral upper bound via a two-term band split #
This file proves the spectral upper bound of the multiplicative ergodic theorem along Viana's
one-sided route: a two-term split of the cocycle quadratic form against the band projector
Pₙ = bandProjector A T 𝟙_{(c,∞)} n x at threshold c = e^{λᵢ} (the fast singular-block
projector of qpow A T n x).
- The slow band contributes
≤ c^{2n}‖w‖²: the slow Gram eigenvalues are≤ c^{2n}, so the slow quadratic form is bounded byc^{2n}‖w‖²with no angle or tilt estimate. This is the restricted-operator-norm bound‖A⁽ⁿ⁾|_{slow}‖ ≤ cⁿ, realized directly through the continuous functional calculus on the positive-semidefiniteqpowrather than through Kingman's theorem on a skew-product. - The fast band contributes
≤ ‖A⁽ⁿ⁾‖²·‖Pₙ w‖², since every spectral valuetofqpowsatisfiest^{2n} ≤ ‖A⁽ⁿ⁾‖².
Adding the two halves yields the master inequality
‖A⁽ⁿ⁾ v‖² ≤ ‖A⁽ⁿ⁾‖²·‖Pₙ v‖² + c^{2n}·‖v‖², from which the spectral upper bound
limsup (1/n) log‖A⁽ⁿ⁾ v‖ ≤ λᵢ follows, conditional on the single angle input
limsup (1/n) log(‖A⁽ⁿ⁾‖·‖Pₙ v‖) ≤ λᵢ. Via the Furstenberg–Kesten top exponent
(1/n)log‖A⁽ⁿ⁾‖ → λ₁, that angle input is in turn equivalent to the tilt-decay
residual limsup (1/n) log‖Pₙ v‖ ≤ λᵢ − λ₁.
Main results #
Oseledets.inner_slow_band_le: the slow-band quadratic form is≤ c^{2n}‖w‖².Oseledets.rpow_mem_spectrum_qpow_le: everyqpowspectral valuethast^{2n} ≤ ‖A⁽ⁿ⁾‖².Oseledets.inner_fast_band_le: the fast-band quadratic form is≤ ‖A⁽ⁿ⁾‖²·‖Pₙ w‖².Oseledets.norm_sq_cocycle_apply_le_split: the master inequality‖A⁽ⁿ⁾ v‖² ≤ ‖A⁽ⁿ⁾‖²·‖Pₙ v‖² + c^{2n}·‖v‖².Oseledets.limsup_log_cocycle_apply_le_of_angle: the spectral upper boundlimsup (1/n) log‖A⁽ⁿ⁾ v‖ ≤ λᵢ, conditional on the angle input.Oseledets.angle_of_tilt_decay: the angle input follows from the Furstenberg–Kesten top exponent together with the tilt-decay residual.Oseledets.lambdaBar_le_of_tilt_decay: the combination, inlambdaBarform — from the tilt-decay residual,lambdaBar A T x v ≤ λᵢ.
Implementation notes #
A naive per-overlap recursion is circular: slow growth g ≤ λᵢ and small overlaps
oⱼ ≤ g − λⱼ (Cauchy–Schwarz) compose vacuously to g ≤ g. The two-term split sidesteps
the per-overlap recursion entirely: the slow half (inner_slow_band_le) gives λᵢ
unconditionally — it never references an overlap, only the spectral ceiling c of the slow
Gram block. The growth is thereby reduced to a single scalar quantity, the fast-band projection
‖Pₙ v‖, whose decay (htilt) is a genuine slow–fast angle estimate about the projector
sequence, independent of g; there is no feedback loop.
The one non-elementary input is htilt: limsup (1/n) log‖Pₙ v‖ ≤ λᵢ − λ₁ for the
limit-slow vector v (P v = 0, P = limₙ Pₙ), the slow–fast angle decay at the full
multi-gap rate λᵢ − λ₁. It is kept as an explicit hypothesis in this file. The summability
estimate summable_norm_bandProjector_succ_sub (in Oseledets.Lyapunov.OseledetsLimit)
supplies ‖Pₙ₊₁ − Pₙ‖ summability at the nearest-gap rate λᵢ − λ_{i−1}; upgrading it to the
full rate is a multi-gap telescope carried out elsewhere in the library.
References #
- Marcelo Viana, Lectures on Lyapunov exponents, Cambridge Studies in Advanced Mathematics 145, Cambridge University Press, 2014
The slow-band growth bound (the clean half of the split) #
A⁽ⁿ⁾ restricted to the SLOW Gram-eigenspace (singular values ≤ c) has operator norm ≤ c.
Concretely, for the complementary band projector Q_n = I - P_n (projection onto eigenvalues
≤ c of qpow), ‖A⁽ⁿ⁾ Q_n v‖² ≤ c^{2n} ‖Q_n v‖² ≤ c^{2n} ‖v‖². This needs NO
angle/tilt estimate.
Slow-band upper bound. For c ≥ 0, the SLOW quadratic form of the cocycle (the part of
‖A⁽ⁿ⁾ w‖² supported on qpow-eigenvalues ≤ c) is ≤ c^{2n} ‖w‖². Formally: applying the
gram quadratic form to the complementary band vector w with P_n w = 0. We package it via the
function g(t) = t^{2n}·(1 - χ(t)) whose cfc is the slow part of gram.
Spectral bound on qpow. Every t in the spectrum of qpow A T n x satisfies
t ^ (2n) ≤ ‖A⁽ⁿ⁾‖². (t^{2n} is a gram eigenvalue, gram = cocycleᵀ·cocycle, and a
PSD eigenvalue is ≤ ‖gram‖ ≤ ‖cocycleᵀ‖·‖cocycle‖ = ‖cocycle‖².)
The fast-band bound (the angle term) #
Fast-band upper bound. The fast quadratic form of the cocycle (part of ‖A⁽ⁿ⁾ w‖²
supported on qpow-eigenvalues > c) is ≤ ‖A⁽ⁿ⁾‖² · ‖P_n w‖², where
P_n w = bandProjector of the fast band applied to w. The eigenvalues t^{2n} on the fast
band are ≤ ‖A⁽ⁿ⁾‖² (rpow_mem_spectrum_qpow_le), so cfc(t^{2n}χ) ⪯ ‖A⁽ⁿ⁾‖²·cfc(χ) (PSD),
and ⟪cfc(χ)w,w⟫ = ‖P_n w‖² (self-adjoint idempotent).
The master inequality: Viana's two-term split #
‖A⁽ⁿ⁾ v‖² ≤ ‖A⁽ⁿ⁾‖² · ‖P_n v‖² + c^{2n} · ‖v‖². The first term is the FAST (angle) part;
the second is the SLOW part (clean).
Master inequality (Viana's split). For c ≥ 0:
‖A⁽ⁿ⁾ v‖² ≤ ‖A⁽ⁿ⁾‖² · ‖P_n v‖² + c^{2n} · ‖v‖², where P_n is the fast band projector
(bandProjector of (c,∞)).
The assembly: master inequality + angle input ⟹ spectral upper bound #
The slow term contributes λᵢ for free. The fast term ‖A⁽ⁿ⁾‖·‖P_n v‖ is the irreducible
ANGLE/tilt input: its normalized-log limsup must be ≤ λᵢ. Given that single hypothesis,
the spectral upper bound limsup (1/n)log‖A⁽ⁿ⁾v‖ ≤ λᵢ follows.
Spectral upper bound (assembled, conditional on the angle input). For v ≠ 0 and
threshold c = exp λᵢ, if the fast term satisfies the angle bound
limsup (1/n) log (‖A⁽ⁿ⁾‖ · ‖P_n v‖) ≤ λᵢ, then
limsup (1/n) log ‖A⁽ⁿ⁾ v‖ ≤ λᵢ.
This is the two-term split: the slow band gives λᵢ unconditionally
(norm_sq_cocycle_apply_le_split, the c^{2n}‖v‖² term); the fast band is controlled by the
angle hypothesis hangle. The only genuinely non-elementary input is hangle (the slow–fast
tempering/angle decay of ‖P_n v‖).
The angle hypothesis from a tilt-decay rate on ‖P_n v‖ #
The angle input hangle of limsup_log_cocycle_apply_le_of_angle is the slow–fast tempering
decay limsup (1/n) log (‖A⁽ⁿ⁾‖ · ‖P_n v‖) ≤ λᵢ. With the Furstenberg–Kesten top
exponent (1/n) log ‖A⁽ⁿ⁾‖ → λ₁, this is equivalent to the tilt-decay rate
limsup (1/n) log ‖P_n v‖ ≤ λᵢ − λ₁. This section packages that equivalence: the
genuine non-elementary content is exactly the tilt-decay htilt.
Angle from tilt-decay. If the Furstenberg–Kesten top exponent is λ₁ (htop) and the
fast-band projection ‖P_n v‖ decays at the tempering rate
limsup (1/n) log ‖P_n v‖ ≤ λᵢ − λ₁ (htilt), then the angle hypothesis of
limsup_log_cocycle_apply_le_of_angle holds.
The spectral upper bound from the tilt-decay residual #
Assembling the two-term split (limsup_log_cocycle_apply_le_of_angle) with the
Furstenberg–Kesten reduction (angle_of_tilt_decay) yields the spectral upper bound in
lambdaBar form: for a Λ-slow vector v (its top fast-band projection decays at the
tempering rate), lambdaBar A T x v ≤ λᵢ.
The single genuinely non-elementary hypothesis is htilt: the slow–fast tempering/angle decay
limsup (1/n) log ‖P_n v‖ ≤ λᵢ − λ₁. Everything else (the slow-band c^{2n} bound,
the Furstenberg–Kesten top exponent, the limsup arithmetic) is supplied here
unconditionally.
The spectral upper bound, lambdaBar form. For a nonzero v, the Furstenberg–Kesten
top exponent λ₁ (htop), and the tilt-decay residual htilt
(limsup (1/n) log ‖P_n v‖ ≤ λᵢ − λ₁, the slow–fast angle estimate), together with the
routine boundedness/positivity side conditions, the growth exponent obeys the spectral upper
bound:
lambdaBar A T x v ≤ λᵢ.
Equivalently limsup (1/n) log ‖A⁽ⁿ⁾ v‖ ≤ λᵢ. The slow vector hypothesis P v = 0
(v orthogonal to the limit fast singular subspace) is what makes htilt hold; reducing
htilt to P v = 0 is the residual multi-gap tilt-decay (see the module docstring).