Median Duality in Fibonacci Triangles: A Link to Lyapunov Exponents of Random Recurrences
Abstract
We study two distinguished families of Fibonacci configurations and connect their median geometry with the asymptotic growth of random second-order recurrences. For the degenerate configuration 𝒞d(n)=(Fn−1,Fn−2,Fn−3), the largest side is the sum of the other two, and Apollonius' theorem yields the exact rational median mad=Fn−4/2 for n≥5. For the isosceles chain 𝒞v(n)=(Fn−1,Fn,Fn), the median to the base is mav=1/2√(4Fn2−Fn−12), and its normalized limit is limn→∞mav/Fn=cos(π/10). Using congruence arguments and Cohn's classical result on Fibonacci numbers that are twice a square, we prove that 4Fn2−Fn−12 is never a perfect square for n≥4; hence the isosceles median is irrational for every n≥4. We then examine three classical sign placements in random second-order Fibonacci recurrences, starting from two mutually independent iid Rademacher sequences. We construct exact, time-dependent diagonal orthogonal gauge transformations relating the two-sided-sign matrix cocycle to the two one-sign cocycles. Each transformed coefficient sequence separately retains the iid Rademacher law. Moreover, under the resulting coupling, the Euclidean operator norms of the three finite matrix products coincide exactly at every time. Consequently, the corresponding top logarithmic Furstenberg--Kesten Lyapunov exponents are equal: γA=γB=γC. This equality is established under the stated symmetric-sign assumptions and does not assert invariance for arbitrary biased sign laws. In particular, the common exponent is γ=log(1.13198824…)=0.12397559…, where 1.13198824… is Viswanath's classical almost-sure exponential growth factor for the random Fibonacci sequence. Thus, the paper brings together two complementary aspects: the arithmetic and geometric distinction between degenerate and non-degenerate isosceles Fibonacci configurations, and the exact gauge invariance of the asymptotic growth of classical random Fibonacci matrix cocycles.
Related
- On the Asymptotic Growth of Constrained Random Matrix Products: A Structural Approach to Random Horadam-Type Recurrences and Their Associated Lyapunov Exponents
- Pointwise Hölder continuity of Lyapunov exponents of random matrix products
- Stably ergodic endomorphisms with positive Lyapunov exponents
- Hausdorff spectrum of Lyapunov exponents for higher-dimensional self-affine systems
- Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative
BibTeX
@misc{basse2026median,
title = {{Median Duality in Fibonacci Triangles: A Link to Lyapunov Exponents of Random Recurrences}},
author = {Moïse Basse and El Hadji Dème and Mohamed Ben Fraj Ben Maaouia},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05763646v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Median Duality in Fibonacci Triangles: A Link to Lyapunov Exponents of Random Recurrences
AU - Moïse Basse
AU - El Hadji Dème
AU - Mohamed Ben Fraj Ben Maaouia
PY - 2026
DA - 2026/09/26
PB - HAL
UR - https://hal.science/hal-05763646v2
AB - We study two distinguished families of Fibonacci configurations and connect their median geometry with the asymptotic growth of random second-order recurrences. For the degenerate configuration 𝒞_d(n)=(F_n−1,F_n−2,F_n−3), the largest side is the sum of the other two, and Apollonius' theorem yields the exact rational median m_a^d=F_n−4/2 for n≥5. For the isosceles chain 𝒞_v(n)=(F_n−1,F_n,F_n), the median to the base is m_a^v=1/2√(4F_n^2−F_n−1^2), and its normalized limit is lim_n→∞m_a^v/F_n=cos(π/10). Using congruence arguments and Cohn's classical result on Fibonacci numbers that are twice a square, we prove that 4F_n^2−F_n−1^2 is never a perfect square for n≥4; hence the isosceles median is irrational for every n≥4. We then examine three classical sign placements in random second-order Fibonacci recurrences, starting from two mutually independent iid Rademacher sequences. We construct exact, time-dependent diagonal orthogonal gauge transformations relating the two-sided-sign matrix cocycle to the two one-sign cocycles. Each transformed coefficient sequence separately retains the iid Rademacher law. Moreover, under the resulting coupling, the Euclidean operator norms of the three finite matrix products coincide exactly at every time. Consequently, the corresponding top logarithmic Furstenberg--Kesten Lyapunov exponents are equal: γ_A=γ_B=γ_C. This equality is established under the stated symmetric-sign assumptions and does not assert invariance for arbitrary biased sign laws. In particular, the common exponent is γ=log(1.13198824…)=0.12397559…, where 1.13198824… is Viswanath's classical almost-sure exponential growth factor for the random Fibonacci sequence. Thus, the paper brings together two complementary aspects: the arithmetic and geometric distinction between degenerate and non-degenerate isosceles Fibonacci configurations, and the exact gauge invariance of the asymptotic growth of classical random Fibonacci matrix cocycles.
ER -