Instability of powers of the golden mean

Abstract

In this paper we determine the Lyapunov exponents (LEs) for some Lebesgue measure zero periodic orbits from the Gauss map. This map generates the integers of a simple continued fractions representation (CFR). Only periodic orbits related to powers of the golden mean ϕ = ( 5 - 1 ) / 2 are considered. It is shown that the LE from the CFR of any power (1/ ϕ i ) ( i = ±1, ±2, …) can be written as a multiple of λ ϕ , which is the LE related to the golden mean. When i is odd, the LEs are given by λ G ( x i ) = iλ ϕ , and when i is even the LEs are λ G ( x i ) = iλ ϕ /2. In general, the LE from the CFR of (1/ ϕ i ) increases as i increases. Additionally, the LE is determined when (1/ ϕ i ) is multiplied by an integer. We also present some examples of the instability of the CFRs related to quark’s mass ratio.