Result: zbMATH Open Web Interface contents unavailable due to conflicting licenses.

It is well known that the asymptotic growth rate of the Fibonacci numbers is \(\ln((1+\sqrt{5})/2)\approx0.4812\). In general, the growth rates of the random Fibonacci type sequences defined by \[ f_k=a_kf_{k-1}+b_kf_{k-2}, \] where \(a_k\) and \(b_k\) are random coefficients, are much more difficult to estimate. This paper looks at a special case where the coefficients are \(2^{n_k}\), and \(\{n_1, n_2,\dots\}\) is an infinite sequence of natural numbers, by using ergodic theory. P. Lévy in 1929 studied a similar case in which the Fibonacci coefficients \(a_k\) are given by a sequence of integers which give the regular continued fraction representation of a real number \(x\in[0,1)\). Lévy showed that, for almost all \(x\), the rate of growth is \(\frac{\pi^2}{12\ln2}\approx1.186569110\). The main result of this paper relies on a property of the generalized Gauss map, which states that, for any integrable function \(f\) on the unit interval, the following holds for almost all \(x\), \[ \lim_{n\to\infty}\frac{1}{n}\sum_k^{n-1}f(T^kx)=\int_0^1\rho(x)f(x)\,dx, \] where \(T\) is a generalized Gauss map, \(\rho(x)=\frac{\rho_0}{(x+1)(x+2)}\) is the probability density of an invariant measure, and \(\rho_0^{-1}=\ln(4/3)\). With this property, this paper shows the rate of growth of the defined Fibonacci numbers is \[ -\rho_0\int_0^1\frac{\ln x}{(x+1)(x+2)}\,dx, \] which is approximately 1.30022988, via numerical integration.