Any relationship between Viswanath's constant and the Khinchine-Lévy constant?

Question

It is well-known that if ${\{{F_n}\}}$ is a random Fibonacci sequence then we have almost certainly $\lim \limits_{n\to\infty}\sqrt[n]{|F_n|}=\tau$ where $\tau\approx 1.554682275$ is Viswanath's constant. There is also a simplified version of Viswanath's proof with some more motivation.

Now, the Khinchine-Lévy constant $\gamma=e^{\pi^2/(12\ln2)}$ (sometimes also called Lévy constant) is obtained in a quite similar way as $\lim \limits_{n\to\infty}\sqrt[n]{q_n}$ for almost all reals $x$, where $\frac{p_n}{q_n}$ is the $n$th convergent of the continued fraction of $x$. (Note that we might as well take $\lim \limits_{n\to\infty}\sqrt[n]{p_n}$.)
So the question:

Is there a relationship between Viswanath's constant and the Khinchine-Lévy constant?

Answer 1

To add some context to Anthony Quas's answer:

The two problems are two special cases of a general question. Suppose you have a map $T: X \to X$ which preserves a measure $\mu$. For simplicity assume that $\mu$ is ergodic, so any invariant set has measure $0$ or $1$. A map $\alpha: \mathbf{Z} \times X \to GL(d,\mathbf{R})$ is called a (dynamical) cocycle if $$\alpha(n+m,x) = \alpha(n,T^m x) \alpha(m,x).$$

The Osceledec multiplicative ergodic theorem says that (in particular) there exists a number $\lambda_1$ called the top Lyapunov exponent for $\alpha$ such that for $\mu$-almost all $x \in X$, $$ \lim_{n \to \infty} \frac{1}{n} \log \|\alpha(n,x)\| = \lambda_1.$$

(The theorem also guarantees the existence of other Lyapunov exponents $\lambda_2, \dots, \lambda_d$. One quick way to define them is to say that $\lambda_1 + \dots + \lambda_k$ measures the exponent of the growth of the norm of the cocycle acting on the $k$'th exterior power of $\mathbf{R}^d$.)

Now both Visvanath's number and the Khinchine-Levy constant are the top Lyapunov exponent for certain cocycles. In Visvanath's case, the space $X$ is the infinite product of $GL(2,\mathbf{R})$'s and $T$ is the shift map. In the Khinchine-Levy case, $T$ is the Gauss map of the interval to itself, and $\alpha$ is closely related to the derivative cocycle, (so $\alpha(n,x)$ is given in terms of the derivative of $T^n$).

In general Lyapunov exponents are virtually impossible to compute, except in the situation where you are dealing with the derivative cocycle of a flow on a homogeneous space. (One exception is derivative cocycles for flows on the moduli space of curves where at least certain combinations of Lyapunov exponents have interpretations in terms of algebraic geometry). But the Gauss map is essentially equivalent to a flow on a homogeneous space, since it is (very closely related to) the coding of the geodesic flow on the modular surface. Thus, it is not surprising that the Khinchine-Levy constant can be evaluated explicitly.

Visvanath's calculation is different: he guesses the stationary measure and then uses the Furstenberg formula for the (top) Lyapunov exponent in terms of the stationary measure. It seems very hard to guess the stationary measure in most cases.

Answer 2

I think there's a substantial difference between the two. First the similarities (probably this is what you had in mind):

The terms of the random Fibonacci sequence $(x_n)$ are obtained from the recurrence

$$ x_{n+1} = a_{n+1}x_n+x_{n-1} $$ $$ x_n = 1 x_n + 0 x_{n-1}$$

where the $(a_n)$ are independently $\pm 1$. Notice that this gives $(x_{n+1},x_n)$ as a matrix multiple of $(x_n,x_{n-1})$, that is the matrix $A_{n+1}$ is (in Mathematica notation - does anyone know how to get MO to display matrices?) $\lbrace\lbrace a_{n+1},1\rbrace,\lbrace 1,0\rbrace\rbrace$

The denominators of the continued fraction are obtained from the same recurrence, but where the $(a_n)$ come from a complicated probability measure taking values on sequences of positive integers.

The respective growth rates come from computing the typical growth rate of the product of matrices $A_n A_{n-1}\ldots A_2$. Computing growth rates of matrix products is known to be hard.

One method (the one I know) for calculating the Khinchine-Levy constant comes from the fact that the growth rate is also given as the derivative at a typical point of the $n$-fold composition of the Gauss Map. This can be estimated using ergodic theory as the invariant density for this map is also known.

In other words, while the Khinchine-Lévy constant can be posed in terms of complicated products of 2$\times$2 matrices, as can the Viswanath constant, the former can be evaluated using other techniques.