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For a class I was teaching, I was demonstrating time-series analysis with the famous discrete-time dynamical system $x_{n+1} = ax_n(1-x_n)$, where $a = 4 - \epsilon$ ($\epsilon$ ``small'') and different initial values of $x_0$.

When looking at the tenths digits of the numbers in the orbit $x_0, x_1, x_2, \ldots$, there was a strong bias towards "0" and "9" over the other digits. This was consistent across a wide range of $x_0$ and $\epsilon$, even though the orbits otherwise appeared to be pure noise,

Is this some numerical/computational phenomenon, or is it actually mathematically valid? In other words, are there results in the literature about the distribution of certain digits in chaotic orbits?

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    $\begingroup$ This dynamical system preserves an ergodic invariant measure which is absolutely continuous with respect to Lebesgue (for "most" parameter values close to 4, anyway) and by the Birkhoff ergodic theorem the proportion of zeroes in the first digit (i.e. the average proportion of time spent in the interval $[0,1/10]$) along a Lebesgue-typical orbit is given by the measure of the interval $[0,1/10]$ with respect to the absolutely continuous invariant measure. So the phenomenon you are reporting is that the absolutely continuous invariant measure gives more mass to $[0,1/10]$ and $[9/10,1]$ than... $\endgroup$
    – Ian Morris
    Commented Jan 6, 2017 at 13:28
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    $\begingroup$ ...would be naively expected if we assumed that the absolutely continuous invariant measure was precisely Lebesgue measure. There are probably some results on the structure of these measures, but I don't know them. I suggest that the "ergodic theory" tag would be useful here. For $a=4$ there should be an exact formula for the density of the invariant measure given by an inverse trigonometric function. $\endgroup$
    – Ian Morris
    Commented Jan 6, 2017 at 13:30
  • $\begingroup$ Thanks for the starting point! It appears that this is well-known enough to be an exercise at this point. (See link and link $\endgroup$
    – Andrew Penland
    Commented Jan 6, 2017 at 14:20
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    $\begingroup$ The case $a=4$ is a well-known exactly solved case. The density is $1/\pi(x(1-x))^{-1/2}$ (this density is obviously higher near 0 and 1 than for other values of $x$). As @IanMorris mentioned, it is proved that for most parameter values close to 4, there is an absolutely continuous invariant measure. This is a deep fact (due to Jakobson). I don't know if it's known whether the density for $a=4-\epsilon$ is typically close to the density for $a=4$, but this would certainly be very far beyond an exercise. $\endgroup$
    – Anthony Quas
    Commented Jan 6, 2017 at 17:58
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    $\begingroup$ @LasseRempe-Gillen The key word is "statistical stability" which means that for a one parameter family $T_a,a\in (-\epsilon,\epsilon)$ the physical measures vary continuously as $a\to0$. Stochastic stability refers to random perturbations. For the specific example which is the logistic family statistical stability does not hold by people.kth.se/~thunberg/full-pdf/natfaksimil.pdf On the other hand if one restricts to subclasses of Collet-Eckman parameters which are the ones Anthony is referring to, then the densities of the measures converge in $L^1$ by a result of Freitas and Todd. $\endgroup$
    – user78465
    Commented Jan 8, 2017 at 21:24

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