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Let $r$ be a real number in $(0,4)$, let $x_0$ be any real number in $(0,1)$, and define a sequence: $$ x_{n+1} = rx_n(1-x_n). $$ This is the logistic map. For some choices for the value of $r$, the resulting sequence will converge to periodic behavior for almost all choices of $x_0 \in (0,1)$. For other choices of $r$, the sequence will not converge to periodic behavior, i.e. the sequence will be chaotic.

Let $S$ be the set of values of $r \in (0,4)$ that result in chaotic behavior. My question is: does the set $S$ have positive measure?

I was looking at pictures of the resulting bifurcation diagrams for the logistic map on the wikipedia article:

https://en.wikipedia.org/wiki/Logistic_map

It looks to me like, in any region where the map looks chaotic, if you zoom in far enough, you can find small intervals where the map again exhibits periodic behavior. This leads me to believe that the set S is a Cantor set. But I can't tell if it's a measure 0 Cantor set or a positive-measure ("fat") Cantor set. It looks like a fat Cantor set but that's just my intuition. Has any result on this been proven? Or am I wrong, and S isn't a Cantor set at all? Maybe S actually provably contains some entire intervals?

Lorioch
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I found this page:

https://sites.google.com/site/fabstellar84/fractals/real_chaos

which claims that the set of values of $r$ such that the resulting sequence is chaotic does indeed have positive measure. I tracked down the original proof cited in that page:

https://projecteuclid.org/euclid.cmp/1103920159

but the proof is way over my head, and even the statement of the theorem is beyond my level of understanding. I'm assuming that when the paper talks about "parameter values $λ$ for which $f$$λ$ has an invariant measure absolutely continuous with respect to Lebesgue measure", that means values $λ$ such that the resulting sequence does not converge to periodic behavior.

Lorioch
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