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Many introductions to chaos start with logistic map $$ x_{n+1}=\lambda x_n(1-x_n) $$ and claim it is chaotic at some values of $\lambda$. Unfortunately, all proofs of chaos I saw were numerical and not rigorous. How does one prove that such a map is chaotic at a particular point in general (I suspect it may be well known in the field, but I couldn't find an anwer easily)?

Pavlo. B.
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  • Which definition of "chaotic" are you using? Is an orbit dense in an interval chaotic enough? Or the orbit from almost all starting points dense? – Yuval Peres Mar 29 '21 at 02:18
  • yes, orbit dense in an interval would be good. Or positive Lyapunov exponent. Or strangeness of the attractor. I am interested in the question, how one proves chaos mathematically. – Pavlo. B. Mar 29 '21 at 02:31
  • It can be proved for $\lambda=4$. For other values of $\lambda$, I'm not sure anyone has a rigorous proof, there may only be compelling numerical evidence. – Gerry Myerson Mar 29 '21 at 02:34
  • Read the book of Haselblatt and A.Katok. – Rick Sanchez C-666 Mar 29 '21 at 03:54
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    Renormalization and the computer assisted proof of Oscar Lanford are rigorous. Here is one beautiful paper – Mittens Mar 29 '21 at 15:02
  • @OliverDiaz, I don't see the connection between the Lanford paper, and the chaotic nature of the logistic map for any values of $\lambda$ other than $4$ (and indeed there are values for which it's known not to be chaotic). Could you expand on your comment to bring out the connection? – Gerry Myerson Mar 31 '21 at 08:05
  • @GerryMyerson, The paper that Oliver linked refers to the quadratic maps in Theorem 5. Lanford uses the parametrization $z \mapsto 1-\mu z^2$ for the quadratic family, This is equivalent via a linear change of variable to the parametrization used by the OP. – Yuval Peres Mar 31 '21 at 23:33
  • @Yuval, thanks. If I understand what's there, it says there's an interval of values of $\mu$ for which the map is chaotic – is that right? But it doesn't specify even one value of $\mu$ for which the map is chaotic, does it? – Gerry Myerson Apr 01 '21 at 12:11
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    Yes, that is my understanding too. Another direction is the work of [1] where another notion of Chaos is proved for a.positive measure of parameters. [1] Jakobson, M.V., 1981. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Communications in Mathematical Physics, 81(1), pp.39-88. – Yuval Peres Apr 01 '21 at 16:07
  • OK, so, it seems that the answer to the question, "How does one prove that such a map is chaotic at a particular point," is, nobody knows. – Gerry Myerson Apr 02 '21 at 12:08

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Here is the proof of typical dense orbits for $\lambda=4$:

Start with the doubling map $\quad \theta_{n+1}=2\theta_n \mod 1$ on $[0,1) \,$. By considering the binary expansion of $\theta_0$, the law of large numbers implies that almost all orbits of the doubling map are dense. Write $x_n=\sin^2(\pi \theta_n)$ and observe that it satisfies the given recursion with $\lambda=4$. It follows that this map has typically dense orbits in $[0,1]$ as well.

Yuval Peres
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