Ergodic measures for the logistic map

Question

$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.

For $c\in (0,4]$ consider the logistic map $$ T_c:[0,1]\to[0,1],\;\;T_c(x)=cx(1-x). $$ Denote by $\Inv_c$ the collection of Borel probability measures on $[0,1]$ that are $T_c$-invariant and by $\Erg_c\subset \Inv_c$ the subset of consisting of ergodic ones.

Question 1. There are simple examples such ergodic measure measure, e.g., measures concentrated on a periodic orbit. Do there exist ergodic measures not concentrated on a periodic orbit?

Question 2. This is a bit more vague. Is it known how the set $\Erg_c$ evolves with changing $c $ inside the set Borel probability measures on $[0,1]$?

Answer 1

When $c=4$, the map $T_4(x)=4x(1-x)$ on the unit interval is semi-conjugate to the transformation $z\mapsto z^2$ of the unit circle via $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$: $$ T_4\left(\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)\right)=\frac{1}{2}-\frac{1}{4}\left(z^2+\frac{1}{z^2}\right). $$ Thus the pushforward of the Lebesgue measure on the unit circle (which is an ergodic measure for $z\mapsto z^2$) by $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$ provides an an absolutely continuous ergodic measure for $T_4:[0,1]\rightarrow [0,1]$.

In general, there is a very deep theorem of Lyubich stating that, aside from a measure zero subset of parameters, for every $c$ the map $T_c$ is either hyperbolic (i.e. has an attracting periodic orbit) or stochastic (i.e. admits an absolutely continuous invariant measure).

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Added: Another well-studied example is $c\approx 3.57$ that happens at the end of period-doubling cascade $-$ the largest parameter for which the topological entropy is zero. For this parameter, the map $T_c$ is infinitely renormalizable (the aforementioned paper of Lyubich shows that the set of such parameters is of measure zero). For this $c$, there exists a Feigenbaum attractor. This is an invariant Cantor set on which the dynamics of $T_c$ is conjugate to a "2-adic adding machine". Ergodic probability measures for such an interval map are classified here; these are supported either on periodic orbits of period $2^n$ or on the Feigenbaum attractor.

Answer 2

There was a large amount of work by many authors on questions of absolutely continuous invariant measures for the members of the logistic family. In particular, a landmark result of Michael Jakobson established that there is a positive measure set of $c$’s for which there is an ergodic absolutely continuous invariant measure.

I will describe my understanding of what happens as $c$ varies. Since I am not an expert, there may be some errors in what I am saying here.

As you vary $c$, there is a dense set of values for which the critical point is in the basin of attraction of an attracting periodic orbit. When that happens it is known that Lebesgue almost every point is attracted to the attracting critical orbit. I believe in that case there are no other measures than the atomic measures concentrated on periodic orbits. Notice also that this is an open condition.

Answer 3

The answers so far focus on physical measures, which have a positive basin of attraction. These could be the measure supported on an attracting periodic orbit, or an absolutely continuous invariant measure. As mentioned already, a deep result by Lyubich shows that one of these cases occurs for almost every parameter, i.e. almost every logistic map is either regular or stochastic (and the measures here are unique). In fact, almost every non-regular map is Collet-Eckmann (see Avila & Moreira, "Statistical properties of unimodal maps: the quadratic family"), and the absolutely continuous invariant measures have nice properties under perturbation (see Baladi & Viana, "strong stochastic stability and rate of mixing for unimodal maps").

But if you ask about ALL invariant (or ergodic invariant) measures, then there will almost always be many of them. Indeed, first of all you have such measures for every periodic orbit, and from the Feigenbaum point onwards, there will be infinitely many such orbits. Moreover, you will have some ergodic invariant measures supported on invariant Cantor sets in the non-wandering set. Some of these - those supported on Cantor repellers, for instance - will persist under perturbations. Others, presumably, may not.