$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$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 $[0,1]$?



Ergodic measures for the logistic map
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](https://annals.math.princeton.edu/2002/156-1/p01) 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).




---


**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](https://www.researchgate.net/publication/228926349_An_ergodic_adding_machine_on_the_Cantor_set); these are supported either on periodic orbits of period $2^n$ or on the Feigenbaum attractor.


