This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It is easy to see that $T$ has 3-periodic attractor $\mathcal A:=\{x_1,x_2,x_3\}$. Moreover, defining $B(T):= \{x\in [0,1]; \mathrm{dist}(T^n(x),\mathcal A) \to 0,\ \text{as }n\to\infty\},$ one can check (via Mañé's theorem for instance) that $\Lambda = [0,1] \setminus B(T)$ is a $T$-invariant set such that there exist $C>0$ and $\lambda>1$ satisfying $$|(T^n)'(x)|> C \lambda^n\ \text{for every }x\in \Lambda.$$
Question: Is there any characterization of the pre-images of $c:=1/2$? In other words, can we characterize any of the sets defined as follows: $$\mathcal P(c):= \overline{\bigcup_{n>0}T^{-n}(c)}\ \text{or } \alpha(c) := \bigcap_{k\in\mathbb N} \overline{\bigcup_{n>k}T^{-n}(c)} ?$$
My guess is that $\alpha(c)\subset \Lambda,$ since the repeller should exhibit attractor-like behaviour through the pre-images of $T$. However, I have read that in several cases, $\mathcal P(c) = [0,1]$. So I am a bit confused. Does anyone knows a reference for this problem or knows about potential approaches to tackle it?