I am trying to describe the following dynamics:
Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies:
- $\forall \, \rho \in [0, 1], \, \exists \, C_{\rho} > 1 \, \text{such that} \, |T_{\rho}'(y)| \geq C_{\rho}$, for $\lambda$-a.e. $y \in [-1, 1]$ wherever it is differentiable;
(i.e., they are "expanding on average")
- $\forall \, \rho \in [0, 1], \, \exists \, \eta_{\rho} > 0 \, \text{such that} \, |\frac{T_{\rho}''}{(T_{\rho}')^2}| \leq \eta_{\rho};$
(a way to "control" the distortions)
- The pre-image of every element by any $T_{\rho}$ is finite, i.e., $\, \forall \rho \in [0, 1], \, \exists N_{\rho}$ positive integer such that $\sup_{y \in [-1, 1]}\#\{T_{\rho}^{-1}(y)\} \leq N_{\rho}$.
I want to understand what happens with the "size" of
$$\Psi_{\theta_0 \dots \theta_{n-1}} = T_{\theta_0} \circ T_{\theta_1} \circ \dots \circ T_{\theta_{n-1}}, \, \forall n \geq 1. $$
In other words, given $\{\theta_n\}_{n \in \mathbb{N}} \subset [0, 1]$, does the sequence $\{\lambda(\Psi^{-1}_{\theta_0 \dots \theta_{n-1}}(B))\}_{n \in \mathbb{N}}$ converges?
This looks almost like some distortion lemmas, except the iterates are among (possibly) distincts $T_{\theta}$'s.
In order to demonstrate a simmilar lemma, it looks like I just need to be careful about the images/domains of each $T_{\theta}$. Is this intuition right, or am I missing something here?