I am currently studying the LSV map, a variant of the class of Pomeau-Manneville maps, which has the form:
$$T(x) =\begin{cases} x+2^\alpha x^{1+\alpha} & 0 \leq x < \frac{1}{2} \\ 2x-1 & \frac{1}{2} \leq x < 1 \\ \end{cases} $$
In the original paper proposing the LSV map in the first paragraph on the second page, the authors say "For $0<\alpha<1$, the map possesses an absolutely continuous [with respect to Lebesgue] probability measure (SRB Measure)..."
Working through the paper it is clear to me that they demonstrate the existence of an invariant probability measure, call it $\nu$ which is absolutely continuous with respect to Lebesgue measure $\mu$. What is not clear to me is why $\nu$ satisfies the definition of an SRB measure, i.e that the measure has a Lebesgue non-trivial basin of attraction. More precisely, it is not clear that there exists a set $V \subset [0,1]$ and $V \subset U$ such that $\mu(V)=\mu(U)>0$ and that for every $x \in V$ and and every continuous function $f: U \to \mathbb{R}$ it is true that:
$$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=0}^n f(T^i(x)) = \int_U f \ d\nu $$
I believe that it has been proved that the invariant measure referred to by Liverani et al. is everywhere positive. Thus, if one could show that the only invariant sets under $T$ have Lebesgue measure one or zero, then one could apply the Birkhoff Ergodic theorem to obtain that the measure is SRB. However, its not clear to me how to show that these are the only invariant sets.
In related literature about interval maps, not necessarily about the LSV map, other authors also seem to take a measure which is invariant and absolutely continuous with respect to Lebesgue measure to be an SRB measure. I am aware of related results for hyperbolic maps, or piecewise hyperbolic maps. However, I can't formulate/prove or find a reference which supports this in the case of non-hyperbolic maps. So here is question:
Under what conditions is an invariant and absolutely continuous with respect to Lebesgue measure also an SRB measure?