We are in the framework of measurable transformations, i.e. let $(X,\mathcal{B},m)$ be a measure space and let $T:X\to X$ be a measurable transformation.
In your opinion, what does the following notation mean?
$$ T^{-n}(A)\qquad\text{for}\quad A\in\mathcal{B} $$
On
The $n$-th preimage of $A$ under $T$, that is $$ T^{-1}(A) = \{x \in X \mid Tx \in A \} $$ denotes the preimage of $A$ under $T$, and we interate this by $$ T^{-n}(A) := T^{-1}\bigl(T^{1-n}(A)\bigr), n \ge 2 $$ or what is the same, the preimage under $T^n = T \circ \cdots \circ T$ of $A$, that is $$ T^{-n}(A) = \{x \in X \mid T^nx \in A\} $$
This is the set of elements $x \in X$ that if you apply $T$ to them $n$ times, you land in $\mathcal{B}$.