state, which we identify as two disjoint sets in the system’s state-space denoted as $A \subset \Omega$ and $B \subset \Omega$ respectively. As mentioned earlier, this calculation will be done by twisting the dynamics in some appropriate way that is dictated by the results of TPT. We begin by introducing the relevant objects that we will consider. If we assume again that we have at our disposal an infinitely long trajectory, $\zz(t)$ with $t>0$, this trajectory will go back and forth between $A$ and $B$ as time goes on, and we can split this trajectory into two pieces, depending on whether it visited last $A$ or $B$. This construction is illustrated in Fig. \[fig:trajAB\], where the trajectory is shown in red if it visited $A$ last and in black if it visited $B$ last: if the trajectory visited $A$ last at time $t$, we will say that it is assigned to $A$ at time $t$, if it visited $B$ last at time $t$, we will say that it is assigned to $B$ at time $t$.

![ Schematic representation of a piece of ergodic trajectory visiting the two sets $A$ and $B$. The pieces of this trajectory for which the last 