# temporal operators: interpreting them topologically in a dynamic topological system

There's a paper that i've been reading recently called "Dynamic Topological Logic" which can be found at: http://individual.utoronto.ca/philipkremer/onlinepapers/DTL.pdf. I have a question about something just two pages into the paper, and was wondering if some kind person here can understand my quesion better than I can. Apologies if this is not the correct place to ask this; any suggestions for re-direction are welcome.

Basic context:

We can interpret the logical system S4 topologically; given a topological space $X$, we call the pair $\langle X,V\rangle$ a $\textit{topological model}$, where $V$ is a valuation function taking the propositional variables of S4 to subsets of $X$. We give certain rules for topologically interpreting general well-formed formulas of S4 which can be found on the 4th page(page 136) of the link. It's a known result that if we look at things this way, S4 is the logical system of topological spaces. (Tarski and McKinsey's paper: http://www.dimap.ufrn.br/~jmarcos/papers/AoT-McKinsey_Tarski.pdf)

Define a $\textit{dynamic topological system}$(DTS) to be a pair $\langle X,f\rangle$, where $X$ is a topological space and $f$ is a continuous function on $X$. We can think of $f$ as moving the points of $X$ from one moment to the next. i.e. $x \in X$ goes to $fx$ goes to $ffx$ ...(etc) The paper studies DTLs (Dynamic Topological Logics) which are the logics of DTSs, just as S4 is the logic of topological spaces. These DTLs are basically versions of S4 with the next modality($\circ$) and the henceforth modality($\ast$).

The part i'm having trouble understanding is the interpretation of $\circ$ and $\ast$ in terms of the function $f$(discussed at the bottom of page 134). The statement at the top of page 135 that $x \in \circ P$ iff $fx \in P$ iff $x \in f^{-1}(P)$ makes intuitive sense to me. However, i'm still struggling to understand the part leading up to that. Say that at moment $m$, the proposition $P$ is true at $fx$, i.e. $fx \in P$. How does that imply that $P$ is true at $x$ after $f$ acts on $x$ once? Wouldn't $P$ be true at $fx$ in that case? Thanks for any advice/suggestions! And of course feel free to ask me to clarify if something doesn't make sense.

Sincerely,

Vien