I asked this at math.stackexchange.com, but got no answers. Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (countable measurable) partition of $X$. Show that $h(ST,A)≤h(S,A)+h(T,A)$. If $S=T$, it's rather easy. I couldnt get any further. Any help/reference will be appreciated. Thanks.
There is a good reason you were having difficulties in proving this.
This was an old question of Rohlin (MR0126526) which was first disproved in the topological setting by Goodwyn (MR0314023) and later independently by Thouvenot and Ornstein. An explicit example appears in the paper by Ornstein and Weiss (MR0910005; page 133); this paper should be relevant to you in many different ways if you are interested in such things.
That being given, there are many (smooth) contexts in which this property does hold (MR2690742); most notably consider commuting automorphisms of the torus (this is a simple exercise).
Let me know if you have trouble finding the papers.