skew product entropy

Question

Suppose, we have a dynamical systen $(X,f)$ and a skew product $(X\times Y)$ with skew product map $$ F(x,y)=(f(x),g_x(y)) $$ with $g_x\colon Y\to Y$ for fixed $x$, do we then have that the topological entropy of $F$, denoted by $h(F)$, is $h(F)=h(f)+h(g_x)$?

Is it possible that $g_x=f_x$ and do we then have that $h(F)=2h(f)$?

Answer 1

Answer to your first question: sometimes yes, sometimes no (this is immediate, since you only need to change the map $g_x$ in the fiber so that it has different entropies for different values of $x$).

As for your second question, we don't know what is $f_x$ and so it is impossible to reply, unless you mean:

Is it true that $h(F)=h(f)+h(g)$ when $g_x=g$ for all $x$?

The answer is yes (simple exercise taking covers, which in this case can always be taken composed of rectangles, which of course generate the topology). A minor comment is that you need to require $X$ and $Y$ to be compact.

A more interesting question would be the following:

Is it true that $h(F)=h(f)+c$ when $h(g_x)=c$ for all $x$?