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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 $h(F)=2h(f)$?

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    Do you mean to write $F=(f, f_{x})$? – 2017-02-17
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    Yes, I think so, this should be more convenient. – 2017-02-17
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    Then I believe the entropy will be $2h(f)$ since for some measure $\mu$ the entropy will be defined in terms of the fibre entropy $h^f(f)$ so that $h_{\mu}=h(f)+h^f(f)$ – 2017-02-17
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    Could you please explain to me what you mean? I do not understand your reasoning (i am new at this stuff). – 2017-02-17
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    @Rhjg Conventionally, a skew product $F$ over a dynamical system $f : X \to X$ is a mapping of the form $F : X \times Y \to X \times Y$ of the form $F(x,y) = (f(x), g_x(y))$, where for each $x \in X$ we have $g_x : Y \to Y$. Would you clarify whether this is what you're talking about, and if so, edit your question to improve the notation? – 2017-02-17
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    @ABlumenthal This is exactly what I mean and I'll add it to my question. – 2017-02-17
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    In the equation $h(F) = h(f)+h(g_x)$, the LHS is a constant, and the RHS depends on $x$... – 2017-02-17

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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 $x$?

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    May I ask another, more General question here? Where does the Name "skew product" come from? In which sense is the product "skew"? – 2017-02-18
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    The proper name is cocycle, which appeared earlier in the Russian school (in the context of dynamics, to which your question belongs). In the West the notion was copied and the name was changed to skew product. Well, in English "skew" could be "oblique", as opposed to "direct" (direct would mean that the second component does not depend on $x$). It seems fair the English name. – 2017-02-18
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    The very last question in your answer whether $h(F)=h(f)+c$ when $h(g_x)=c$ for all x: Does this hold? – 2017-02-20
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    By the way, is it always true, that for each x we have $h(F(x,y))=h(f)+h(g_x)$? – 2017-02-20
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    You can ask this in a separate question, much better to see who can say something about it. On your second question, it is false in general. – 2017-02-20