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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 $x\in X$.

I have a questions concerning topological entropy $h(F)$ of $F$:

Do we have that $$ h(F)=h(f)+c\qquad\text{ if } h(g_x)=c\text{ for all }x\in X? $$

(For example, if $h(g_x)=0$ for all $x\in X$, do we then have $h(F)=h(f)$?)

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    The answer is no in general. You can find in the paper "A Note on the Entropy of Skew Product Transformations" by R. Adler an example and a formula of the entropy of skew product map. – 2017-02-20
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    Hm, okay. Unfortunately, I do not have a measure or sth like this. – 2017-02-20
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    You can convert the example into a pure topological one in fact. – 2017-02-20
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    You can ignore the measurable structure and change it to a topological space and measure preserving transform to a homeomorphism, and measure entropy to topological entropy. I think it still works. – 2017-02-20
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    And the integral turns into what? – 2017-02-20
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    You mean you want a formula for the entropy? – 2017-02-20
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    Yes, but i do not know if this is possible. – 2017-02-20
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    I am not sure if there is a nice formula, but since topological entropy is the supermum of the measure entropy over all invariant measures, you can get something. But I am not sure the formula, even in measurable case, can be easily used to calculate entropy except some special case. – 2017-02-20
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    @SimingTu The question is about the topological entropy. For the metric entropy you may know that the most general result is due to Ledrappier and Young with a far reaching Pesin formula for measures that are absolutely continuous on conditionals. They also gave a "formula" in the nonabsolutely continuous case. – 2017-02-20
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    @Rhjg I don't know the answer to your question. My personal guess is that the answer is yes (since there is uniform continuity on $x$). – 2017-02-20

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