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Let $(X,d), (Y,e)$ be compact metric spaces and $T\colon X\to X, S\colon Y\to Y, \pi\colon X\to Y$ (surjective) be continuous maps with $\pi\circ T=S\circ\pi$. Then, as Bowen Show in Theorem 17 of this paper: $$ h(T)\leqslant h_e(S)+\sup_{y\in Y}h_d(T,\pi^{-1}(y)). $$

Now, in this paper, on page 3, there is also a lower "Bowen inequality":

$$ \max\left\{h(\sigma),\sup_{(s_n)\in\mathcal{A}}h(\varphi,X,(s_n))\right\}\leqslant h(\varphi)\leqslant h(\sigma)+\sup_{(s_n)\in\mathcal{A}}h(\varphi,X,(s_n)) $$

The upper inequality seems to be a direct application of the Bowen Theorem above on the maps $\sigma\colon\mathcal{A}\to\mathcal{A}, \varphi\colon\mathcal{A}\times X\to\mathcal{A}\times X$ and $\pi\colon\mathcal{A}\times X\to\mathcal{A}$ being the projection of the "first component". This should give $$ h(\varphi)\leqslant h(\sigma)+\sup_{(s_n)\in\mathcal{A}}h(\varphi,X,(s_n)) $$ since for each $(s_n)\in\mathcal{A}$ it shall be that $\pi^{-1}((s_n))=\left\{(s_n)\right\}\times X$.

But where does the lower estimation $$ \max\left\{h(\sigma),\sup_{(s_n)\in\mathcal{A}}h(\varphi,X,(s_n))\right\}\leqslant h(\varphi) $$ come from? Is this a general result also given by Bowen or is this a special estimation for the concrete case in the last mentioned paper?

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    Factors cannot have larger entropy and so the lower bound is immediate. **But you really need to use the notion for noncompact spaces!** Be careful since there are various nonequivalent notions in the noncompact case, Bowen's definition being only a possible one. – 2017-02-23
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    Ok! I now do understand the lower boundary. But I am not sure I do understand what you mentioned afterwards about noncompact sets. The Bowen definition requires metric spaces not necessarily being compact. So what do you mean? – 2017-02-23
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    Fibers need not be compact. – 2017-02-23
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    But where is the problem? Sorry, I do not understand. :-) – 2017-02-23
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    Since Bowen introduced his definition in the general noncompact case, many more (nonequivalent) notions appeared. You need to decide which one you use (**in the second component**), although the inequalities that you describe should work for all of them. In the compact case all these notions coincide. – 2017-02-23
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    Best reference that I can suggest: http://emerald.tufts.edu/as/math/Preprints/HasselblattNiteckiPropp.pdf – 2017-02-23
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    Ok, now I got your point. You are saying that in the noncompact case I should always say which definition of topological entropy I am meaning. So in my example, since the fibre maybe is non-compact, I should say that I am usind the Bowen-definition. I thought that maybe is redundant since I use Bowen-inequality (maybe already implying that I meant Bowen's top. entropy definition). – 2017-02-23
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    It is not really standard to refer to it as "Bowen inequality", and many people really don't know Bowen's definition (nowadays it is not the canon, more common the Carathéodory approach). Do say which is the notion, better like that. – 2017-02-23

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