Let $(X,T)$ and $(Y,G)$ be topological dynamical systems. If $(Y,G)$ is a factor of $(X,T)$ it is well known and easy to proof that $h(G)\le h(T)$ , where $h$ denotes the topological entropy. If the systems are conjugated we have $h(T)=h(G)$. My question is under what (weaker) conditions we have $h(T)=h(G)$. I guess that this is true if the semi-conjugation is finite to one. Perhaps finite to one on a topological large set is enough, or? Where do I find proofs of such results?

## 1 Answer

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There is formula due to Rufus Bowen which can be stated as follows:

$$ h(T)\leq h(S) + \sup\left\{h\big(T,\pi^{-1}(y)\big) : y\in Y\right\},$$

where $\pi\colon X\to Y$ denotes the semi-conjugacy between $T$ and $G$, and $h(T,K)$ denotes the entropy of a non-necessary $T$-invariant set, as defined in the very same paper.