Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?



Topological dynamical systems with only zero-entropy factors
This question is very related to the question of **lowering topological entropy**, introduced in ``Can one always lower topological entropy?'' by Shub and Weiss and then very nearly solved by Lindenstrauss in "Lowering topological entropy" and "Mean Dimension, Small Entropy Factors, and an Embedding Theorem." There's too much to state everything here, but here's a quick overview:


The question is: does every $(X,T)$ of positive entropy possess a nontrivial (i.e. not one point) factor $(Y,S)$ with $h(Y) < h(X)$?


1. Shub and Weiss showed that the answer is "yes" for subshifts and more generally, systems with the so-called **small boundary property**.
2. Lindenstrauss showed that the answer is "yes" for systems $(X,T)$ where $X$ is finite-dimensional. He then showed that the same is true when $(X,T)$ has zero **mean dimension** and a nontrivial minimal factor. Mean dimension is too long to define here, but in particular, $(X,T)$ has zero mean dimension if $(X,T)$ has finite entropy!
3. Germane to your question: Lindenstrauss also gives examples of $(X,T)$ (with infinite entropy and $X$ infinite-zero dimension) for which every nontrivial factor $(Y,S)$ factors onto the original system $(X,T)$! The easiest is $X = [0,1]^{\mathbb{Z}}$ and $T$ the shift $\sigma$, but he also constructs a minimal example.


To summarize: any finite-entropy system (with a nontrivial minimal factor) has a nontrivial factor of smaller entropy, thus not conjugate to the original. So for systems with a nontrivial minimal factor, the only possibilities for your condition are zero or infinite entropy.


For infinite entropy, we can't yet rule out your condition, since technically the example above doesn't satisfy your criteria (the nontrivial factors are not isomorphic to the original system). However, the factors are in a way "bigger" than the original system, which is perhaps even more surprising!


