The factor entropy–mean dimension equivalence conjecture

Let $(X,T)$ be a dynamical system. A factor is nontrivial if it is not a one-point dynamical system. Topological entropy measures orbit complexity, while mean dimension measures the asymptotic dimensional complexity of the system. Factor entropy–mean dimension equivalence conjecture. The following conditions are equivalent:

• every nontrivial factor of $(X,T)$ has infinite topological entropy;
• every nontrivial factor of $(X,T)$ has positive mean dimension.

Positive mean dimension always implies infinite topological entropy, but the converse fails for general dynamical systems. The conjecture asks whether the two properties become equivalent after requiring them for every nontrivial factor; it is related to the Gutman–Lindenstrauss–Tsukamoto conjecture on zero mean-dimensional systems.