Two continuous maps are topologically conjugate if a homeomorphism between their state spaces satisfies the displayed intertwining identity. It is conjugate maps with a topology-preserving conjugating map. On compact metric spaces, pulling back a compatible metric through preserves every Bowen metric, and hence topological entropy.
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Topological conjugacy is a concept from dynamical systems that deals with the relationship between two dynamical systems which are "the same" in a certain topological sense. Specifically, two dynamical systems are said to be topologically conjugate if there exists a continuous bijective map (called a conjugacy) between their state spaces that preserves the dynamics of the systems.