I posted this on math stack exchange some 10 days ago, but received no answers (https://math.stackexchange.com/q/4773194/1223994). Let $X$ be a compact metric space and denote by $d$ the metric on $X$. I wondered whether the following metric $d_\infty : C(X,X)\times C(X,X)\rightarrow \mathbb{R}_0^+$ given by $$d_\infty (f,g)=\sup_{x\in X, k\in\mathbb{N}} d(f^k(x),g^k(x))$$ has been considered and if it has a name in the literature. Thanks
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$\begingroup$ I suppose if X has a transitive group action, conjugation by shifts is continuous at f iff f is equicontinuous (as a dynamical system)? $\endgroup$– Ville SaloCommented Oct 10, 2023 at 16:40
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$\begingroup$ Or something to that effect... $\endgroup$– Ville SaloCommented Oct 10, 2023 at 16:41
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1$\begingroup$ A finite version with min instead of sup is studied in V. Barros, L. Liao, J. Rousseau, On the shortest distance between orbits and the longest common substring problem, Adv. Math. 344 (2019), 311–339. Maximal orbit distance would be a reasonable name? $\endgroup$– Jochen TrumpfCommented Oct 11, 2023 at 0:04
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