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I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for ...

Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,...