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YCor
YCor
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Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.

Then it is true that $T$ is bijective.

Question: Is there an elementary proof of this fact? (Injectivity clearly follows, surjectivity is the issue.) The two proofs I know go through the enveloping semigroup, and the Stone Cech-Čech compactification $\beta \mathbb N$ respectively.

Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.

Then it is true that $T$ is bijective.

Question: Is there an elementary proof of this fact? The two proofs I know go through the enveloping semigroup, and the Stone Cech compactification $\beta \mathbb N$ respectively.

Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.

Then it is true that $T$ is bijective.

Question: Is there an elementary proof of this fact? (Injectivity clearly follows, surjectivity is the issue.) The two proofs I know go through the enveloping semigroup, and the Stone-Čech compactification $\beta \mathbb N$ respectively.

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Nate River
Nate River
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Is there an elementary proof that distal maps are invertible?

Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.

Then it is true that $T$ is bijective.

Question: Is there an elementary proof of this fact? The two proofs I know go through the enveloping semigroup, and the Stone Cech compactification $\beta \mathbb N$ respectively.