Wikipedia says if γ≤1, it’s a “non-expansive map”. But yes, contraction maps have some Lipschitz constant γ that enforces the behavior you describe. However, notice we still have the math ⟹ “intuitive contraction” here, so it has the reverse-direction correspondence. Intriguingly, we’re missing part of “intuitive contraction = bring things closer together” for the γ≤1 case, as you point out, so we don’t have the forward direction fulfilled.
I guess it has a bunch of names: the link at the top of the wikipedia page is on the words “non-expansive map”, at the bottom it’s “short map”, and the title of the wikipedia page for the thing it calls it a “metric map”, and also lists the name “weak contraction”. So strange that this simple definition would be so little-used and often-named!
Wikipedia says if γ≤1, it’s a “non-expansive map”. But yes, contraction maps have some Lipschitz constant γ that enforces the behavior you describe. However, notice we still have the math ⟹ “intuitive contraction” here, so it has the reverse-direction correspondence. Intriguingly, we’re missing part of “intuitive contraction = bring things closer together” for the γ≤1 case, as you point out, so we don’t have the forward direction fulfilled.
I guess it has a bunch of names: the link at the top of the wikipedia page is on the words “non-expansive map”, at the bottom it’s “short map”, and the title of the wikipedia page for the thing it calls it a “metric map”, and also lists the name “weak contraction”. So strange that this simple definition would be so little-used and often-named!
Just like how open maps turn out to be way less useful in topology than continuous maps.