I’m actually really bothered by this one because that’s not what a contraction mapping is. A contraction mapping isn’t just something that brings points closer together, it’s a mapping where there’s some factor γ<1 such that for any pair of points, their distance gets multiplied by a factor of at most γ. So, if your function brings all points closer together but e.g. the distance between points 1 and 2 gets multiplied by 0.9, the distance between 2 and 3 gets multiplied by 0.99, the distance between 3 and 4 gets multiplied by 0.999, etc, then that’s called a “short map” or a “metric map”, not a contraction, and the contraction mapping theorem fails to hold (counterexample left to the reader’s imagination).
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!
I’m actually really bothered by this one because that’s not what a contraction mapping is. A contraction mapping isn’t just something that brings points closer together, it’s a mapping where there’s some factor γ<1 such that for any pair of points, their distance gets multiplied by a factor of at most γ. So, if your function brings all points closer together but e.g. the distance between points 1 and 2 gets multiplied by 0.9, the distance between 2 and 3 gets multiplied by 0.99, the distance between 3 and 4 gets multiplied by 0.999, etc, then that’s called a “short map” or a “metric map”, not a contraction, and the contraction mapping theorem fails to hold (counterexample left to the reader’s imagination).
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.