Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and let $f:X\to Y$ be a function. Recall that a metric takes nonnegative real values (Nonnegativity of a metric is a consequence of the other axioms, not an axiom).

• $f$ is Lipschitz with constant $L$, where $L\in \mathbb{R}$ and $L\ge 0$, if $d_Y(f(x),f(x^{\mathrm{\prime }}))\le L{d}_X(x,x^{\mathrm{\prime }})\text{for all }x,x^{\mathrm{\prime }}\in X\mathrm{.}$ $f$ is Lipschitz if it is Lipschitz with some such constant.
• Let $\alpha \in \mathbb{Q}$ with $0<\alpha \le 1$ (Order on the rationals). $f$ is $\alpha$-Hölder with constant $C$, where $C\in \mathbb{R}$ and $C\ge 0$, if $d_Y(f(x),f(x^{\mathrm{\prime }}))\le C{d}_X(x,x^{\mathrm{\prime }}{)}^{\alpha }\text{for all }x,x^{\mathrm{\prime }}\in X,$ the power being the rational power of a nonnegative base (Rational powers $a^r$ of a positive base). $f$ is $\alpha$-Hölder if it is so with some such constant, and Hölder if it is $\alpha$-Hölder for some rational $\alpha \in (0,1]$.
• $f$ is a contraction with constant $q$ if it is Lipschitz with constant $q$ and $0\le q<1$. The number $q$ is then called a contraction constant for $f$.

The power $d_X(x,x^{\mathrm{\prime }}{)}^{\alpha }$ is defined at every pair, including $x=x^{\mathrm{\prime }}$. The base $d_X(x,x^{\mathrm{\prime }})$ is a nonnegative real, and Rational powers $a^r$ of a positive base defines $a^{\alpha }$ for every $a>0$ and, by its supplementary clause, sets $0^{\alpha }=0$ for every rational $\alpha >0$. Since $\alpha >0$ is required here, the displayed inequality at $x=x^{\mathrm{\prime }}$ reads $0\le C\cdot 0$, which holds; so no separate clause and no restriction to $x\mathrm{\ne }{x}^{\mathrm{\prime }}$ is needed. Note that this does not by itself explain the strict inequality $\alpha >0$: if one extended the formula to $\alpha =0$ using the convention $0^0=1$ of Integer powers $a^m$, the equal-point inequality would still be the automatic $0\le C$. Globally, however, that extension would reduce to the bounded-diameter condition $d_Y(f(x),f(x^{\mathrm{\prime }}))\le C$, outside the standard Hölder range adopted here.

Why the exponent is rational on this page and why it is at most $1$. At this point in the reading order only rational powers are available (Rational powers $a^r$ of a positive base), so $\alpha$ ranges over the rationals. The upper bound $\alpha \le 1$ is the standard convention, and it is where the notion is useful: the classical theory reserves the name for $\alpha \in (0,1]$, and nothing in this library uses an exponent outside that range. No claim is made here about what an exponent $\alpha >1$ would do.

Constants are not unique and are not part of the data. If $f$ is Lipschitz with constant $L$ it is Lipschitz with every constant $L^{\mathrm{\prime }}\ge L$, and likewise for Hölder constants; the adjectives above are existential statements. A contraction, by contrast, requires a constant strictly below $1$, and that is a real restriction: exhibiting the constant is part of exhibiting a contraction, and a map that shrinks every distance without admitting one uniform constant is not a contraction here.

• The three conditions are ranked, and the ranking is a theorem. Contraction implies Lipschitz by definition; Lipschitz and Hölder each imply uniform continuity (Uniform continuity of a map of metric spaces: one $\delta$ serving every point), and uniform continuity implies continuity. That is Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent, where the one implication that needs a hypothesis, namely Lipschitz implies Hölder, is stated with the boundedness hypothesis it actually needs.
• A Lipschitz map with constant $0$ is constant when $X$ is nonempty, since $d_Y(f(x),f(x^{\mathrm{\prime }}))\le 0$ forces $f(x)=f(x^{\mathrm{\prime }})$ by the separation axiom (M1) (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric). So the constant maps are exactly the maps admitting every nonnegative constant, and they are contractions with constant $0$.
• The rational exponent is a position in the reading order, not a limitation of the notion. Real exponents are constructed later, in Real powers for positive bases, with the zero-base positive-exponent convention ↗, and that construction removes the ambient restriction; the definition above is retained as its rational-exponent version. Nothing on this page uses the later construction.
• Naming forks. Many texts call a Lipschitz map with constant $L<1$ a contraction and one with $d(fx,fy)<d(x,y)$ for $x\mathrm{\ne }y$ a contractive or weakly contractive map. This library uses contraction only in the first sense; the second condition is strictly weaker and does not force a fixed point (FALSE: $d(fx,fy)<d(x,y)$ for all $x\mathrm{\ne }y$ on a complete metric space forces a fixed point), witnessed by $x\mapsto x+1\mathrm{/}x$ on $[1,\mathrm{\infty })$ strictly decreases every distance and has no fixed point ↗, which is precisely why the two names are kept apart here.