Bilipschitz equivalence

The concept of Bilipschitz equivalence is used in mathematics to investigate the "rough" geometry of metric spaces .

definition

A bijection

${\displaystyle f\colon M_{1}\to M_{2}}$

between metric spaces and is a Bilipschitz equivalence if there is a constant such that ${\displaystyle (M_{1},d_{1})}$ ${\displaystyle (M_{2},d_{2})}$ ${\displaystyle L\geq 1}$

${\displaystyle {\frac {1}{L}}d_{1}(x,y)\leq d_{2}(f(x),f(y))\leq Ld_{1}(x,y)}$

applies to all . ${\displaystyle x,y\in M_{1}}$

Examples

• A linear mapping is a Bilipschitz equivalence if and only if holds. ${\displaystyle A\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle \det(A)\neq 0}$
• ${\displaystyle \left\{0,1\right\}^{\mathbb {N} }}$is bilipschitz-equivalent to the Cantor set , the Bilipschitz-equivalence is given by . ${\displaystyle f((x_{n})_{n})=\sum _{n=1}^{\infty }{\frac {x_{n}}{3^{n}}}}$
• The Cayley graphs assigned to different finite generating systems S ₁ , S _{2 of} a group are bilipschitz-equivalent. ${\displaystyle G}$
• There are Bilipschitz equivalences that are not quasi-isometrics .
• If two equally discrete, non-mediate metric spaces are quasi-isometric, then they are also bilipschitz equivalent.

Individual evidence

1. ↑ D. Burago, B. Kleiner: Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps. Geom. Funct. Anal. 8 (1998), no. 2, 273-282. on-line
2. ↑ T. Dymarz: Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups. Duke Math. J. 154 (2010), no. 3, 509-526. on-line
3. ↑ A metric space is called uniformly discrete if there is a constant such that the inequality holds for all . It is called non-indirect if there are no Følner episodes . ${\displaystyle r>0}$ ${\displaystyle x\not =y}$ ${\displaystyle d(x,y)>r}$
4. ↑ K. Whyte: amenability, bi-Lipschitz equivalence, and the conjecture of Neumann. Duke Math. J. 99 (1999), no. 1, 93-112. on-line