Dimension-raising maps in a large scale (Q2856666)

For the classical Hurewicz dimension-raising theorem and finite-to-one mapping theorem, the authors formulate a dimension-raising type theorem and a finite-to-one mapping type theorem for asymptotic dimension and asymptotic Assouad-Nagata dimension.NEWLINENEWLINEFor any map \(f:X\to Y\) and natural number \(n\), they consider the following properties: (B) There exists \(d>0\) such that for each \(r>0\) and for each \(B\subset Y\) with diam\(\,(B)\leq r\), there exists \(A\subset X\) with diam\(\,(A)\leq dr\) and \(f(A)=B\); (B)\(_n\) For each \(r<\infty\), there exists \(d<\infty\) such that for each \(B\subset Y\) with diam\(\,(B)\leq r\), \(f^{-1}(B)=\bigcup_{i=1}^nA_i\) for some subsets \(A_i\subset X\) with diam\(\,(A_i)\leq d\) for \(i=1,\ldots,n\); (C)\(_n\) There exist \(c,r_0>0\) such that for each \(r\geq r_0\) and for each \(B\subset Y\) with diam\((B)\leq r\), \(f^{-1}(B)=\bigcup_{i=1}^n A_i\) for some \(A_i\subset X\) with diam\(\,(A_i)\leq cr\) for \(i=1,\ldots,n\).NEWLINENEWLINEIn this paper, the author show the following: Let \(X\) and \(Y\) be metric spaces. If \(f:X\to Y\) is a Lipschitz onto map such that \(|f^{-1}(y)|\leq n\) for each \(y\in Y\) and \(f\) has property (B), then \(\dim_{\text{AN}}Y\leq (\dim_{\text{AN}}X+1)n-1\). If \(f:X\to Y\) is a coarse map with property (B)\(_n\), then \(\text{asdim}\,Y\leq(\text{asdim}\,X+1)n-1\). If \(f:X\to Y\) be an asymptotically Lipschitz map with property (C)\(_n\), then \(\text{asdim}_{\text{AN}}\,Y\leq(\text{asdim}_{\text{AN}}\,X +1)n-1\). Furthermore, they show that for a metric space \(X\), if \(\text{asdim}\,X\leq n\) then \(X\) admits an \((n+1)\)-precode structure for asymptotic dimension, and if \(\text{asdim}_{\text{AN}}\,X\leq n\) then \(X\) admits an \((n+1)\)-precode structure for asymptotic Assouad-Nagata dimension.