Note on local-to-global properties of BLD-mappings (Q2789858)

The author studies maps of bounded length distortion. He studies limits of them, preimages, and asymptotic values.NEWLINENEWLINERoughly speaking, a continuous, open, and discrete map is of bounded length distortion (BLD) if it preserves length of paths up to a constant. Another important class of mappings is the one of Lipschitz quotient (LQ) maps. Roughly, these are maps that map balls to a shape that is close to the one of the original ball.NEWLINENEWLINEThe first result establishes an equivalence between maps of bounded length distortion and maps of discrete Lipschitz quotient maps (that BLD-mappings are LQ-mappings was already known, so the author proves the other part of the equivalence). The next result is roughly that sequences of LQ-mappings preserve the cardinality of preimages of points. This result together with the equivalence of BLD- and LQ-mappings is used to prove that certain limits of BLD-mappings are BLD as well. The next result establishes the existence of small enough balls such that the mapping behaves well on their preimages.NEWLINENEWLINEThe paper concludes with a proof that BLD-mappings do not have asymptotic values and with a sufficient condition for being a covering map.