Hausdorff and box dimensions of self-affine sets in a locally compact non-Archimedean field

Abstract

Abstract We consider affine iterated function systems in a locally compact non-Archimedean field F . We establish the theory of singular value decomposition in F and compute the box and Hausdorff dimensions of self-affine sets in F n , in generic sense, which is an analogy of Falconer’s result for the real case. In R n , the box and Hausdorff dimensions of self-affine sets can be obtained only when the norms of linear parts of affine transformations are strictly less than 1 2 . However, in a locally compact non-Archimedean field, the same result can be obtained without the restriction of the norms.