The Measure-preserving Parametrization Of Self-affine Sets Generated By Single-matrix IFS

In this thesis,we mainly study the measure-preserving parametrization of self-affine sets generated by single-matrix IFS.The so-called measure-preserving parametriza-tion,preserves the Lebesgue measure on the interval[0,1]and the self-affine measure given on self-affine sets.First,we discuss the graph-directed measure(?1,…,?m)on the invariant sets {Ej}j=1m of the graph-directed IFS(GIFS in short).We prove that,given a linear GIFS satisfying the strong open set condition,and given the graph-directed measure,then Ej has a parametrization of preserving measure ?j.We also call(Ej,?i)can be measure-preserving parametrization.(Our result gener-alizes the results in[H.Rao,S.Q.Zhang,Space-filling curves of self-similar sets(?):iterated function systems with order structures,Nonlinearity,29(2016)],which concerns the Hausdorff measure).Second,for a self-affine set F generated by single-matrix IFS,we construct skeletons,edge-to-trail substitution and linear GIFS by-using the same method in[X.R.Dai,H.Rao,and S.Q.Zhang,Space-filling curves of self-similar sets(?):edge-to-trail substitution rule,Nonlinearity,32(2019)].In this paper,we transform a self-affine set into a graph-directed set,and we show that the self-affine measure ? can be transformed into graph-directed measure.Hence we prove that,(F,?)can be measure-preserving parametrization when the self-affine sets possesing a skeleton and satisfying the open set condition.Finally,we give an example of measure-preserving parametrization of a connected McMullen set.