Intersection of a Moran type Sierpinski carpet and a line with rational slope

Abstract

In 2005, Liu et al. calculated the dimensionality of the intersection of Sierpinski carpet and a straight line with rational slope in the sense of Lebesgue measure.Sierpinski carpet is a self-similar set in two-dimensional planes obtained by an iterative function system, so each layer has the same structure. While the Sierpinski carpet set with Moran structure is the limit set obtained by the action of two iterative function systems in the two-dimensional plane, which we denote as ~$F_σ$~. And the structure of each layer may be different, controlled by 0,1 sequence ~$σ$~ and controlled by the set. In this paper, the upper and lower box dimensions of the set ~$F_σ$~ and the straight line ~$L_{a}$~ with rational slope are calculated, where ~$a$~ is the intercept of the straight line. In addition, we consider some related problem. The main difficulty in the research is that the structure of each layer of the set ~$F_σ$~ may be different, so the structure of each layer needs to be considered with the help of the sequence ~$σ$~ in the calculation process.