Narrow Results

The k-SAT (k ≥ 3) is a typical NP-complete problem in computer science. To visualize k-SAT, we embed the Boolean expressions in k-CNF, with variables in an infinite list, into cube [0, 1]^k of Euclidean space. In this paper, we find fractal structure of the visualized image of satisfiable expressions, we also prove that the image has full Hausdorff dimension k under some reasonable condition, by constructing some Moran subset. The results show this image is quite different from that with respect to a finite list of Boolean variables as in Refs. 1 and 2.

We prove that the irregular set is either empty or full of Hausdorff dimension in a class of one-dimensional non-uniformly hyperbolic dynamic systems.

The quasi-Lipschitz mappings, weaker than the bi-Lipschitz mappings, preserve Hausdorff, packing and box dimensions, but change Assouad dimension ${dim}_A(\cdot )$. In this paper, for Moran fractals, we investigate the change of their Assouad dimension under the quasi-Lipschitz mappings. We study a class of Moran set which is quasi-Lipschitz Assouad-minimal, i.e. for any $E$ in the class, ${dim}_{A}f(E)\ge {dim}_A(E)$ for all quasi-Lipschitz mappings $f$ defined on $E$. For another class of Moran sets, we prove that for any $F$ in the class, ${inf}_g{dim}_{A}g(F)={dim}_{qA}F,$ where the infimum is taken over all quasi-Lipschitz mappings $g$ defined on $F$, and ${dim}_{qA}(\cdot )$ is the quasi-Assouad dimension introduced in [F. Lü and L. F. Xi, Quasi-Assouad dimension of fractals, J. Fractal Geom.3 (2016) 187–215].

In this paper, we discuss the lower type dimensions for some Moran sets. On one hand, for Moran set $E$ with $c_{\ast }>0$, we prove that ${dim}_{qL}E={sup}_f{dim}_{L}f(E)$, where the supremum is taken over all quasi-Lipschitz mappings $f$. On the other hand, we obtain the lower spectrum formula for homogeneous Moran sets. In the proof a lower spectrum formula for a large class of fractal sets is established.

In this paper, we prove that a large class of Moran sets on the line with packing dimension 1 is quasisymmetrically packing-minimal.

In this paper, two large classes of Moran sets with packing dimension 1 are shown to be quasisymmetrically minimal for packing dimension.

We obtain the Assouad dimensions of two Moran classes with zero infimum contraction under suitable conditions.