Multifractal analysis in non-uniformly hyperbolic interval maps

In this paper, we establish a framework for the construction of Moran set driven by dynamics. Under this framework, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic measure in a class of non-uniformly hyperbolic interval maps with finitely many branches.

LOWER TYPE DIMENSIONS OF SOME MORAN SETS

In this paper, we discuss the lower type dimensions for some Moran sets. On one hand, for Moran set [Formula: see text] with [Formula: see text], we prove that [Formula: see text], where the supremum is taken over all quasi-Lipschitz mappings [Formula: see text]. 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.

INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

ASSOUAD-MINIMALITY OF MORAN SETS UNDER QUASI-LIPSCHITZ MAPPINGS

The quasi-Lipschitz mappings, weaker than the bi-Lipschitz mappings, preserve Hausdorff, packing and box dimensions, but change Assouad dimension [Formula: see text]. 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 [Formula: see text] in the class, [Formula: see text] for all quasi-Lipschitz mappings [Formula: see text] defined on [Formula: see text]. For another class of Moran sets, we prove that for any [Formula: see text] in the class, [Formula: see text] where the infimum is taken over all quasi-Lipschitz mappings [Formula: see text] defined on [Formula: see text], and [Formula: see text] 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].

Multifractal Structure of the Divergence Points of Some Homogeneous Moran Measures

The pointxfor which the limitlimr→0⁡(log⁡μBx,r/log⁡r)does not exist is called divergence point. Recently, multifractal structure of the divergence points of self-similar measures has been investigated by many authors. This paper is devoted to the study of some Moran measures with the support on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists; the Moran measures associated with this kind of structure are neither Gibbs nor self-similar and than complex. Such measures possess singular features because of the existence of so-called divergence points. By the box-counting principle, we analyze multifractal structure of the divergence points of some homogeneous Moran measures and show that the Hausdorff dimension of the set of divergence points is the same as the dimension of the whole Moran set.