Continuity of sub-additive topological pressure with matrix cocycles *

Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.

On the group of isometries of planar IFS fractals*

For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps

In this paper, we introduce some definitions of generalized affine set-valued maps: affinelike, preaffinelike, nearaffinelike, and prenearaffinelike maps. We present examples to explain that our definitions of generalized affine maps are different from each other. We derive a theorem of alternative of Farkas–Minkowski type, discuss Lagrangian multipliers for constrained set-valued optimization problems, and obtain some optimality conditions for weakly efficient solutions.

CONNECTEDNESS OF SELF-AFFINE SETS WITH PRODUCT DIGIT SETS

Let [Formula: see text] be an expanding lower triangular matrix and [Formula: see text]. Let [Formula: see text] be the associated self-affine set. In the paper, we generalize some connectedness results on self-affine tiles to self-affine sets and provide a necessary and sufficient condition for [Formula: see text] to be connected.

The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.

On the Boundary of Self-Affine Sets

This paper is devoted to studying the boundary behavior of self-affine sets. We prove that the boundary of an integral self-affine set has Lebesgue measure zero. In addition, we consider the variety of the boundary of a self-affine set when some other contractive maps are added. We show that the complexity of the boundary of the new self-affine set may be the same, more complex, or simpler; any one of the three cases is possible.

Dimension Spectrum for Sofic Systems

We study the dimension spectrum of sofic system with the potential functions being matrix valued. For finite-coordinate dependent positive matrix potential, we set up the entropy spectrum by constructing the quasi-Bernoulli measure and the cut-off method is applied to deal with the infinite-coordinate dependent case. We extend this method to nonnegative matrix and give a series of examples of the sofic-affine set on which we can compute the spectrum concretely.