Nontrivial fractals in the plane and linear operators with joint spectral radius equal to 1
Properties of invariant bodies of a family of linear operators \(A_1,\dots,A_k:\mathbb{R}^n\to\mathbb{R}^n\) have been studied in the paper [\textit{V. Yu. Protasov}, Fundam. Prikl. Mat. 2, No. 1, 205-231 (1996; Zbl 0899.47002)], which also contains an algorithm that calculates the joint spectral radius of these operators. In the examples considered in the paper of Protasov (loc. cit), the dimensions (both topological and Hausdorff) of the invariant sets on the plane are equal to either 0 or 1. The main goal of this paper is to construct a class of operators with invariant sets having fractional Hausdorff dimension.
scientific article; zbMATH DE number 1262513