Doubling Property Of Self-affine Measures On Carpets Of Bedford And McMullen

In this paper, we mainly discuss a class of self-affine set named Bedford and Mc-Mullen carpet, completely characterize doubling self-affine measures on it. A carpet of Bedford and McMullen is a set in the plane R2 defined as where n≥m≥2 are integers and Ω (?){0,1,..., n-1} ×{0,1,..., m - 1}. We always assume that #(Ω)> 1. Hereafter # denotes the cardinality. For each w=(i, j) ∈Ω let fw be the affine mapThen S satisfies so S is a self-affine set. In this paper, we call the triple (n, m,Ω) the defining data of the carpet S and denote by M the family of carpets of Bedford and McMullen. When n=m, it is a self-similar carpet of Sierpinski.As it is well known, when a self-similar set satisfies the strong separation condi-tion, all self-similar measures are doubling. We further prove that all Markov measures are doubling on a self-similar set with the strong separation condition. Subsequently, we focus on self-similar measures and Markov measures on Sierpinski carpets. With-out the strong separation condition, Sierpinski carpets can be divided into different types. In each case, we fully characterize doubling self-similar measures and doubling Markov measures on a Sierpinski carpet.Given a Bedford and McMullen carpet S ∈ M., we obtain an equivalent condition for a general Borel measure to be doubling on S. After that, according to the geometric characteristics of the carpet, we encounter several cases. In each case, we completely characterize doubling self-affine measures on S.Moreover, we find a new phenomenon that, there are self-affine carpets in M. which do not carry any doubling self-affine measure. Denote by M1 the family of such carpets, the following question arises:When does a carpet in M1 carry a doubling self-affine measure? We give a geometric characterization for those "good" carpets.