Upper convex density and similarly contractive fixed points

Abstract

In this work, for the self similar sets satisfying the open set condition, we obtain an interesting character that the upper convex density at every similarly contractive fixed point equalling 1 implies that the upper convex densities at all the points are equal to 1. By using this result, we sufficiently answer the open problem 6 posed by Z. Zhou and L. Feng in [Twelve open problems on the exact value of the Hausdor measure and on topological entropy: A brief survey of recent results, Nonlinearity, 17(2004) 493–502.].