Abstract: The study of Bedford-McMullen carpets dates back to their introduction in 1984 by Tim Bedford and Curtis McMullen and is concentrated mainly on dimension theory, invariant measures and multifractal analysis. We consider the topological structure of these carpets. We prove that if a B.-M. carpet K is simply-connected and is not the filled rectangle, then K is a dendrite. We show that it has 3, 4 or 6 boundary points, which are arranged in pairs. The minimal subtree of K containing these boundary points is called the main tree of K. This main tree defines completely the topology of the dendrite K. There are 7 types of possible main trees for simply-connected B.-M. carpets. Taking into account the relations between the ramification points of the main tree and the boundary points of K, we obtain 16 possible types of simply-connected B.-M. carpets.

Cite: Tetenov A.V. , Drozdov D.A.
The topology of simply-connected Bedford-McMullen carpets
4th International Conference on Nonlinear Applied Analysis and Optimization 17-19 окт. 2024