Differential calculus on topological spaces with weak Markov structure I

Abstract

The concept of a weak Markov set takes its origin from Whitney problems for differentiable functions on ${\mathbb{R}}^n$. In the present paper we study a version of the first Whitney problem of characterizing families of continuous functions satisfying certain differential relations on weak Markov sets. To this end we develop differential calculus on weak Markov sets similar to that on open subsets of ${\mathbb{R}}^n$. Then we show that some classical results for smooth functions and differential forms (such as Poincaré lemma, de Rham and Hartogs theorems, Künneth formulas, etc.) are valid also on certain weak Markov sets and more generally certain topological spaces with weak Markov structures. The class of such spaces includes, in particular, $C^{\infty }$ manifolds with boundaries and some Lipschitz and fractal topological manifolds. Thus the paper offers yet another approach to analysis on fractals, a developing area of modern mathematics that focuses on geometric and dynamical aspects of fractals.