How to construct totally disconnected Markov sets?

Abstract

We deduce a polynomial estimate on a compact planar set from a polynomial estimate on its circular projection, which enables us to prove Markov and Bernstein-Walsh type inequalities for certain sets. We construct -totally disconnected Markov sets that are scattered around zero in different directions; -a Markov set E ⊂ ℝ such that neither E ∩ [0, +∞] nor E ∩ [-∞, 0] admit Markov's inequality;-a Markov set that is not uniformly perfect.Finally, we propose a construction based on a generalization of iterated function systems: a way of obtaining a big family of uniformly perfect sets. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.