Chebyshev polynomials on a system of curves

This paper is devoted to the problem of how close can one get with the n-th Chebyshev numbers of a compact set ?? to the theoretical lower bound cap(??) ⁿ . It is shown that for a system of m ?? 2 analytic curves, there is always a subsequence for which the Chebyshev numbers are at least (1 + ??)cap(??) ⁿ , while for another subsequence they are at most (1 + O(n ^?1/(m?1)))cap(??) ⁿ . It is also shown that the last estimate is optimal. We also discuss how well a system of curves can be approximated by lemniscates in Hausdorff metric. The proofs are based on potential theoretical arguments. Simultaneous Diophantine approximation of harmonic measures lies in the background. To achieve the correct rate, a perturbation of the multi-valued complex Green??s function is introduced which makes the n-th power of its exponential single-valued and which allows the construction of Faber-like polynomials on multiply connected domains.