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In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials. # 2002 Elsevier Science (USA)

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We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.