An orbit of the shift σ: t ↦ 2t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle Cμ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in Cμ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.
We prove that there exists a homeomorphism χ between the connectedness locus MΓ for the family Fa of (2 : 2) holomorphic correspondences introduced by Bullett and Penrose [BP], and the parabolic Mandelbrot set M1. We prove that the homeomorphism χ is dynamical (Fa is a mating between P SL(2, Z) and P χ(a)), that it is conformal on the interior of MΓ, and that it extends to a homeomorphism between pinched neighbourhoods of MΓ and M1 in the natural 1-parameter moduli spaces which contain them.
Abstract. Holomorphic correspondences are multivalued maps f = Q + Q −1 − : Z → W between Riemann surfaces Z and W , where Q − and Q + are (single-valued) holomorphic maps from another Riemann surface X onto Z and W respectively. When Z = W one can iterate f forwards, backwards or globally (allowing arbitrarily many changes of direction from forwards to backwards and vice versa). Iterated holomorphic correspondences on the Riemann sphere display many of the features of the dynamics of Kleinian groups and rational maps, of which they are a generalization. We lay the foundations for a systematic study of regular and limit sets for holomorphic correspondences, and prove theorems concerning the structure of these sets applicable to large classes of such correspondences.
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