1995/12/18 by Pilgrim, Kelvin, Lei, Tan
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/9512221
Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let f(z) be a rational map and suppose that the postcritical set P(f) is finite. Let α be an embedded closed arc in the sphere and suppose that f|α is a homeomorphism. Define a branched covering g as follows. Cut the sphere open along α. Glue in a closed disc D. Map S2 - \Int (D) via f and \Int (D) by a homeomorphism to the complement of f(α). We prove theorems which give combinatorial conditions on f and α for g to be equivalent in the sense of Thurston to a rational map. The main idea in our proofs is a general theorem which forces a possible obstruction for g away from the disc D on which the new dynamics is defined.