Reducible Pure Braids. Irreducible Nodal Components, Irreducible Braid Components, and the Proof of the Main Theorem
摘要
We describe the decomposition of conjugacy classes of reducible pure braids into irreducible components and show that the conformal module of any conjugacy class of braids equals the minimum of the conformal modules of the irreducible components. Similarly, we define the irreducible nodal components of the conjugacy class of mappings associated to the braid, and show that the entropy of the conjugacy class of mappings is equal to the maximum of the entropies of the irreducible nodal components. The Main Theorem for irreducible pure braids implies the Main Theorem for reducible pure braids. The conjugacy class of a reducible pure braid can be recovered from its irreducible components. The conjugacy class of its mapping class can be recovered up the products of commuting Dehn twists from the irreducible nodal components.