Braids, conformal module and entropy

In the paper we discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which occurred in connection with the interest in the 13th Hilbert problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphisms. We prove that these invariants are related: They are inverse proportional. This allows to use known results on entropy for applications to the concept of conformal module. In particular, we give a short conceptional proof of a theorem which appeared in connection with research on the Thirteen's Hilbert Problem. We also give applications of the concept of conformal module to the problem of isotopy of continuous objects involving braids to the respective holomorphic objects. The objects considered here are quasipolynomials of degree three as well as elliptic fiber bundles. A byproduct of the proof is a systematic treatment of reducible braids and of the entropy of mapping classes on Riemann surfaces of second kind, as well as expressions of entropy and conformal module of conjugacy classes of reducible braids in terms of the respective invariants of the irreducible components.