Smale's mean value conjecture for finite Blaschke products

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.Comment: To appear in an issue of Journal of Analysis denoted to the Proceedings of the Conference on Modern Aspects of Complex Geometry (MindaFest)

Tischler graphs of critically fixed rational maps and their applications

A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article we study properties of a combinatorial invariant, called Tischler graph, associated with such a map. More precisely, we show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. We also discuss the relevance of this result for classical open problems in holomorphic dynamics, such as combinatorial classification problem and global curve attractor problem

Dynamical Belyi maps

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics.Comment: 21 page

Finitely ramified iterated extensions

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.Comment: 19 page

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.A challenging conjecture of Stephen Smale on geometry of polynomials is under discussion. We consider an interpretation which turns out to be an interesting problem on equilibrium of an electrostatic field that obeys the law of the logarithmic potential. This interplay allows us to study the quantities that appear in Smale’s conjecture for polynomials whose zeros belong to certain specific regions. A conjecture concerning the electrostatic equilibrium related to polynomials with zeros in a ring domain is formulated and discussed.Research supported by the Brazilian foudations CNPq under Grant 304830/2006-2 and FAPESP under Grant 03/01874-2