From local to global analytic conjugacies

Let $f_1$ and $f_2$ be rational maps with Julia sets $J_1$ and $J_2$, and let $\Psi:J_1\to \mathbb{P}^1$ be any continuous map such that $\Psi\circ f_1=f_2\circ \Psi$ on $J_1$. We show that if $\Psi$ is $\mathbb{C}$-differentiable, with non-vanishing derivative, at some repelling periodic point $z_1\in J_1$, then $\Psi$ admits an analytic extension to $\mathbb{P}^1\setminus {\mathcal E}_1$, where ${\mathcal E}_1$ is the exceptional set of $f_1$. Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if ${\mathcal E}_1=\emptyset$ then the extended map $\Psi$ is rational, and in this situation $\Psi(J_1)=J_2$ and $\Psi^{-1}(J_2)=J_1$, provided that $\Psi$ is not constant. On the other hand, if ${\mathcal E}_1\neq \emptyset$ then the extended map may be transcendental: for example, when $f_1$ is a power map (conjugate to $z\mapsto z^{\pm d}$) or a Chebyshev map (conjugate to \pm \text{Х}_d with \text{Х}_d(z+z^{-1}) = z^d+z^{-d}), and when $f_2$ is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof

B\"ottcher coordinates

A well-known theorem of B\"ottcher asserts that an analytic germ f:(C,0)->(C,0) which has a superattracting fixed point at 0, more precisely of the form f(z) = az^k + o(z^k) for some a in C^*, is analytically conjugate to z->az^k by an analytic germ phi:(C,0)->(C,0) which is tangent to the identity at 0. In this article, we generalize this result to analytic maps of several complex variables

Quasiconformal variation of slit domains

We use quasiconformal variations to study Riemann mappings onto variable single slit domains when the slit is the tail of an appropriately smooth Jordan arc. In the real analytic case our results answer a question of Dieter Gaier and show that the function κ in Löwner's differential equation is real analytic

The set of maps F_{a,b}: x -> x+a+{b/{2 pi}} sin(2 pi x) with any given rotation interval is contractible

Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set

Minnesota: Decades of Decisions and Impact on Sports Law

The purpose of this Article is to demonstrate that Minnesota provides one of the most substantial examples of how sport and the law intersect. The Article begins with the 1970s and explores, decade-by-decade, many of the major sports law claims, cases, judgments, and incidents associated with Minnesota. The state’s flagship institution—the University of Minnesota Twin Cities—is the epicenter of many of these cases, providing examples of impropriety within institutional rules or the bylaws of the National Collegiate Athletic Association. Despite the many instances that the University of Minnesota Twin Cities has run afoul of NCAA rules, it is not the only campus or university in the state that provides sports law material. Minnesota has had a lasting effect on the Eighth Circuit Court of Appeals, professional sports, and labor relations of the sports world. Ultimately, this Article demonstrates that Minnesota must continue to be a part of the discussion in sports law