arXiv subjects
Publications and source records attributed to Adam Epstein.
We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to the set of angles of external rays landing on the real axis.
We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_\lambda:\mathbb{R}/\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ defined by \[F_\lambda(x) := 2x + a+ \frac{b}{\pi} \sin(2\pi x){\quad\text{with}\quad} \lambda:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] We prove that if $F_\lambda^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in $\mathbb{R}/\mathbb{Z}$, then there is a system of local coordinates $(\alpha,\beta):W\to \mathbb{R}^2$ defined in a neighborhood $W$ of $\lambda$, such that $\alpha(\lambda) =\beta(\lambda)=0$ and $F_\mu^{\circ n} - {\rm id}$ has a multiple zero with $\mu\in W$ if and only if $\beta^3(\mu) = \alpha^2(\mu)$. This shows that the tips of tongues are regular cusps.
Let f be an entire function that has only finitely many critical and asymptotic values. Up to topological equivalence, the function $f$ is determined by combinatorial information, more precisely by an infinite graph known as a "line-complex". In this note, we discuss the natural question whether the order of growth of an entire function is determined by this combinatorial information. The search for conditions that imply a positive answer to this question leads us to the "area property", which turns out to be related to many interesting and important questions in conformal dynamics and function theory. These include a conjecture of Eremenko and Lyubich, the measurable dynamics of entire functions, and pushforwards of quadratic differentials. We also discuss evidence that invariance of order and the area property fail in general.
Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map (P^1, P_f) \to (P^1, P_f). We give explicit examples of such maps f showing that the following cases may occur: (1) the basepoint is an attracting fixed point, the image of \sigma_f is open and dense, and the map \sigma_f is a covering map onto its image; (2) the basepoint is a superattracting fixed point, \sigma is surjective, and \sigma is a ramified Galois covering, (3) \sigma_f is constant.
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.
We give an arithmetic proof of rigidity for postcritically finite polynomials.
We prove a refinement of the Fatou-Shishikura Inequality - that the total count of nonrepelling cycles of a rational map is less than or equal to the number of independent infinite forward critical orbits - from a suitable application of Thurston's Rigidity Theorem - the injectivity of $I-f_*$ on spaces of meromorphic quadratic differentials.
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.