Related papers: On the postcritical set of a rational map

Define z to be the smallest cardinality of a function f:X->Y with X and Y sets of reals such that there is no Borel function g extending f. In this paper we prove that it is relatively consistent with ZFC to have b<z where b is, as usual,…

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and…

Recent work proposed $\delta$-relevant inputs (or sets) as a probabilistic explanation for the predictions made by a classifier on a given input. $\delta$-relevant sets are significant because they serve to relate (model-agnostic) Anchors…

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or,…

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic…

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our…

We obtain a criterion for when the specialization of the iterated Galois group for a post-critically finite (PCF) rational map is as large as possible, i.e., it equals the generic iterated Galois group for the given map.

We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or…

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of…

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of…

It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical…

A polynomial skew product of C^2 is a map of the form f(z,w) = (p(z), q(z,w)), where p and q are polynomials, such that f is regular of degree d >= 2. For polynomial maps of C, hyperbolicity is equivalent to the condition that the closure…

For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…

The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in…

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak$^*$ topology, is a non-empty metrizable Choquet simplex. We show that…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…

The paper deals with Newton maps of complex exponential functions and a surgery tool developed by P. Ha\"issinsky. The concept of "Postcritically minimal" Newton maps of complex exponential functions are introduced, analogous to…

The present note studies \emph{surjective rational endomorphisms} $f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f \ne \emptyset$. We develop an experimental approach, based on some…