Related papers: Cut and semi-conjugate
A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…
We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most one-dimensional sets. They are…
We characterize pairs of rational functions $A$, $B$ such that $A$ is semiconjugate to $B$, and $B$ is semiconjugate to $A$.
A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\mathbb R C\to\mathbb{RP}^1$. Let $c_1,\dots,c_r$ be connected components of $\mathbb R…
Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \,…
An important question is to describe topological conjugacy classes of dynamical systems. Here we show that within the space of real analytic one-dimensional maps with critical points of prescribed order, the conjugacy class of a map is a…
We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection…
Using dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension…
A new class of rational parametrization has been developed and it was used to generate a new family of rational functions B-splines $\displaystyle{{\left({}^{\alpha}{\mathbf B}_{i}^{k} \right)}_{i=0}^{k}}$ which depends on an index $\alpha…
Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we…
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex…
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF…
We consider the order parameter $u=\left<{\rm Tr}\phi^2\right>$ as function of the running coupling constant $\tau \in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets.…
Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be a double cover of $\Sigma$ and let $\mu : S \to S'$ be the canonical resolution. Put $f = f'\circ\mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting…
A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…