Related papers: On Rational Maps with Two Critical Points
We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed…
In this article, we show that all admissible rational maps with fixed or period two cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials which make up the matings that construct these rational…
We give quantitative bounds for the number of quasi-integral points in orbits of semigroups of rational maps under some conditions, generalizing previous work of L. C. Hsia and J. Silverman (2011) for orbits generated by the iterations of…
This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast…
Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the…
The dynamics of one dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a…
In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…
Using a renormalization method, we study the critical behavior for intermittency in two coupled one-dimensional (1D) maps. We find two fixed maps of the renormalization transformation. They all have common relevant eigenvalues associated…
Critical points mark locations in the domain where the level-set topology of a scalar function undergoes fundamental changes and thus indicate potentially interesting features in the data. Established methods exist to locate and relate such…
This note will discuss the dynamics of iterated cubic maps from the real or complex line to itself, and will describe the geography of the parameter space for such maps. It is a rough survey with few precise statements or proofs, and…
These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first…
A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study…
A systematic mapping is a way to identify, evaluate and interpret all relevant research available to a matter of particular research. One of the reasons for conducting systematic reviews is that it summarizes the existing evidence regarding…
In this paper we describe the dynamics of certain rational maps of the form $k \cdot (x+x^{-1})$ over finite fields of odd characteristic.
The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.
Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points…
It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set. R. Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point.…
A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period…
This is a study of the Wittner capture construction for critically finite quadratic rational maps for which one critical point is periodic, and the second critical point is in the backward orbit of the first. This construction gives a way…
We define dual-critical graphs as graphs having an acyclic orientation, where the indegrees are odd except for the unique source. We have very limited knowledge about the complexity of dual-criticality testing. By the definition the problem…