Dynamics of rational functions over different fields

Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A(x)-A(y)=0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A^{\circ d}, d\geq 1, into compositions of rational functions can be obtained from decompositions of a single iterate A^{\circ N} for N large enough.

Let f(z) be a rational function of degree at least 2 with coefficients in a number field K, and assume that the second iterate f^2(z) of f(z) is not a polynomial. The second author previously proved that for any b in K, the forward orbit O_f(b) contains only finitely many quasi-S-integral points. In this note we give an explicit upper bound for the number of such points.

Let T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

We study a question on characterizing polynomials among rational functions of degree $$>1$$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If /' is in L log L on a finite interval, then / can be approximated in the uniform norm by rational functions of degree n to an error 0(l/n) on that interval.

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Let be the Hardy space of functions that are analytic in the disk and let be the derivative of of order in the sense of Weyl. It is shown, for example, that if is a rational function of degree with all its poles in the domain , then , where , , and depends only on and .Bibliography: 32 titles.

Let be a Lebesgue measurable subset of a -dimensional cube (), let , where , and let be the least deviation of , in the metric of , from the rational functions of degree . If , then, for , has a local differential of order in the -metric at each point , except perhaps points of some set of metric dimension (this inequality is sharp). In addition, has a global differential of order in the metric of for any .Bibliography: 15 titles.

Denote by the least uniform deviation of the function , defined in a subset of -dimensional Euclidean space, from the rational functions of degree . It is shown that if , then, a.e. on , has a total differential. The case was previously treated by E. P. Dolženko. Bibliography: 9 titles.

J-stability in non-archimedean dynamics

We consider rational maps of degree d ≥ 2 on the Riemann sphere and obtain large deviation results for Hölder observables under the measure of maximal entropy.

Given a rational map $\\phi: {\\mathbb P}^1\\to {\\mathbb P}^1$ defined over a\nnumber field $K$, we prove a finiteness result for $\\phi$-preperiodic points\nwhich are $S$-integral with respect to a non-preperiodic point $P$, provided\n$P$ satisfies a certain local condition at each place. This verifies a special\ncase of a conjecture of S. Ih.\n

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

ABC implies a Zsigmondy principle for ramification

The presented method is an extension of the well known autoregressive or Maximum-Entropy-Method. Here the power spectrum of the given process is fitted to a rational function of degree d. A motivation is given why it is better to use rational function instead of functions of the Form 1/ | Polynom (λ)|2. Some examples illustrate the use of the method. No information is given about the statistics of the estimate.