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Register a new userDivergence and convergence of conjugacies in non-Archimedean dynamics
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Authors
Karl-Olof Lindahl
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We continue the study in [21] of the linearizability near an indif- ferent fixed point of a power series f, defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegels linearization theorem [27] is true also for non- Archimedean fields. However, they also showed that the condition in Siegels theorem is usually not satisfied over fields of prime character- istic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analyt- ically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.
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We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show: - a quantified version of Silvermans and Ingrams recent comparison between the critical height and any ample height on the moduli space $mathcal{M}_d(bar{mathbb{Q}})$, - two improvements of McMullens finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, - a characterization of non-affine isotrivial rational maps defined over the function field $mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.
This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $beta$-expansions. More precisely, let $([0,1),T_{beta})$ be the $beta$-dynamical system for a general $beta>1$ and $psi:[0,1]mapstomathbb{R}$ be a continuous function. Denote by $textsf{A}(psi,x)$ all the accumulation points of $Big{frac{1}{n}sum_{j=0}^{n-1}psi(T^jx): nge 1Big}$. The Hausdorff dimensions of the sets $$Big{x:textsf{A}(psi,x)supset[a,b]Big}, Big{x:textsf{A}(psi,x)=[a,b]Big}, Big{x:textsf{A}(psi,x)subset[a,b]Big}$$ i.e., the points for which the Birkhoff averages of $psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $psi$.
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^alpha$ that the appropriate change of variables reduces equations with $D^alpha$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator $I^alpha$, and study a related analog of the Laplace transform.
We prove equidistribution of Weierstrass points on Berkovich curves. Let $X$ be a smooth proper curve of positive genus over a complete algebraically closed non-Archimedean field $K$ of equal characteristic zero with a non-trivial valuation. Let $L$ be a line bundle of positive degree on $X$. The Weierstrass points of powers of $L$ are equidistributed according to the Zhang-Arakelov measure on the analytification $X^{an}$. This provides a non-Archimedean analogue of a theorem of Mumford and Neeman. Along the way we provide a description of the reduction of Weierstrass points, answering a question of Eisenbud and Harris.
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