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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.
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Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a emph{finest monotone map $ph$ onto a locally connected continuum $J_{sim_P}$}, i.e. a monotone map $ph:Jto J_{sim_P}$ such that for any other monotone map $psi:Jto J$ there exists a monotone map $h$ with $psi=hcirc ph$. Then we extend $ph$ onto the complex plane $C$ (keeping the same notation) and show that $ph$ monotonically semiconjugates $P|_{C}$ to a emph{topological polynomial $g:Cto C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwis fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $ph$ not to collapse $J$ into a point.
We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-circles but the Julia set is non-locally connected.
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.
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.
We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.
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