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Register a new userEstimates of linearization discs in $p$-adic dynamics with application to ergodicity
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Authors
Karl-Olof Lindahl
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We give lower bounds for the size of linearization discs for power series over $mathbb{C}_p$. For quadratic maps, and certain power series containing a `sufficiently large quadratic term, we find the exact linearization disc. For finite extensions of $mathbb{Q}_p$, we give a sufficient condition on the multiplier under which the corresponding linearization disc is maximal (i.e. its radius coincides with that of the maximal disc in $mathbb{C}_p$ on which $f$ is one-to-one). In particular, in unramified extensions of $mathbb{Q}_p$, the linearization disc is maximal if the multiplier map has a maximal cycle on the unit sphere. Estimates of linearization discs in the remaining types of non-Archimedean fields of dimension one were obtained in cite{Lindahl:2004,Lindahl:2009,Lindahl:2009eq}. Moreover, it is shown that, for any complete non-Archimedean field, transitivity is preserved under analytic conjugation. Using results by Oxtoby cite{Oxtoby:1952}, we prove that transitivity, and hence minimality, is equivalent the unique ergodicity on compact subsets of a linearization disc. In particular, a power series $f$ over $mathbb{Q}_p$ is minimal, hence uniquely ergodic, on all spheres inside a linearization disc about a fixed point if and only if the multiplier is maximal. We also note that in finite extensions of $mathbb{Q}_p$, as well as in any other non-Archimedean field $K$ that is not isomorphic to $mathbb{Q}_p$ for some prime $p$, a power series cannot be ergodic on an entire sphere, that is contained in a linearization disc, and centered about the corresponding fixed point.
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We find the exact radius of linearization disks at indifferent fixed points of quadratic maps in $mathbb{C}_p$. We also show that the radius is invariant under power series perturbations. Localizing all periodic orbits of these quadratic-like maps we then show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics of polynomials over $mathbb{C}_p$ where the boundary of the linearization disk does not contain any periodic point.
We consider a family of $(2,1)$-rational functions given on the set of $p$-adic field $Q_p$. Each such function has a unique fixed point. We study ergodicity properties of the dynamical systems generated by $(2,1)$-rational functions. For each such function we describe all possible invariant spheres. We characterize ergodicity of each $p$-adic dynamical system with respect to Haar measure reduced on each invariant sphere. In particular, we found an invariant spheres on which the dynamical system is ergodic and on all other invariant spheres the dynamical systems are not ergodic.
A rational map with good reduction in the field $mathbb{Q}_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $mathbb{P}^1(mathbb{Q}_p)$ over $mathbb{Q}_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $mathbb{P}^1(mathbb{Q}_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and $1$-Lipschitz. It is also proved that a rational map having good reduction of degree $2$, $3$ and $4$ can never be minimal on the whole space $mathbb{P}^1(mathbb{Q}_2)$.
Monomial mappings, $xmapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.
We describe the set of all $(3,1)$-rational functions given on the set of complex $p$-adic field $mathbb C_p$ and having a unique fixed point. We study $p$-adic dynamical systems generated by such $(3,1)$-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each $(3,1)$-rational function on $mathbb C_p$ there is a point $hat x=hat x(f)in mathbb C_p$ which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of $f$) of the points after a finite step come to $hat x$. For a class of $(3,1)$-rational functions defined on the set of $p$-adic numbers $mathbb Q_p$ we study ergodicity properties of the corresponding dynamical systems. We show that if $pgeq 3$ then the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For $p=2$, under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the $(3.1)-$rational functions.
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