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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.
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
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 f
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 de
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 pro
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 po