Let K be a function field in one variable over an arbitrary field F. Given a rational function φ ∈ K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of φ all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial φ, such points exist only if φ is isotrivial. In fact, such K-rational points exist only if φ is defined over the constant field of K after a K-rational change of coordinates.
In this paper we study the dynamics of a rational function \phi\in K(z) defined over some finite extension K of \mathbb{Q}_p. After proving some basic results, we define a notion of ‘components’ of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps.
We present a p -adic and non-archimedean version of some classical complex holomorphic function theory. Our main result is an analogue of the Five Islands Theorem from Ahlfors' theory of covering surfaces. For non-archimedean holomorphic maps, our theorem requires only two islands, with explicit and nearly sharp constants, as opposed to the three islands without explicit constants in the complex holomorphic theory. We also present non-archimedean analogues of other results from the complex theory, including theorems of Koebe, Bloch, and Landau, with sharp constants.
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