M. Nevins and .D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers"

[abstract:] "We describe the trajectories of the successive iterates of the square map and its perturbations on the field of p-adic numbers. We show that the cycles of the square map on Qp arise from cycles of the square map on Fp, and that all nonperiodic trajectories in the unit disk densely define a compact open subset. We find that the perturbed maps x |-> x2 + a, with a inside the unit circle, have similar dynamics to x |-> x2, but that each fundamental cycle arising from Fp can further admit harmonic cycles, for different choices of p and a. In contrast, the cycles of the maps x |-> x2 + a, with a on the boundary of the unit circle, are no longer tied to those of the square map itself. In all cases we give a refined algorithm for computing the finitely many periodic points of the map."

S. Ben-Menahem, "p-Adic iterations", preprint, Tel-Aviv UP (1988) 1627-88

R. Benedetto, "p-Adic dynamics and Sullivan’s no wandering domain theorem", Compositio Mathematica 122 (2000) 281-298

[abstract:] "In this paper we study dynamics on the Fatou set of a rational function f(z) defined over a finite extension Qp, the field of pp-adic dynamics", we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points."

R. Benedetto, "Examples of wandering domains in p-adic polynomial dynamics" (Comptes Rendus Mathématique. Académie des Sciences. Paris , 335 (2002), 615--620.

[abstract:] "For any prime p > 0, we construct p-adic polynomial functions in Cp[z] whose Fatou sets have wandering domains." DVI available on his homepage:

R. Benedetto, "Non-archimedean holomorphic maps and the Ahlfors Islands Theorem" (American Journal of Mathematics, accepted.)

[abstract:] "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."

R. Benedetto, "Components and periodic points in non-archimedean dynamics" Proceedings of the London Mathematical Society 84 no. 3 (2002) 231-256

[abstract:] "We expand the notion of non-archimedean connected components introduced in "Hyperbolic maps in p-adic dynamics". We define two types of components and discuss their uses and applications in the study of dynamics of a rational function f in K(z) defined over a non-archimedean field K. Using this theory, we derive several results on the geometry of such components and the existence of periodic points within them. Furthermore, we demonstrate that for appropriate fields of definition, the conjectures stated in "p-Adic dynamics and Sullivan's No Wandering Domains Theorem", including the No Wandering Domains conjecture, are equivalent regardless of which definition of 'component' is used. We also give a number of examples of p-adic maps with interesting or pathological dynamics."

R. Bendetto, "Hyperbolic maps in p-adic dynamics", Ergodic Theory and Dynamical Systems 21 (2001) 1-11

[abstract:] "In this paper we study the dynamics of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. 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."

S. De Smedt, A. Khrennikov, "A p-adic behaviour of dynamical systems", Rev. Mat. Comput. 12 (1999) 301-323

L. Hsia, "A weak Néron model with applications to p-adic dynamical systems", Composito Math. 100 (1996) 277-304

Hua-Chieh Li, "p-adic periodic points and Sen’s theorem", J. Number Theory 56 no. 2 (1996) 309-318

J. Lubin, "Nonarchimedean dynamical systems", Compositio Math. 94 no,. 3 (1994) 321-346

J. Lubin, "Formal flows on the nonarchimedean open unit disk", Compositio Math. 124 (2000) 123-136

D. Chistyakov, "Fractal geometry for images Of continuous map Of p-adic numbers and p-adic solenoids into Euclidean spaces"

[abstract:] "Explicit formulas are obtained for a family of continuous mappings of p-adic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in $\Qn{2}$ and $\Qn{3}$. In each of the families, the subset of the embeddings is found. For these embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure on the image of $\Qp$ coincides with the Haar measure on $\Qp$. It is proved that under certain conditions, the image of the p-adic solenoid is an invariant set of fractional dimension for a dynamic system. Computer drawings of some fractal images are presented."

D. Chistyakov, "Fractal measures, p-adic numbers and continuous transition between dimensions"

[abstract:] "Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of 'anomalous' fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in Rn, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp in C continuously dependent on "complex dimension" d in C is obtained. This family is such that: 1) if d = 2(1), then the image of b>Qp is C (real axis in C.); 2) the fractal measures coincide with the images of the Haar measure in Qp, and at d = 2(1) they also coincide with the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures of images for any compact set in Qp are holomorphic in d, similarly to the dimensional regularization method in QFT."

V. Anashin, "Uniformly distributed sequences of p-adic integers, II"

[abstract:] "The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In some cases (and especially for p = 2) the descriptions are given by explicit formulae. Some of the results may be viewed as descriptions of ergodic isometric dynamical systems on p-adic unit disk."

E. Thiran, D. Verstegen and J. Weyers, "p-adic dynamics", Journal of Statistical Physics 54 nos. 3-4 (1989) 893-913

D. Verstegen, "p-adic dynamical systems" from Number Theory and Physics (J.-M. Luck, P. Moussa and M. Waldschmidt, eds.), Springer Proceedings in Physics 47 (Springer, 1990) 235-242