Non-archimedean Dynamics Research Articles

How Number Theory Elucidates the Mysteries of Complex Dynamics--Viewed through Non-Archimedean Dynamics

How Number Theory Elucidates the Mysteries of Complex Dynamics--Viewed through Non-Archimedean Dynamics

Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

Abstract The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics. A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston’s rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen’s theorem (and therefore a new proof of Thurston’s theorem) without using quasiconformal maps or Teichmüller theory. We show that, aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen’s theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds: (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field, and (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.

A Survey of Non-Archimedean Dynamics

A Survey of Non-Archimedean Dynamics

J-stability in non-archimedean dynamics

Let Cv be a complete, algebraically closed non-archimedean field, and let f∈Cv(z) be a rational function of degree d≥2. If f satisfies a bounded contraction condition on its Julia set, we prove that small perturbations of f have dynamics conjugate to those of f on their Julia sets.

Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

AbstractWe show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

On a Characterization of Polynomials Among Rational Functions in Non-Archimedean Dynamics

We study a question on characterizing polynomials among rational functions of degree $$>1$$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.

Geometric formulas on Rumely’s weight function and crucial measure in non-archimedean dynamics

We introduce the $f$-crucial function $\operatorname{Crucial}_f$ associated to a rational function $f\in K(z)$ of degree $>1$ over an algebraically closed field $K$ of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely's (resultant) function $\operatorname{ordRes}_f$ in terms of the hyperbolic metric $\rho$ on the Berkovich upper half space $\mathsf{H}^1$ in the Berkovich projective line $\mathsf{P}=\mathsf{P}(K)$. We also obtain geometric formulas for Rumely's weight function $w_f$ and crucial measure $\nu_f$ on $\mathsf{P}^1$ associated to $f$, as well as improvements of Rumely's principal results. As an application to dynamics, we obtain a quantitative equidistribution of the sequence $(\nu_{f^n})_n$ of $f^n$-crucial measures towards the $f$-equilibrium (or canonical) measure $\mu_f$ on $\mathsf{P}^1$.

Rescaling limits in non-Archimedean dynamics

Suppose $\{f_t\}$ is an analytic one-parameter family of rational maps defined over a non-Archimedean field $K$. We prove a finiteness theorem for the set of rescalings for $\{f_t\}$. This complements results of J. Kiwi.

-stability of expanding maps in non-Archimedean dynamics

The aim of this paper is to show $J$-stability of expanding rational maps over an algebraically closed, complete and non-Archimedean field of characteristic zero. More precisely, we will show that for any expanding rational map, there exists a neighborhood of it such that the dynamics on the Julia set of any rational map in the neighborhood is the same as the dynamics of the expanding rational map as a non-Archimedean analogue of a corollary of Mañé, Sad and Sullivan’s result [On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4)16 (1983), 193–217] in complex dynamics.

Equidistribution of the crucial measures in non-Archimedean dynamics

TextLet K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ∈K(z) with deg⁡(ϕ)≥2. In this paper we consider the family of functions ordResϕn(x), which measure the resultant of ϕn at points x in PK1, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function gμϕ(x,x) attached to the canonical measure of ϕ. Following this, we are able to prove an equidistribution result for Rumely's crucial measures νϕn, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ. VideoFor a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00.

Rescaling limits of complex rational maps

We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space. It is shown that any given sequence of maps of degree d ≥ 2 has at most 2 d − 2 dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained by employing tools from nonarchimedean dynamics.

A criterion for potentially good reduction in nonarchimedean dynamics

Let K be a non-archimedean eld, and let 2 K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sucient condition, involving only the xed points of and their preimages, that determines whether or not the dynamical system : P 1 ! P 1 has potentially good reduction. Fix the following notation throughout this paper. K: a eld K: an algebraic closure of K j j v: a non-archimedean absolute value on K OK: the ring of integersfx2 K :jxjv 1g of K MK: the maximal ideafx2 K :jxjv < 1g ofOK k: the residue eld

Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to Lyubich in the archimedean case and due to Favre and Rivera-Letelier for constant targets in the non-archimedean and any characteristic case and for moving targets in the non-archimedean and 0 characteristic case.

Periodic points, linearizing maps, and the dynamical Mordell–Lang problem

Under suitable hypotheses, we prove a dynamical version of the Mordell–Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ : X → X . We also prove a version of the Mordell–Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.

On the chaotic behavior of cubic p-adic dynamical systems

-adic dynamical systems has mainly been developed during the last 15 years [1], [2].Other studies of non-Archimedean dynamics in the neighborhood of a periodic and an almost periodicpoint over global fields using local fields appeared in [3], [4]. In [5], a problem about the completedescription of attractors, Siegel discs of perturbated monomial dynamical systems, was formulated.In [2], certain properties of the perturbated dynamical systems defined by the functions

Wandering domains and nontrivial reduction in non-Archimedean dynamics

Let K be a non-archimedean field with residue field k, and suppose that k is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions φ ∈ K(z) and Rivera-Letelier’s notion of nontrivial reduction. First, if φ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of φ has wandering components by any of the usual definitions of “components of the Fatou set”. Second, we show that if k has characteristic zero and K is discretely valued, then the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate. The theory of complex dynamics in dimension one, founded by Fatou and Julia in the early twentieth century, concerns the action of a rational function φ ∈ C(z) on the Riemann sphere P(C) = C∪{∞}. Any such φ induces a natural partition of the sphere into the closed Julia set Jφ, where small errors become arbitrarily large under iteration, and the open Fatou set Fφ = P (C) Jφ. There is also a natural action of φ on the connected components of Fφ, taking a component U to φ(U), which is also a connected component of the Fatou set. In 1985, using quasiconformal methods, Sullivan [32] proved that φ ∈ C(z) has no wandering domains; that is, for each component U of Fφ, there are integers M ≥ 0 and N ≥ 1 such that φ(U) = φ(U). We refer the reader to [1, 13, 24] for background on complex dynamics. Recall that a non-archimedean metric on a space X is a metric d which satisfies the ultrametric triangle inequality d(x, z) ≤ max{d(x, y), d(y, z)} for all x, y, z ∈ X. In the past two decades, beginning with a study of linearization at fixed points by Herman and Yoccoz [19], there have been a number of investigations of non-archimedean dynamics; for a small sampling, see [4, 5, 6, 10, 20, 27, 28, 31]. It is natural to ask which properties of complex dynamics extend to the non-archimedean setting and which do not. We fix the following notation throughout this paper. K a complete non-archimedean field with absolute value | · | K an algebraic closure of K CK the completion of K OK the ring of integers {x ∈ K : |x| ≤ 1} of K k the residue field of K OCK the ring of integers {x ∈ CK : |x| ≤ 1} of CK k the residue field of CK Date: July 30, 2003. 2000 Mathematics Subject Classification. Primary: 11S80; Secondary: 37F10, 54H20.

On Siegel's linearization theorem for fields of prime characteristic

In 1981, Herman and Yoccoz (1983 Generalizations of some theorems of small divisors to non Archimedean fields Geometric Dynamics (Lecture Notes in Mathematics) ed J Palis Jr, pp 408–47 (Berlin: Springer) Proc. Rio de Janeiro, 1981) proved that Siegel's linearization theorem (Siegel C L 1942 Ann. Math. 43 607–12) is true also for non-Archimedean fields. However, the condition in Siegel's theorem is usually not satisfied over fields of prime characteristic. We consider the following open problem from non-Archimedean dynamics. Given an analytic function f defined over a complete, non-trivial valued field of characteristic p > 0, does there exist a convergent power series solution to the Schröder functional equation (2) that conjugates f to its linear part near an indifferent fixed point? We will give both positive and negative answers to this question, one of the problems being the presence of small divisors. When small divisors are present this brings about a problem of a combinatorial nature, where the convergence of the conjugacy is determined in terms of the characteristic of the state space and the powers of the monomials of f, rather than in terms of the diophantine properties of the multiplier, as in the complex case. In the case that small divisors are present, we show that quadratic polynomials are analytically linearizable if p = 2. We find an explicit formula for the coefficients of the conjugacy, and applying a result of Benedetto (2003 Am. J. Math. 125 581–622), we find the exact size of the corresponding Siegel disc and show that there is an indifferent periodic point on the boundary. In the case p > 2 we give a sufficient condition for divergence of the conjugacy for quadratic maps as well as for a certain class of power series containing a quadratic term (corollary 2.1).

Components and Periodic Points in Non-Archimedean Dynamics

In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.