Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.

The case of non-expanding dynamics (the ones that satisfy a Lipschitz condition with a constant 1, a 1-Lipschitz for short) on the ring Z p of p-adic integers is sufficiently well studied [4,5,18,19], see also [3] and references therein. However, it is not so much known about the dynamics in domains different from Z p although the later dynamics can be useful in applications to computer science (e.g. in computer simulations, numerical methods like Monte-Carlo, cryptography) and to mathematical physics, see [3], [7] and [24]. Dynamical systems on p-adic spheres are an interesting and nontrivial example of the dynamics. The first result in this direction, namely, the ergodicity criterion for monomial dynamical systems on p-adic spheres, was obtained in [22,23]. It deserves a note that although these dynamical systems are a p-adic counterpart of a classical dynamical systems, circle rotations, in the p-adic case the dynamics exhibit quite another behavior than the classical one. Later the case of monomial dynamical systems on padic spheres was significantly extended: In [6], ergodicity criteria for locally The first of the authors is supported in parts by Russian Foundation for Basic Research grant 12-01-00680-a and by Chinese Academy of Sciences visiting professorship for senior international scientists grant No 2009G2-11. The second and the third of the authors are supported by the joint grant of Swedish and South-African Research Councils "Non-Archimedean analysis: from fundamentals to applications." analytic dynamical systems on p-adic spheres were obtained, for arbitrary prime p.
In the current paper, we consider an essentially wider class of dynamics than in [6], namely, the class of all 1-Lipschitz dynamical systems; however, on 2-adic spheres only: We find necessary and sufficient conditions for ergodicity of these dynamical systems, see further Theorem 3.1. Then with the use of the criterion we find necessary and sufficient conditions for ergodicity of perturbed monomial dynamical systems on 2-adic spheres around 1 in the case when perturbations are 1-Lipschitz and 2-adically small (Theorem 4.1). Earlier similar results were known only under additional restriction (however, for arbitrary prime p) that perturbations are smooth, cf. [6] and [3,Section 4.7]. In this connection it should be noticed that transition of results of the present paper for arbitrary prime p seems to be a non-trivial task: It is well known that in p-adic analysis cases of even and odd primes differ essentially.
Our basic technique is van der Put series. The van der Put series were primarily known only as a tool to find antiderivatives (see [29,28,34]); recently by using the series the authors in [8,9,39] developed a new technique to determine whether a 1-Lipschitz transformation is measure-preserving and/or ergodic on Z p . Our approach seems to be fruitful: The analog of the techniques was successfully applied to determine ergodicity of 1-Lipschitz transformations on another complete non-Archimedean ring, the ring F 2 [[X]] of formal power series over a two-element field F 2 , see recent paper [26].
We remark that as the mappings under consideration are in general not differentiable, it is impossible to apply the technique based on expansion into power series to the case under consideration as the said technique can be used for analytical and smooth dynamical systems only, see e.g. [6]. The van der Put basis is much better adapted to studies of non-smooth dynamics: A special collection of step-like functions, characteristic functions of balls, constitutes the basis. The van der Put basis reflects the ultrameric (non-Archimedean) structure of p-adic numbers, [28,34]. We note that in the p-adic case the linear space consisting of linear combinations of step-like functions is a dense subspace of the space of continuous functions, [34].
The 2-adic spheres are a special case of p-adic spheres; as a matter of fact, 2-adic spheres are 2-adic balls: Denote the p-adic absolute value via | | p ; then, as a p-adic sphere S p −r (a) = {z ∈ Z p : |z − a| p = p −r } of radius p −r centered at a ∈ Z p is a disjoint union of p − 1 balls B p−r−1 (b) = {z ∈ Z p : |z−b| p ≤ p −r−1 } = b+p r+1 Z p of radii p −r−1 cantered at b ∈ {a+p r s : s = 1, 2, . . . , p−1}, in the case p = 2 we get that S 2 −r (a) = B 2 −r−1 (a+2 r ). Fortunately, the problem to determine ergodicity of a 1-Lipschitz transformation on a ball B p −k (a) = a + p k Z p ⊂ Z p can be reduced to the same problem on the whole space Z p .
Indeed, if f is a 1-Lipschitz transformation such that f (a + p k Z p ) ⊂ a + p k Z p , then necessarily f (a) = a + p k y for a suitable y ∈ Z p . Thus, f (a + p k z) = f (a) + p k · u(z) for any z ∈ Z p ; so we can relate to f the following 1-Lipschitz transformation on Z p : It can be shown that the transformation f is ergodic on the ball B p −k (a) if and only if the transformation u is ergodic on Z p . To determine ergodicity of a transformation on the space Z p various techniques may be used, see [3] for details. In the paper, we exploit a version of the idea described above to reduce the case of ergodicity on 2-adic spheres to the case of ergodicity on the whole space Z 2 (cf. further Proposition 3), and we use van der Put series for the latter study since the series turned out to be the most effective technique in the case when p = 2, cf. [8,9,26,39].

Preliminaries
We remind that p-adic absolute value satisfies strong triangle inequality: The p-adic absolute value induces (p-adic) metric on Z p in a standard way: given a, b ∈ Z p , the p-adic distance between a and b is |a − b| p . Absolute values (and also metrics induces by these absolute values) that satisfy strong triangle inequality are called non-Archimedean. Although the strong triangle inequality is the only difference of the p-adic metric from real or complex metrics it results in dramatic differences in behaviour of p-adic dynamical systems compared to that of real or complex counterparts. The space Z p is equipped with a natural probability measure, namely, the Haar measure µ p normalized so that µ p (Z p ) = 1: Balls B p −r (a) of nonzero radii constitute the base of the corresponding σ-algebra of measurable subsets, µ p (B p −r (a)) = p −r . The measure µ p is a regular Borel measure, so all continuous transformations f : Z p → Z p are measurable with respect to µ p . As usual, a measurable mapping f : implies either µ p (S) = 0 or µ p (S) = 1 (in the paper, speaking of ergodic mapping we mean that the mappings are also measure-preserving).
Let a transformation f : Z p → Z p be non-expanding with respect to the p-adic metric; that is, let f be a 1-Lipschitz with respect to the p-adic metric: for all x, y ∈ Z p . The 1-Lipschitz property may be re-stated in terms of congruences rather than in term of inequalities, in the following way.
Given a, b ∈ Z p and k ∈ N = {1, 2, 3, . . .}, the congruence a ≡ b (mod p k ) is well defined: the congruence just means that images of a of b under action of the ring epimorphism modp k : Z p → Z/p k Z of the ring Z p onto the residue ring Z/p k Z modulo p k coincide. Remind that by the definition the epimorphism modp k sends a p-adic integer that has a canonic representa- Note also that we treat if necessary elements from Z/p k Z as numbers from {0, 1, . . . , p k − 1}.
Now it is obvious that the congruence a ≡ b (mod p k ) is equivalent to the inequality |a − b| p ≤ p −k . Therefore the transformation f : Z p → Z p is 1-Lipschitz if and only if it is compatible; that is, The compatibility property implies that given a 1-Lipschitz transformation f : The mapping f mod p k does not depend on the choice of representative z in the ball z + p k Z p (the latter ball is a coset with respect to the epimorphism modp k ); that is, the following diagram commutes: Now we remind definition and basic properties of van der Put series following [28]. Given a continuous p-adic function f : Z p → Z p defined on Z p and valuated in Z p , there exists a unique sequence B 0 , B 1 , B 2 , . . . of p-adic integers such that for all x ∈ Z p , where therefore n = log p m + 1 for all m ∈ N 0 ; we put log p 0 = 0 by this reason. Recall that ⌊α⌋ for a real α denotes the integral part of α, that is, the nearest to α rational integer which does not exceed α.
The ergodic 1-Lipschitz transformations of Z 2 are completely characterized by the following theorem, see [8]: for suitable b m ∈ Z 2 that satisfy the following conditions:

Ergodicity of 1-Lipschitz dynamical systems on 2-adic spheres
In this section we prove ergodicity criterion for 1-Lipschitz dynamics on 2-adic spheres, so further p = 2 and f : Z 2 → Z 2 is a 1-Lipschitz function. Let S 2 −r (a) be a sphere of radius 2 −r with a center at the point a ∈ {0, . . . , 2 r − 1}, and let the sphere S 2 −r (a) be invariant under action of f ; that is, let f (S 2 −r (a)) ⊂ S 2 −r (a). As p = 2, the sphere S 2 −r (a) coincides with the ball B 2 −r−1 (a + 2 r ) of radius 2 −r−1 centered at the point a + 2 r : S 2 −r (a) = a + 2 r + 2 r+1 x : x ∈ Z 2 = B 2 −r−1 (a+2 r ). Therefore the sphere S 2 −r (a) is f -invariant if and only if f (a + 2 r + 2 r+1 Z p ) ⊂ a + 2 r + 2 r+1 Z p ; that is, if and only if (3.1) f (a + 2 r ) ≡ a + 2 r (mod 2 r+1 ) as a 1-Lipschitz function maps a ball of radius 2 −ℓ into a ball of radius 2 −ℓ .
Further, as f is 1-Lipschitz, we can represent f : then g : Z 2 → Z 2 is a 1-Lipschitz function. The following proposition holds: The function f is ergodic on the sphere S 2 −r (a) if and only if f (a + 2 r ) ≡ a + 2 r (mod 2 r+1 ) and the function g defined as Proof. The first of conditions of the proposition just means that the sphere S 2 −r (a) is f -invariant, see (3.1). In view of that condition, Condition 2 of Lemma 1 holds then if and only if the function g is transitive modulo 2 t for all t = 1, 2, 3, . . . However, the latter condition is equivalent to the ergodicity of the function g on Z 2 by Theorem 2.1. Now given a 1-Lipschitz function f : Z 2 → Z 2 , in view of Theorem 2.2 and (2.6), f has a unique representation via van der Put series: (1) f (a + 2 r ) ≡ a + 2 r + 2 r+1 (mod 2 r+2 ); (2) b f (a + 2 r + m · 2 r+1 ) 2 = 1, for m ≥ 1; As g is 1-Lipschitz, Theorem 2.2 yields that for every k = 0, 1, 2, . . . there exists b k ∈ Z p such that B k = b k 2 ⌊log 2 k⌋ . So from (3.7) it follows that if m ≥ 2 and n = ⌊log 2 m⌋ then By Remark 2 to Theorem 2.3, (3.9) proves condition 3 (as well as condition 1 for m = 1) of the theorem under proof.