Group structure of the $p$-adic ball and dynamical system of isometry on a sphere

Abstract

In this paper the group structure of the p-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations ⊕ and ⊙ on a ball and sphere respectively, and prove that this sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in Zp is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere, and show that the trajectory of any initial point that is not a fixed point isn't convergent. We study ergodicity of this p-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For p ≥ 3 we prove that the dynamical systems are not ergodic. But for p = 2 under some conditions the dynamical system may be ergodic.

Ergodicity and periodic orbits of $p$-adic $(1,2)$-rational dynamical systems with two fixed points

“…In this section we are interested in periodic trajectories and their characteristics. Since our function is an isometry on an invariant sphere, we get the following result about periodic trajectories from [16]: Theorem 7. If the dynamical system (S r (x i ), f ), i = 1, 2 has n-periodic orbit y 0 → y 1 → ... → y n → y 0 , then the following statements hold:…”

“…So, we denote ρ(r) = |f (x) − x| p for all x ∈ S r (x i ), i = 1, 2, r = |c| p . In that case, we have the following assertions from [16]:…”

“…If this compact set has some algebraic structure, then can we look at the natural Haar measure? If the considered compact set is a ball or a sphere, the answer to this question is positive, which is given as follows in [16].…”

“…In [16], given an important results about the dynamics of isometric maps, and since the function (2.3) we are considering is also an isometry, the results obtained in [16] are also relevant for the dynamics of the function (2.3), i.e., if S r (x i ), i = 1, 2 is invariant sphere for the function f given by (2.3), then we have the following:…”

“…• For any initial point x ∈ S r (x i ), i = 1, 2 (except fixed point) the orbit {f n (x)| n ∈ N} isn't convergent. The result of the following Lemma is given as a condition in [16]. Let S r (x i ), i = 1, 2 be invariant sphere for the function f given by (2.3), then we denote ρ(r, x) = |f (x) − x| p for x ∈ S r (x i ).…”