Not The Discrete Group Of Archimedes Domain And Power System

The dynamic systems of polynomials, rational functions and iterated function systems have been studied for a long time in the complex plane. Nowadays, it is very popular to construct the counterpart theories in the p-adic rational number fields Qp and p-adic complex number field Cp which is the completion of the algebraic closure of Qp. Some theories can be easily got on Cp by the nature of non-archimedean fields, but generally speaking, it is more difficult to deal with the theory on Cp, since Qp is not algebraic closed and totally disconnected, and Cp is algebraic closed, but is not locally compact and totally disconnected.The first part includes the discreteness of subgroups in non-archimedean space and the J(?)rgensen inequality of discrete subgroups. Many mathematicians and re-searchers are interested in studying discrete subgroups of non-archimedean fields. The study of discrete subgroups of special linear group of dimension2over Cp is very crucial to the study of algebraic curves whose universal covering space is Mum-ford curve, e.t.c, p-adic Schottky curve. Many mathematicians, such as Mumford, Gerritzen, Manin, Myers, Marius, Voskuil, Kato and Cornelissen made a lot of con-tribution to this field. More details can be found in [39,40,41,68,89,96,97].In [68], Kato classified elements in SL(2, Cp) into three classes:one is elliptic, another is parabolic and the other is loxodromic, and he proved that if a subgroup G(?)SL(2, Cp) is discrete, then G does not contain any parabolic elements and elliptic elements of infinite order. We generalize the classification of Kato to the special linear group of high dimension SL(m, Cp), where m≥2, and we prove that if a subgroup G (?) SL(m, Cp) is discrete, then the subgroup G does not contain any parabolic element and any elliptic element of infinite order. There is a basic theorem of discrete groups in the Klein group, namely if a non-elementary subgroup is discrete, then the subgroup generated by any two elements is discrete. For SL(2, R), J(?)rgensen proved that the subgroup G of SL(2, R) is discrete if and only if any cyclic group of G is discrete in [66]. However, this is not true for SL(2,C) or high dimension. In Cp, we point out that the result is true even for SL(m, Cp), namely G (?) SL(m, Cp) is discrete if and only if any cyclic group of G is discrete. This result shows that the converse proposition of Kato’s result is also true, namely if a group G (?) SL(m, Cp) does not contain any parabolic elements and elliptic elements of infinite order, then G is discrete.In the theory of Klein group, a discrete group G contains elliptic elements only, then G is a finite group. We can generalize the result to the finite extension Kp of Qp, namely if G (?) SL(2, Kp) is discrete and contains elliptic elements only, then G is a finite group. But we construct a counterexample in SL(2, Cp), namely there exists a subgroup G (?) SL(2, Cp) containing infinitely many elements and the norm of all elements in G have a uniformly upper bound, but G is discrete.J(?)rgensen inequality is a necessary condition for the discreteness of subgroups of SL(2,C). J(?)rgensen inequality has been widely applied in many aspects such as the algebraic and geometric convergence of subgroups and the estimation of hyper-bolic manifolds, namely the volume of hyperbolic manifolds have a uniformly lower bound. J(?)rgensen’s inequality was generalized by many authors in various cases. The methods used to prove this inequality have been generalized to a wide variety of dif-ferent contexts but, generally, the statements look rather different from that given by J(?)rgensen. For example, a geometrical interpretation says there is always an embed-ded tubular neighborhood of a very short geodesic in a hyperbolic manifold and that this neighborhood, or "collar" has volume uniformly bounded away from zero. More details can be found in [64,70,72,73,71,92]. Armitage and Parker were the first one who introduced the J(?)rgensen inequality into the non-Archimedean space, and they gave the J(?)rgensen inequality of discrete subgroups of SL(2, QP). They give a geometric interpretation of the J(?)rgensen inequality, namely the translation length of any element in SL(2, Qp) has a uniformly lower bound. Our result improve the result of Armitage and Parker. Furthermore, we make use of the J(?)rgensen inequality and discreteness of subgroups to prove that the quotient space of Berovich hyperbolic space by the action of the discrete subgroup has a ball whose radius has a uniformly lower bound, namely the quotient space is incompressible.In the second part, we study the property of the limit sets of discrete subgroups of SL(2, Cp) and IFSs and Julia sets of rational functions in Cp. With the rapid development of the arithmetic dynamics, the study of Julia sets of rational functions over P1(Cp), limit sets of IFSs, and limit sets of discrete subgroups of SL(2,CP) is a hot spot in mathematics. A lot of mathematicians made a lot of distribution to this field such as Anashin, Baker, Bezivin, Benedetto, DeMarco, Favre, Fan, Hsia, Khrennikov, Rivera-Letelier, Rumely, Silverman and Wang. More details can be found in [23,24,42,43,44,45,46,74,75,76,77,78,79,80,106,107,108,110].In the complex analysis, the limit sets of discrete subgroups and Julia sets of rational functions are important subjects of the theory of Klein group and the complex dynamical system. There are a lot of similarities between limit sets of discrete groups and Julia sets of rational functions. In complex dynamic systems, Julia sets of rational functions of degree more than2are the closure of repelling periodic points; in the Klein group, limit sets of non-elementary discrete groups are the closure of repelling fixed points of loxodromic elements. Naturally, the same questions can be proposed in arithmetic dynamical systems.(1) Are Julia sets of rational functions of degree more than2over P1(CP) the closure of repelling periodic points?(1) Are limit sets of non-elementary discrete subgroups of SL(2, Cp) the repelling fixed points of loxodromic elements?These two questions are two open problems. For the question (1), Bezivin pointed out that if a rational function of degree more than2over P1(CP) has a repelling periodic point, then the Julia set is the closure of repelling periodic points. Okuyama pointed out that if the Lyapunov expotent of a rational function of degree more than2over P1(CP) is positive, then the Julia set is the closure of repelling periodic points. We partially answer this question.In the process of researching SL(2, C), the action of elements of SL(2, C) can be extended to the action of hyperbolic manifold H3={(x,y,t)|t>0,x∈R,y∈R}, namely SL(2,C) is the isometric group of H3={(x, y, t)|x E R, y∈R}. The action of elements of SL(2, Cp) on the P1(Cp) can also be extended into the Berkovic space. Especially, the action of elements of SL(2, Cp) is an isometric action on the Berkovich hyperbolic space HBerk:=PBerk\P1(Cp), namely SL(2,CP) is the isometric group of HBerk.We say that G acts discontinuously on HBerk if and only if for any point x E HBerk of type Ⅱ, and any distinct elements g1,g2,…, the sequence g1(x),g2(x), can not converge to any y in HBerk with respect to the Berkovich topology.We prove that any G (?) SL(2, Cp) which acts discontinuously on HBerk is a discrete subgroup of SL(2, Cp). Furthermore, we prove that a non-elementary subgroup of SL(2,Cp) which acts discontinuously on HBerk should have at least one loxodromic, and its limit set is the closure of repelling fixed points. Thus, we partially affirmatively answer the question (2). We give an example that an elementary discrete subgroup which contains elliptic elements only does not act discontinuously on HBerk.Hence, the condition of " non-elementary "is necessary. If we can prove that a non-elementary discrete subgroup of SL(2,Cp) acts discontinuously on HBerk, then we can give an affirmative answer to the question (2). We point out that if G (?) SL(2, Qp) is discrete, then G acts discontinuously on HBerk.As an application, we prove that limit sets of non-elementary discrete subgroups of SL(2, Qp) are compact.In the Berkovich space, We can define the limit set of discontinuous subgroup of HBerk over the Berkovich space which is denoted by∧, namely the closure of the limit set over P1(CP) with respect to the topology of the Berkovich space. We say that x E PBerk is dynamically stable, if there exists a neighborhood U such that∪n≥{fn(∪)} omits infinitely many points in Berkovich space. We prove that PBerk\∧is dynamical stable, but any point in A is not dynamical stable, and we also get that PBerk\∧is the equicontinuity locus of G. In the complex plane, it is a very important subject to study the geometric metric property of Julia sets of rational functions and limit sets of IFSs. Similarly, we can study the geometric metric property of Julia sets of rational functions over P1(Cp) and limit sets of IFSs over Cp. In the complex plane, it is very important to know whether a set is uniformly perfect. A uniformly perfect set and a domain with uniformly perfect boundary have many excellent analytic and geometric characteristics. More details can be found in [8,100,117,111,113,117,105,104,125]. We prove that Julia sets of rational functions over P1(Cp) which are not empty are uniformly perfect, and limit sets of bi-Lipschitz contractive strictly differentiable IFSs and contractive non-degenerating analytic IFSs with more than two points are uniformly perfect. The alien specialist pointed out that we should prove the doubling property of these sets, since if a complete metric space is uniformly perfect, uniformly disconnected, compact, bounded and doubling, then the metric space is the quasisymmetric image of the standard Cantor set. We partially answer this question. We prove that if a polynomial of degree more than2with the compact Julia set, then the Julia set of the polynomial is doubling and uniformly disconnected, and thus the Julia set of the polynomial is the quasisymmetric image of the standard Cantor set. If the IFS satisfies the condition of separation, namely for any f, g which are the generated elements in IFS,f(∧)∩g(A)=(?). If bi-Lipschitz contractive strictly differentiable IFSs and contractive non-degenerating analytic IFSs satisfy this condition, then limit sets of bi-Lipschitz contractive strictly differentiable IFSs and contractive non-degenerating analytic IFSs with more than two points are doubling which yields that the limit sets are the quasisymmetric images of the standard Cantor set.