Aihua Fan

Shilei Fan
email: slfan@mail.ccnu.edu.cn

AND Lingmin Liao
email: lingmin.liao@u-pec.fr

Yuefei Wang
email: wangyf@math.ac.cn

MINIMALITY OF p-ADIC RATIONAL MAPS WITH GOOD REDUCTION

Keywords: 2010 Mathematics Subject Classification. Primary 37P05; Secondary 11S82, 37B05 p-adic dynamical system, minimal decomposition, projective line, good reduction, rational map

published or not. The documents may come

INTRODUCTION

The present study contributes to the theory of p-adic dynamical systems which has recently been intensively and widely developed. For this development, one can consult the books [START_REF] Anashin | Applied Algebraic Dynamics[END_REF][START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF][START_REF] Silverman | The arithmetic of dynamical systems[END_REF] and their bibliographies therein.

For a prime number p, let Q p be the field of p-adic numbers and Z p be the ring of integers in Q p . Ergodic theory on the ring Z p is extremely important for applications to automata theory, computer science and cryptology, especially in connection with pseudorandom numbers and uniform distribution of sequences. The minimality, or equivalently the ergodicity with respect to the Haar measure, of non-expanding dynamical systems on Z p are extensively studied in [START_REF] Anashin | Ergodic transformations in the space of p-adic integers[END_REF][START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF][START_REF] Coelho | Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers[END_REF][START_REF] Durand | Minimal polynomial dynamics on the set of 3-adic integers[END_REF][START_REF] Fan | Strict ergodicity of affine p-adic dynamical systems on Zp[END_REF][START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF][START_REF] Fan | Dynamics of the square mapping on the ring of p-adic integers[END_REF][START_REF] Fan | Dynamics of Chebyshev polynomials on Z 2[END_REF][START_REF] Gundlach | On ergodic behavior of p-adic dynamical systems[END_REF][START_REF] Jeong | Toward the ergodicity of p-adic 1-Lipschitz functions represented by the van der Put series[END_REF][START_REF] Larin | Transitive polynomial transformations of residue rings[END_REF][START_REF] Oselies | Ergodische Eigenschaften der Automorphismen p-adischer Zahlen[END_REF], and so on.

The dynamical properties of the fixed points of the rational maps have been studied in the space C p of p-adic complex numbers [START_REF] Albeverio | Sattarov, p-adic (2, 1)-rational dynamical systems[END_REF][START_REF] Khamraev | On a class of rational p-adic dynamical systems[END_REF][START_REF] Mukhamedov | On rational p-adic dynamical systems[END_REF][START_REF] Sattarov | p-adic (3, 2)-rational dynamical systems. p-Adic Numbers Ultrametric[END_REF] and in the adelic space [START_REF] Dragovich | Linear fractional p-adic and adelic dynamical systems[END_REF]. The Fatou and Julia theory of the rational maps on C p , and on the Berkovich space over C p , are also developed [START_REF] Benedetto | Hyperbolic maps in p-adic dynamics[END_REF][START_REF] Baker | Potential theory and dynamics on the Berkovich projective line[END_REF][START_REF] Hsia | Closure of periodic points over a non-Archimedean field[END_REF][START_REF] Rivera-Letelier | Dynamique des fonctions rationnelles sur des corps locaux[END_REF][START_REF] Silverman | The arithmetic of dynamical systems[END_REF]. However, the global dynamical structure of rational maps on Q p remains unclear, though the rational maps of degree one are totally characterized in [START_REF] Fan | On minimal decomposition of p-adic homographic dynamical systems[END_REF] .

Let φ ∈ Q p (z) be a rational map of degree d ≥ 2. Then φ induces a dynamical system on the projective line P 1 (Q p ) over Q p , denoted by (P 1 (Q p ), φ). Let E ⊂ P 1 (Q p ) be a subset such that φ(E) ⊂ E. Then restricted to E, φ defines a subsystem (E, φ| E ). The subsystem (E, φ| E ) is called minimal if for any point x ∈ E, the orbit of x under φ is dense in E. In this article, we suppose that φ has good reduction (see the definition below). The minimality of (P 1 (Q p ), φ) and its subsystems will be fully investigated. As we will see, any rational map φ having good reduction is 1-Lipschitz continuous on P 1 (Q p ) which is equipped with its spherical metric. This suggests that (P 1 (Q p ), φ) shares many properties with the polynomial dynamics on Z p , which are 1-Lipschtz with respect to the metric induced by the p-adic absolute value.

Let denote the reduction modulo p from Z p to Z p /pZ p such that a → a with a ≡ a (mod p). For a polynomial f (z) = d i=0 a i z i ∈ Z p [z], the reduction of f modulo p is defined as

f (z) = d i=0 a i z i .
Note that φ can be written as a quotient of polynomials f, g ∈ Z p [z] having no common factors, such that at least one coefficient of f or g has absolute value 1. Let F p = Z p /pZ p be the finite field of p elements. The degree of a rational map φ, denoted by deg φ, is the maximum of the degrees of its denominator and numerator without common factors. The reduction of φ modulo p is the rational function (of degree at most deg φ) Assume that φ has good reduction. Let ρ(•, •) be the spherical metric on P 1 (Q p ) (see the definition in Section 2). Then the map φ is 1-Lipschitz continuous (everywhere nonexpanding [27, p.59]): ρ(φ(P 1 ), φ(P 2 )) ≤ ρ(P 1 , P 2 ) for all P 1 , P 2 ∈ P 1 (Q p ).

φ(z) = f (z) g(z) ∈ F p (z),
The 1-Lipschitz continuity of φ leads us to investigate the minimality and minimal decomposition of the dynamical system (P 1 (Q p ), φ), as in the case of polynomial dynamical systems on Z p studied in [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF].

Let P 1 (F p ) be the projective line over F p . The reduction φ induces a transformation from P 1 (F p ) into itself. We denote by φ k the k-th iteration of φ. A criterion of the minimality of the system (P 1 (Q p ), φ) is given in the following theorem.

Theorem 1.1. Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction. Then the dynamical system (P 1 (Q p ), φ) is minimal if and only if the following conditions are satisfied:

(1) The reduction φ is transitive on P 1 (F p ).

(

) (φ (p+1) ) (0) ≡ 1 (mod p) and |φ (p+1) (0)| p = 1/p. ( 2
) For the case p = 2 or 3, additionally, |φ (p+1)p (0)| p = 1/p 2 . 3
Under the spherical metric ρ(•, •), P 1 (Q p ) can be considered as an infinite symmetric tree (see Figures 1 and2). This tree is in fact an infinite (p + 1)-Cayley tree. The centered vertex which is the root of the tree, is called Gauss point. Other vertices correspond to balls with radius strictly less than one (with respect to the spherical metric). The points in P 1 (Q p ) are the boundary points of the tree. A vertex is said to be at level n (n ≥ 1), if there are n edges between the vertex and the Gauss point. Then for n ≥ 1, there are (p + 1)p n-1 vertices at level n. Since a rational map with good reduction is 1-Lipschitz continuous with respect to the spherical metric, it will induce an action on the tree under which the Gauss point is fixed and the sets of vertices at the same level are invariant.

Remark that if we consider φ as action on the tree, the condition (1) in Theorem 1.1 means that φ is transitive on the set of vertices at level 1, the conditions (1) and ( 2) imply that φ is transitive on the set of vertices at level 2, while the condition (3) implies that φ 1 FIGURE 1. Tree structure of P 1 (Q 2 ). The points of P 1 (Q 2 ) are considered as the boundary points of the infinite tree. is transitive on the set of vertices at level 3 if conditions (1) and ( 2) are also satisfied. We will see in Theorem 4.1 that the transitivity of φ on these finite levels is sufficient for the minimality of φ on the whole space. The conclusion of Theorem 1.1 can be interpreted as follows.

The dynamical system (P 1 (Q p ), φ) is minimal if and only φ is transitive on the set of vertices at level 2 for p ≥ 5, and at level 3 for p = 2 or 3.

When the system (P 1 (Q p ), φ) is not minimal, we describe the dynamical structure of the system by showing all its minimal subsystems. Theorem 1.2. Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction. We have the following decomposition

P 1 (Q p ) = P M B
where P is the finite set consisting of all periodic points of φ, M = i M i is the union of all (at most countably many) clopen invariant sets such that each M i is a finite union of balls and each subsystem φ : M i → M i is minimal, and points in B lie in the attracting basin of a periodic orbit or of a minimal subsystem. Moreover, the length of a periodic orbit has one of the following forms:

k or k , if p ≥ 5, k or k or kp, if p = 2 or 3, where 1 ≤ k ≤ p + 1 and | (p -1).
The decomposition in Theorem 1.2 will be referred to as the minimal decomposition of the system φ :

P 1 (Q p ) → P 1 (Q p ).
We remark that the possible lengths of periodic orbits were investigated in [START_REF] Zieve | Cycles of Polynomial Mappings[END_REF], see also [27, pp.62-64].

A finite periodic orbit of φ is by definition a minimal set. But for the convenience of the presentation of the paper, only the sets M i in the above decomposition are called minimal components. What kind of set can be a minimal component of a good reduction rational map? In a recent work [START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF], the authors showed that for any 1-Lipschitz continuous map, a minimal component must be a Legendre set and that any Legendre set is a minimal component of some 1-Lipschitz continuous map. Recall that E ⊂ P 1 (Q p ) is a Legendre set (see [8, p. 778]) if for every integer n ≥ 0, every non-empty intersection of E with a ball of radius p -n contains the same number of balls of radius p -(n+1) .

We will show that for a rational map with good reduction, the minimal components M i are Legendre sets of special forms. Further, the dynamics of the minimal subsystems are adding machines. Let (p s ) s≥1 be a sequence of positive integers such that p s |p s+1 for every s ≥ 1. We denote by Z (ps) the inverse limit of Z/(p s Z), called an odometer. The sequence (p s ) s≥1 is called the structure sequence of the odometer. The map z → z + 1 on Z (ps) is the adding machine on Z (ps) . The structures of minimal components are determined as follows.

Theorem 1.3. Let φ ∈ Q p (z) be a rational map of deg φ ≥ 2 with good reduction.

Assume that E ⊂ P 1 (Q p ) is a minimal component of φ. Then φ : E → E is conjugate to the adding machine on an odometer Z (ps) , where

(p s ) = (k, k , k p, k p 2 , • • • )
with integers k and satisfying that 1 ≤ k ≤ p + 1 and | (p -1).

In applications, one usually would like to construct or determine a minimal rational (polynomial) map by its coefficients. However, this is far from easy, although a criterion of the minimality of rational maps with good reduction is given in Theorem 1.1. For rational maps of degree one, a complete description is given in [START_REF] Fan | On minimal decomposition of p-adic homographic dynamical systems[END_REF]. Here we give the description of rational maps of degree at least two with good reduction, but only for p = 2. It seems to be a much more harder task for p ≥ 3.

For a rational map φ ∈ Q p (z) of degree at least 2, the number of periodic points of a fixed period must be finite. Hence, there exists a z 0 ∈ Q p such that z 0 , φ(z 0 ), φ 2 (z 0 ) are distinct. Consider the linear fraction

g(z) = (z -z 0 )(φ 2 (z 0 ) -φ(z 0 )) (z -φ(z 0 ))(φ 2 (z 0 ) -z 0 ) .
Then g(z 0 ) = 0, g(φ(z 0 )) = ∞ and g(φ 2 (z 0 )) = 1. Let ψ = g •φ•g -1 be the conjugation of φ by g. Then we have ψ(0) = ∞ and ψ(∞) = 1 and the rational function ψ can be written as

ψ(z) = a 0 + a 1 z + • • • + a d-1 z d-1 + z d b 1 z + • • • + b d-1 z d-1 + z d where a i , b j ∈ Q p (0 ≤ i < d, 1 ≤ j < d) and d ≥ 2
is the degree of φ. Therefore, without loss of generality, we can always assume that φ(0) = ∞ and φ(∞) = 1.

The following theorem provides, in some sense, a principle for constructing minimal rational maps on P 1 (Q 2 ). For a degree d ≥ 2 rational map of form

φ(z) = a 0 + a 1 z + • • • + a d-1 z d-1 + a d z d b 1 z + • • • + b d-1 z d-1 + b d z d (1.1) with a i , b j ∈ Q 2 and a d = b d = 1, we set A φ := i≥0 a i , B φ := j≥1 b j , A φ,1 := i≥0 a 2i+1 , A φ,2 := i≥0 a 4i+1 and A φ,3 := i≥0 a 4i+3 .
Theorem 1.4. Let φ be defined as (1.1). If φ has good reduction and is minimal on

P 1 (Q 2 ), then                                  a i , b j ∈ Z 2 , for 0 ≤ i ≤ d -1 and 1 ≤ j ≤ d -1, a 0 ≡ 1 (mod 2), B φ ≡ 1 (mod 2), A φ ≡ 2 (mod 4), A φ,1 ≡ 1 (mod 2), b 1 ≡ 1 (mod 2), a d-1 -b d-1 ≡ 1 (mod 2), a 0 b 1 (a d-1 -b d-1 )(A φ,2 -A φ,3 )B φ + 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 + A φ,3 ) ≡ 1 (mod 4). (1.2)
Conversely, the condition ( 1.2) implies that φ is 1-Lipschitz continuous and minimal on

P 1 (Q 2 ).
Corollary 1.5. Let φ ∈ Q 2 (z) be a rational map of degree 2, 3 or 4 with good reduction. Then the dynamical system (P 1 (Q 2 ), φ) is not minimal.

A complete characterization of minimal polynomial maps on Z p in terms of their coefficients are given in [START_REF] Larin | Transitive polynomial transformations of residue rings[END_REF] for p = 2 and in [START_REF] Durand | Minimal polynomial dynamics on the set of 3-adic integers[END_REF] for p = 3. However, there is still no complete description for p ≥ 5. On the other hand, the characterizations of minimal Mahler series and van der Put series by their coefficients are investigated in [START_REF] Anashin | Uniformly distributed sequences of p-adic integers[END_REF][START_REF] Anashin | Applied Algebraic Dynamics[END_REF] and in [START_REF] Anashin | Ergodicity criteria for non-expanding transformations of 2adic spheres[END_REF][START_REF] Jeong | Toward the ergodicity of p-adic 1-Lipschitz functions represented by the van der Put series[END_REF] respectively.

The above condition (1.2) could be quickly checked by computer, since there are only arithmetic operations '+, -, ×' in the equations of (1.2). By condition (1.2) and a quick computer computation, we obtain a series of minimal rational maps in Q 2 (z) of degree 4 which is 1-Lipschitz but do not have good reduction. See the table in Section 5.2.

The paper is organized as follows. In Section 2, we give some preliminaries of P 1 (Q p ), including the spherical metric and the tree structure of P 1 (Q p ). Section 3 is devoted to the induced dynamical systems on the vertices. The proofs of Theorems 1.1-1.3 are given in Section 4. In Section 5, we characterize the minimal rational maps with good reduction in terms of their coefficients for the case p = 2. Theorem 1.4 and Corollary 1.5 will be proved in this section. To illustrate our result, in the last section, we present two rational maps with good reduction whose exact minimal decompositions are described when p = 3.

PROJECTIVE LINE

Any point in the projective line P 1 (Q p ) may be given in homogeneous coordinates by a pair [x 1 : x 2 ] of points in Q p which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor λ ∈ Q * p :

[x 1 : x 2 ] = [λx 1 : λx 2 ].
The field Q p may be identified with the subset of P 1 (Q p ) given by

[x : 1] ∈ P 1 (Q p ) | x ∈ Q p .
This subset contains all points in P 1 (Q p ) except one: the point of infinity, which may be given as

∞ = [1 : 0]. 1 FIGURE 2. Tree structure of P 1 (Q 3 )
. The points of P 1 (Q 3 ) are considered as the boundary points of the infinite tree.

The spherical metric defined on P 1 (Q p ) is analogous to the standard spherical metric on the Riemann sphere.

If P = [x 1 , y 1 ] and Q = [x 2 , y 2 ] are two points in P 1 (Q p ), we define ρ(P, Q) = |x 1 y 2 -x 2 y 1 | p max{|x 1 | p , |y 1 | p } max{|x 2 | p , |y 2 | p } or, viewing P 1 (Q p ) as Q p ∪ {∞}, for z 1 , z 2 ∈ Q p ∪ {∞} we define ρ(z 1 , z 2 ) = |z 1 -z 2 | p max{|z 1 | p , 1} max{|z 2 | p , 1} if z 1 , z 2 ∈ Q p , and
ρ(z, ∞) = 1, if |z| p ≤ 1, 1/|z| p , if |z| p > 1.
Remark that the restriction of the spherical metric on the ring Z p := {x ∈ Q p , |x| ≤ 1} of p-adic integers is same as the metric induced by the absolute value | • | p .

A rational map φ ∈ Q p (z) induces a transformation on P 1 (Q p ). Rational maps are always Lipschitz continuous on P 1 (Q p ) with respect to the spherical metric (see [27, Theorem 2.14.]). In particular, rational maps with good reduction are 1-Lipschitz continuous.

Lemma 2.1 ([27], p.59). Let φ ∈ Q p (z) be a rational map with good reduction. Then the map φ :

P 1 (Q p ) → P 1 (Q p ) is 1-Lipschitz continuous: ρ(φ(P ), φ(Q)) ≤ ρ(P, Q), ∀P, Q ∈ P 1 (Q p ).
For a ∈ P 1 (Q p ) and an integer n ≥ 1, denote by

B n (a) := {x ∈ P 1 (Q p ) : ρ(x, a) ≤ p -n } the ball of radius p -n centered at a. The projective line P 1 (Q p ) consists of p + 1 disjoint balls of radius p -1 , P 1 (Q p ) = B 1 (∞) 0≤i≤p-1 B 1 (i) .
For each integer n ≥ 1, every ball of radius p -n consists of p disjoint balls of radius

p -n-1 . For example, B 1 (∞) consists of B 2 (1/p), B 2 (2/p), • • • , B 2 ((p -1)/p), B 2 (∞).
As mentioned in Section 1, the projective line P 1 (Q p ) over Q p could be considered as an infinite (p + 1)-Caylay tree: the branch index of each vertex is p + 1, i.e. each vertex is an endpoint of p + 1 edges. There is a centered vertex which is called Gauss point. Excluding Gauss point, each vertex corresponds to a ball in P 1 (Q p ) of radius strictly less then one (with respect to the spherical metric). The Gauss point can be considered as the ball of radius 1 centered at the origin which is actually the whole space P 1 (Q p ). The set of edges of the tree consists of the pairs of balls (B, B ) of radius ≤ 1 such that

B ⊂ B, r(B) = p • r(B )
where r(B) and r(B ) are the radii of balls B and B respectively.

The points in P 1 (Q p ) are the boundary points of the tree. Remind that a vertex is said to be at level n(n ≥ 1) if there are n edges between Gauss point and the vertex. A vertex at level n corresponds a ball of radius p -n . The projective line P 1 (Q p ) consists of (p + 1)p n-1 disjoint balls of radius p -n . There are (p + 1)p n-1 vertices at level n. For example, see Figures 1 and2 for the tree structures of P 1 (Q 2 ) and P 1 (Q 3 ).

INDUCED DYNAMICS

For each positive integer n, P 1 (Q p ) consists of (p + 1)p n-1 disjoint balls of radius p -n . Denote by B n the set of the (p + 1)p n-1 disjoint balls of radius p -n . In general, any 1-Lipschitz continuous map φ :

P 1 (Q p ) → P 1 (Q p ) induces a transformation on B n . Denote by φ n the induced map of φ on B n , i.e. φ n (B n (x)) = B n (φ(x)), ∀x ∈ P 1 (Q p ).
As P 1 (Q p ) is considered as an infinite tree, for each positive integer n, there is a one-toone correspondence between B n and the vertices of the tree at level n. The 1-Lipschitz continuous map φ induces a transformation on the tree under which the set of vertices of each level is invariant.

Many properties of the dynamics φ on P 1 (Q p ) are linked to those of φ n on B n .

Proposition 3.1. Let φ : P 1 (Q p ) → P 1 (Q p ) be a 1-Lipschitz continuous map. Then the system (P 1 (Q p ), φ) is minimal if and only if the finite system (B n , φ n ) is minimal for all integers n ≥ 1.

The equivalence in Proposition 3.1 is similar to that for 1-Lipschitz continuous maps on the ring Z p or on a discrete valuation domain (see [1, p. 111], [2, Theorem 1.2] and [8, Corollary 4]).

Proof. The " only if " part of the statement is obvious, we prove only the " if " part. Suppose that (P 1 (Q p ), φ) is not minimal. Then there exist an open set S and a point z ∈ P 1 (Q p ) such that φ k (z) / ∈ S for all integers k ≥ 1. Since the set S is open, there exists a ball B n0 (z 0 ) ⊂ S for some z 0 ∈ P 1 (Q p ) and some integer n 0 large enough. Recall that (B n , φ n ) is minimal for all integers n > 0. So there exists an integer k 0 such that φ k0 n0 (B n0 (z)) = B n0 (z 0 ). This implies ρ(φ k0 (z), z 0 ) ≤ p -n0 , which contradicts the fact that φ k0 (z) / ∈ S.

Recall that a rational map with good reduction is 1-Lipschitz continuous. For a rational map φ ∈ Q p (z) with good reduction, to study the minimality of the system (P 1 (Q p ), φ), it suffices to study the induced dynamics (B n , φ n ) for all integers n ≥ 1. Now let us first study the dynamics (B 1 , φ 1 ) at level 1. Note that each point in P 1 (F p ) corresponds to a ball in B 1 . The induced map φ 1 of φ can be considered as the reduction φ acting on P 1 (F p ). The map φ : P 1 (F p ) → P 1 (F p ) admits some periodic orbits. A periodic orbit is call a cycle of φ. The points outside any cycle will be mapped into some cycle after several iterations.

Let

(x 1 , • • • , x k ) ⊂ P 1 (F p ) be a cycle of φ.
To simplify notation, we consider a transformation f ∈ PGL 2 (Z p ) with f (x 1 ) = 0, where x 1 is a point in the ball corresponding to x 1 . Replacing x 1 and φ with 0 and f • φ • f -1 respectively, we may assume that x 1 = 0. Under this assumption, we will see in the following that φ k acting on pZ p is conjugate to a power series acting on Z p .

For two rational maps θ, ω ∈ Q p (z) with good reduction, the composition θ•ω has good reduction, and θ • ω = θ • ω (see [27, p.59, Theorem 2.18]). So φ k has good reduction and 0 is a fixed point of φ k . We write φ k in the form

φ k (z) = a 0 + a 1 z + • • • a n z n b 0 + b 1 z + • • • b n z n with coefficients a 0 , • • • a d , b 0 , • • • b d ∈ Z p and at least one coefficient in Z p \ pZ p .
The fact that 0 is a fixed point of φ k implies that φ k (0) = a 0 /b 0 ≡ 0 (mod p). Since φ k has good reduction, we have a 0 ∈ pZ p and b 0 ∈ Z p \ pZ p . Otherwise, the numerator and denominator of the reduction have the common factor z, which implies deg

φ k < deg φ k .
Multiplying numerator and denominator by b -1 0 , we may thus write φ k in the form

φ k (z) = a 0 + a 1 z + • • • a n z n 1 + b 1 z + • • • b n z n .
Then by the following Lemma 3.2, φ k is 1-Lipschitz from pZ p to itself and

φ k (z) = a 0 + λz + λ 2 z 2 + λ 3 z 3 + • • • , with λ i ∈ Z p for all i ≥ 2.
Lemma 3.2. Let φ ∈ Q p (z) be a rational map of the form

a 0 + a 1 z + • • • a n z n 1 + b 1 z + • • • b n z n , with a j , b j ∈ Z p .
Then φ is 1-Lipschitz on pZ p . Furthermore,

φ(z) = a 0 + λz + λ 2 z 2 + λ 3 z 3 + • • • (3.1)
with λ i ∈ Z p for all i ≥ 2.

Proof. The Taylor expansion of φ around z = 0, which in this case can be obtained by simple long division, gives

φ(z) = a 0 + λz + A(z) 1 + zB(z) z 2
with A(z), B(z) ∈ Z p [z] and λ = φ (0) ∈ Z p . Observe that on pZ p , we have

1 1 + zB(z) = 1 -zB(z) + z 2 (B(z)) 2 -z 3 (B(z)) 3 + • • • .
Hence we can write φ(z) as the form (3.1). Obviously, φ induces a 1-Lipschitz map on pZ p . Now we study the dynamical system (pZ p , φ k ). For simplicity, we write ϕ instead of φ k . Note that every power series

ϕ(z) = ∞ i=0 λ i z i ∈ Z p [[z]]
converges on pZ p . If λ 0 ∈ pZ p , then ϕ(pZ p ) ⊂ pZ p and ϕ induces a 1-Lipschitz map from pZ p to itself.

In the remainder of this section we assume

ϕ(z) = ∞ i=0 λ i z i ∈ Z p [[z]
] with λ 0 ∈ pZ p . The dynamical system (pZ p , ϕ) is conjugate to a system (Z p , χ) by the transformation f (z) = z/p, i.e.

pZ p ϕ / / z/p pZ p z/p Z p χ / / Z p ,
where

χ(z) = ∞ i=0 p i-1 λ i z i ∈ Z p [[z]
] converges on Z p and induces a map from Z p to itself.

The dynamical system of χ on Z p and its induced dynamics on Z p /p n Z p was studied in [START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF]. Actually, the convergent series with coefficient in the integral ring O K of a finite extension K of Q p were studied as dynamical systems on O K in [START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF]. The system (Z p , θ) is a special case when K = Q p .

In the following we shall translate results about θ in the language of results about ϕ. For more details on this subject, the reader may consult [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF][START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF].

For each n ≥ 1, denote by ϕ n the induced map of ϕ on pZ p /p n Z p , i.e.,

ϕ(x mod p n ) ≡ ϕ(x) (mod p n ),
for all x ∈ pZ p . In the sprit of Proposition 3.1, the minimality and minimal decomposition of the system (pZ p , ϕ) can be deduced form its induced dynamics (pZ p /p n Z p , ϕ n ). So we need to study the finite systems (pZ p /p n Z p , ϕ n ). To this end, the main idea, which comes from [START_REF] Desjardins | Polynomial mappings mod p n[END_REF][START_REF] Zieve | Cycles of Polynomial Mappings[END_REF], is to study the behaviour of systems (pZ p /p n Z p , ϕ n ) by induction. We refer the reader to [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF][START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF] for the application of this idea to give a minimal decomposition theorem for any convergent power series in

Z p [[z]]. A collection σ = (x 1 , • • • , x k ) of k distinct points in pZ p /p n Z p is called a cycle of ϕ n of length k or a k-cycle at level n, if ϕ n (x 1 ) = x 2 , • • • , ϕ n (x i ) = x i+1 , • • • , ϕ n (x k ) = x 1 . Set X σ := k i=1 X i where X i := {x i + p n t + p n+1 Z p ; t = 0, 1, • • • , p -1} ⊂ pZ p /p n+1 Z p . Then ϕ n+1 (X i ) ⊂ X i+1 (1 ≤ i ≤ k -1) and ϕ n+1 (X k ) ⊂ X 1 .
In the following we are concerned with the behavior of the finite dynamics ϕ n+1 on the set X σ and determine all cycles in X σ of ϕ n+1 , which will be called lifts of σ. Remark that the length of any lift σ of σ is a multiple of k. Let X i = x i + p n Z p be the closed disk of radius p -n corresponding to x i ∈ σ and

X σ := k i=1 X i
be the clopen set corresponding to the cycle σ.

Let ψ := ϕ k be the k-th iteration of ϕ. Then, any point in σ is fixed by ψ n , the n-th induced map of ψ. For any point x ∈ X σ , we have (ψ(x) -x)/p n ∈ Z p . Let

α n (x) := ψ (x) = k-1 j=0 ϕ (ϕ j (x)) (3.2) β n (x) := ψ(x) -x p n = ϕ k (x) -x p n . ( 3.3)
One can check that α n (x) (mod p) is always constant on X σ . We should remark that under the condition that α n ≡ 1 (mod p), β n (mod p) is also constant on X σ [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF][START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF] . We distinguish the following four behaviours of ϕ n+1 on X σ : (a) If α n ≡ 1 (mod p) and β n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves a single cycle of length pk. In this case we say σ grows.

(b) If α n ≡ 1 (mod p) and β n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves p cycles of length k. In this case we say σ splits.

(c) If α n ≡ 0 (mod p), then ϕ n+1 restricted to X σ preserves one cycle of length k and the remaining points of X σ are mapped into this cycle. In this case we say σ grows tails.

(d) If α n ≡ 0, 1 (mod p), then ϕ n+1 restricted to X σ preserves one cycle of length k and p-1 cycles of length k , where is the order of α n in the multiplicative group F p \{0}. In this case we say σ partially splits.

For n ≥ 1, let σ = (x 1 , . . . , x k ) ⊂ pZ p /p n Z p be a k-cycle and let σ be a lift of σ which is a of ϕ n+1 contained in X σ . To illustrate the change of nature for a cycle to its lifts, we shall show the relationship between (α n , β n ) and (α n+1 , β n+1 ). For the calculation of (α n+1 mod p, β n+1 mod p) from (α n mod p, β n mod p), the interested reader may consult to [14, Lemma 2] and [16, p.1636]. The following propositions predict the behaviours of the lifts of a cycle σ. (1) If σ grows or splits, then any lift σ grows or splits.

(2) If σ grows tails, then the single lift σ also grows tails.

(3) If σ partially splits, then the lift σ of the same length as σ partially splits, and the other lifts of length k grow or split. Proposition 3.4 ( [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF][START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF]). Let σ be a growing cycle of ϕ n and σ be the unique lift of σ.

(1) If p > 3 and n ≥ 1, then σ also grows.

(2) If p = 3 and n ≥ 2, then σ also grows.

(3) If p = 2 and σ grows, then the lift of σ grows.

For more details of Proposition 3.4, we refer the reader to [14, Proposition 2] for the first two assertion, and to [14, ). Then by Proposition 3.4, a criterion of the minimality of the system ϕ on pZ p can be obtained.

Corollary 3.5. The dynamical system (pZ p , ϕ) is minimal if and only if (1) the finite system (pZ p /p 3 Z p , ϕ 3 ) is minimal for p = 2 or 3;

(2) the finite system (pZ p /p 2 Z p , ϕ 2 ) is minimal for p ≥ 5.

As we investigate the induced dynamical systems level by level, the possible periods of ϕ on pZ p can also be obtained, as a consequence of Proposition 3.4.

Corollary 3.6. Let ϕ(z) = ∞ i=0 λ i z i ∈ Z p [[z]
] with λ 0 ∈ pZ p and consider the dynamical system (pZ p , ϕ).

(1) If p > 3, the period of a periodic orbit must be a factor of p -1.

(2) If p = 3, the period of a periodic orbit must be 1, 2 or 3.

(3) If p = 2, the period of a periodic orbit must be 1 or 2.

Furthermore, the dynamical structure of ϕ on pZ p is fully illustrated by the following minimal decomposition.

Theorem 3.7 ([16], Theorem 1.1). Let ϕ(z) = ∞ i=0 λ i z i ∈ Z p [[z]] with λ 0 ∈ pZ p .
Suppose ϕ n = id for all n ≥ 1. We have the following decomposition

pZ p = P M B,
where P is the finite set consisting of all periodic points of ϕ, M = i M i is the union of all (at most countably many) clopen invariant sets such that each M i is a finite union of balls and each subsystem ϕ : M i → M i is minimal, and the points in B lie in the attracting basin of P M.

The sets M i in the above decomposition are called minimal components. To completely characterize the dynamical system (pZ p , ϕ), each minimal component is described as the adding machine on an odometer by giving the structure sequence of the odometer.

Theorem 3.8 ([16], Theorem 1.1). Let ϕ(z) = ∞ i=0 λ i z i ∈ Z p [[z]] with λ 0 ∈ pZ p . If E ⊂ pZ p
is a minimal clopen invariant set of ϕ, then ϕ : E → E is conjugate to the adding machine on the odometer Z (ps) , where

(p s ) = ( , p, p 2 , • • • )
where ≥ 1 is an integer dividing p -1.

PROOF OF THEOREMS 1.1-1.3

Proof of Theorem 1.1. We begin with proving the " if " part of the theorem. The reduction φ is minimal on P 1 (F p ) which implies that φ 1 is minimal on B 1 . Let ψ = φ p+1 be the p+1-th iteration of φ. Then the ball B 1 (0) = pZ p is an invariant set of ψ. By the argument of Section 3, ψ can be written as a power series with coefficients in Z p which converges on pZ p , i.e.

ψ(z) = λ 0 + λ 1 z + λ 2 z 2 + λ 3 z 3 + • • • with λ i ∈ Z p
for all i ≥ 1 and λ 0 ∈ pZ p . Noting that φ is 1-Lipschitz, ψ is also 1-Lipschitz. The minimality of the dynamical system (pZ p , ψ) leads to ψ is isometric on pZ p , see [START_REF] Anashin | Applied Algebraic Dynamics[END_REF][START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF]. So φ is isometric on P 1 (Q p ) with respect to the spherical metric. Hence, ψ is minimal on each ball of radius 1/p which implies the minimality of (P 1 (Q p ), φ). It suffices to show that the dynamical system (pZ p , ψ) is minimal.

For n = 1, the cycle (0) is the unique cycle of ψ 1 . The condition (φ (p+1) ) (0) ≡ 1 (mod p) implies that the cycle (0) grows or splits. By the condition |φ (p+1) (0)| p = 1/p, the cycle (0) grows which then means that the dynamical system (pZ p /p 2 Z p , ψ 2 ) is minimal.

For the cases p > 3, it follows from Corollary 3.5 that the dynamical system (pZ p , ψ) is minimal.

Note that (φ p(p+1) ) (0) ≡ (φ (p+1) ) (0) ≡ 1 (mod p). For the cases p = 2 or 3, the additional condition implies that the unique lift of (0) at level 2 grows. Thus the system (pZ p /p 3 Z p , ψ n ) is minimal. By Corollary 3.5, the dynamical system (pZ p , ψ) is minimal. Now we prove the " only if " part. Let φ be a minimal rational map with good reduction. By Proposition 3.1, the system (B n , φ n ) is minimal for all integers n ≥ 1. The minimality of system (B 1 , φ 1 ) implies that the reduction φ is minimal on P 1 (F p ), while the minimality of system (B 2 , φ 2 ) implies that the cycle (0) of ψ 1 = φ p+1 1 grows. Hence we obtain (φ (p+1) ) (0) ≡ 1 (mod p) and |φ (p+1) (0) -0| p = 1/p. Furthermore, by the minimality of system (B 3 , φ 3 ), the unique lift of (0) of φ (p+1) at level 2 grows. So we have |φ p(p+1) (0

)| p = 1/p 2 .
In the above proof of Theorem 1.1, we have indeed established the following result.

Theorem 4.1. Let φ : P 1 (Q p ) → P 1 (Q p ) be a rational map of deg φ ≥ 2 with good reduction. Then the dynamical system (P 1 (Q p ), φ) is minimal if and only if the following condition is satisfied:

(1) the system (B 3 , φ 3 ) is minimal for p = 2 or 3, (2) the system (B 2 , φ 2 ) is minimal for p ≥ 5.

Proof of Theorem 1.2. The space P 1 (Q p ) is a union of p + 1 balls with radius p -1 . Each ball can be identified with a point in P 1 (F p ). The reduction map φ on P 1 (F p ) admits some cycles. By iteration of φ, the points outside the cycles are attracted by the cycles. The ball corresponding to such a point will be put into the third part B in the minimal decomposition.

Let (x 1 , • • • , x k ) ⊂ P 1 (F p ) be a cycle of φ. Without loss of generality, we assume x 1 = 0. Let ψ = φ k be the k-th iteration of φ. Then the ball B 1 (0) = pZ p is an invariant set of ψ. Noting that φ is a rational map of deg φ ≥ 2, φ n is a rational map of degree (deg φ) n . So φ n = id for all n ≥ 1. By Theorem 3.7, the dynamical system (pZ p , ψ) has a minimal decomposition which then gives a minimal decomposition of the system

( k i=1 B 1 (x i ), φ).
Applying the same argument to all the cycles of φ, we obtain the minimal decomposition of the whole system (P 1 (Q p ), φ).

Moreover, by Corollary 3.6, the possible lengths of periodic orbits are obtained.

Proof of Theorem 1.3. Let k denote the length of the induced periodic orbit of φ on the minimal set E at the first level. Each point at this first level corresponds to a point in

P 1 (F p ). So k ≤ p + 1.
Without loss of generality, we assume that 0 ∈ E. The set B 1 (0) ∩ E is invariant by φ k . Consider the dynamical system (B 1 (0) ∩ E, φ k ). Noting φ have good reduction, by (3.1) and Theorem 3.8, we deduce that φ k : B 1 (0) ∩ E → B 1 (0) ∩ E is conjugate to the adding machine on the odometer Z (ps) , where

(p s ) = ( , p, p 2 , • • • ) with | (p -1)
, which implies φ : E → E is conjugate to the adding machine on the odometer Z (ps) , where

(p s ) = (k, k , k p, k p 2 , • • • ).

MINIMAL RATIONAL MAP FOR THE

CASE p = 2 5.1. Minimal Conditions. Let φ(z) = a 0 + a 1 z + • • • + a d-1 z d-1 + a d z d b 1 z + • • • + b d-1 z d-1 + b d z d be a rational map of degree d ≥ 2 with a i , b j ∈ Q 2 and a d = b d = 1. Recall that A φ = i≥0 a i , B φ = j≥1 b j , A φ,1 = i≥0 a 2i+1 , A φ,2 = i≥0 a 4i+1 and A φ,3 = i≥0 a 4i+3 .
Since φ(0) = ∞, for the convenience of calculation, we shall select a suitable coordinate. Let

ψ(z) := 1 φ(z) = b 1 z + • • • + b d-1 z d-1 + b d z d a 0 + a 1 z + • • • + a d-1 z d-1 + a d z d (5.1) and ϕ(z)
:= φ( 1 z ) = a d + a d-1 z + • • • + a 1 z d-1 + a 0 z d b d + b d-1 z + • • • + b 1 z d-1 . (5.2)
Then we have

φ 3 = φ • ϕ • ψ. Lemma 5.1. If φ has good reduction and (B 1 , φ 1 ) is minimal, then        a i , b j ∈ Z 2 , for 0 ≤ i ≤ d -1 and 1 ≤ j ≤ d -1, a 0 ≡ 1 (mod 2), A φ ≡ 0 (mod 2), B φ ≡ 1 (mod 2). (5.3)
Conversely, the condition (5.3) implies that φ is 1-Lipschitz continuous and (B 1 , φ 1 ) is minimal. Moreover, the Taylor expansion of φ 3 at 0 is of the form

φ 3 (z) = φ(1) + λz + λ 2 z 2 + λ 3 z 3 + • • • , where λ, λ 2 , λ 3 • • • ∈ Z 2 .
Proof. Assume that φ has good reduction, the coefficients a i and b j are in Z 2 . Otherwise, the degree of the reduction map of φ is strictly less than d, which implies that φ has bad reduction. Let

φ(z) = f (z) g(z) = a 0 + a 1 z + • • • + a d-1 z d-1 + z d b 1 z + • • • + b d-1 z d-1
+ z d be the reduction of φ modulo p. The condition that φ has good reduction implies that the polynomials f , g have no common zero. As we have already indicated above, we can assume φ(0) = ∞ and φ(∞) = 1. So we have a 0 ≡ 1 (mod 2). The minimality of the system (B 1 , φ 1 ) means that φ(1) = 0. So we have f (1) = 0 which implies A φ ≡ 0 (mod 2). Since the polynomials f , g have no common zero, we have g(1) = 0 which means that B φ ≡ 1 (mod 2).

Since

A φ = i≥0 (2i + 1)a 2i+1 + i≥1 2ia 2i ≡ i≥0 a 2i+1 (mod 2),
we have A φ,1 ≡ 1 (mod 2).

By the last condition φ(1) ≡ 2 (mod 4) in (5.5), we immediately get

A φ ≡ 2 (mod 4).
For the proof of Theorem 1.4, we need calculate the second derivative of φ 3 at 0. For simplicity, denote A φ := i≥2 i(i -1)a i and B φ := i≥2 i(i -1)b i . Before the proof let us first calculate the second derivatives ψ (0), ϕ (0) and φ (1):

ξ 1 := ψ (0) = 2b 2 a 2 0 -2a 1 b 1 a 0 a 3 0 , ξ 2 := ϕ (0) = 2(a d-2 -b d-2 ) + 2(b 2 d-1 -a d-1 b d-1 ), ξ := φ (1) = A φ B 2 φ -B φ A φ B φ + 2(A φ (B φ ) 2 -A φ B φ B φ ) B 3 φ .
Now we are ready to prove Theorem 1.4.

Proof of Theorem 1.4. Under the condition that φ has good reduction, Theorem 4.1 implies that (P 1 (Q p ), φ) is minimal if and only if (B 3 , φ 3 ) is minimal which is equivalent to the following three conditions (B 2 , φ 2 ) is minimal, (φ 6 ) (0) ≡ 1 (mod 2), and φ 6 (0) ≡ 4 (mod 8).

(5.6)

In the following, we will characterize the three conditions of (5.6). Firstly, the minimality of the system (B 2 , φ 2 ) has already been characterized by coefficients in Lemma 5.2. Then we obtain all the conditions in (1.2) except the last one.

Secondly, if the first condition of (5.6) is satisfied then by (5.5),

(φ 6 ) (0) ≡ ((φ 3 ) (0)) 2 ≡ 1 (mod 2).
Thus the second condition of (5.6) is in fact included in the first condition of (5.6). Finally, we are going to investigate the relation among the coefficients by using the last condition φ 6 (0) ≡ 4 (mod 8) of (5.6) which will lead to the last condition of (1.2).

Note that all the Taylor's coefficients of φ 3 expanded at z = 0 belong to Z 2 . Hence for z ∈ 2Z 2 , we have

φ 3 (z) ≡ φ(1) + (φ 3 ) (0)z + (φ 3 ) (0) 2 z 2 (mod 8).
Thus φ 6 (0) ≡ φ(1) + (φ 3 ) (0)φ(1) + (φ 3 ) (0) 2 φ(1) 2 (mod 8).

(5.7)

Notice that φ(1) ≡ 2 (mod 4) and (φ 3 ) (0) ≡ 1 (mod 2). Then dividing both sides of (5.7) by φ(1), we deduce that the condition φ 6 (0) ≡ 4 (mod 8) is equivalent to

1 + (φ 3 ) (0) + (φ 3 ) (0) 2 φ(1) ≡ 2 (mod 4),
which in turn, by noting that (φ 3 ) (0)/2 ∈ Z 2 , is equivalent to (φ 3 ) (0) + (φ 3 ) (0) ≡ 1 (mod 4).

(5.8)

To continue, let us calculate (φ 3 ) (0) (mod 4), (φ 3 ) (0) (mod 4) and simplify their expressions. Using a 0 ≡ 1 (mod 2) and B φ ≡ 1 (mod 2), we have 1/a 0 ≡ a 0 (mod 4) and B 2 φ ≡ 1 (mod 4). Then

(φ 3 ) (0) ≡ a 0 b 1 (a d-1 -b d-1 )(A φ B φ -A φ B φ ) (mod 4).
Since A φ ≡ 2 (mod 4) and α 1 ≡ α 2 ≡ 1 (mod 2), we have

(φ 3 ) (0) ≡ a 0 b 1 (b d-1 -b d-1 )A φ B φ -A φ B φ (mod 4).
For the second derivative of φ 3 at 0, by the chain rule, we have

(φ 3 ) (0) = ηη 2 ξ 1 + ηη 2 1 ξ 2 + ξη 2 1 η 2 2 .
(5.9)

Since η ≡ η 1 ≡ η 2 ≡ 1 (mod 2), we have η 2 ≡ η 2 1 ≡ η 2 2 ≡ 1 (mod 4). Consequently, we obtain (φ 3 ) (0) ≡ ηη 2 ξ 1 + ηξ 2 + ξ (mod 4).

(5.10)

Apply the expressions of ξ, ξ 1 and ξ 2 to Equation (5.10). Using the facts that a 0 ≡ b 1 ≡ a d-1 -b d-1 ≡ 1 (mod 2) and B φ ≡ 1 (mod 2), we deduce

(φ 3 ) (0) ≡ 2ηη 2 (a 0 b 2 -a 1 b 1 ) + 2η(a d-2 -b d-2 + b 2 d-1 -a d-1 b d-1 )+ + A φ B φ -2A φ B φ -A φ B φ (mod 4) ≡ 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 ) + A φ B φ -2A φ B φ -A φ B φ (mod 4).
Observing that B φ ≡ 0 (mod 2) and A φ ≡ 0 (mod 2), we have

(φ 3 ) (0) + (φ 3 ) (0) ≡ a 0 b 1 (a d-1 -b d-1 )A φ B φ -A φ B φ + 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 ) + A φ B φ -2A φ B φ (mod 4).
Since A φ ≡ 2 (mod 4) and A φ ≡ A φ,1 ≡ 1 (mod 2), we get

(φ 3 ) (0) + (φ 3 ) (0) ≡ a 0 b 1 (a d-1 -b d-1 )A φ B φ + 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 ) + A φ B φ (mod 4).

Note that

A φ ≡ i≥0 a 4i+1 + 2 i≥0 a 4i+2 - i≥0 a 4i+3 (mod 4) and A φ ≡2 i≥0 a 4i+2 + 2 i≥0 a 4i+3 (mod 4). Then (φ 3 ) (0) + (φ 3 ) (0) ≡a 0 b 1 (a d-1 -b d-1 )(A φ,2 -A φ,3 )B φ + 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 ) + 2 i≥0 a 4i+2 + 2( i≥0 a 4i+2 + A φ,3 ) (mod 4) ≡a 0 b 1 (a d-1 -b d-1 )(A φ,2 -A φ,3 )B φ + 2(b 2 -a 1 + a d-2 -b d-2 + b d-1 + A φ,3 ) (mod 4).
By (5.8), we obtain the last condition of (1.2). Hence, we obtain the first part of Theorem 1.4.

For the other part of Theorem 1.4, note that the first four conditions of (1.2) imply the 1-Lipschitz continuity of φ. Further, φ 3 can be written as a form of power series with coefficients in Z p . Thus from the proof of Theorem 1.1, we deduce that the minimality of φ is equivalent to the conditions (5.6). Hence the above proof of the first part leads to the conclusion of the second part of the theorem.

Proof of Corollary 1.5. Remark that the good reduction property and minimality of φ imply that φ is isometric on P 1 (Q 2 ).

For a rational map φ of degree 2 which has good reduction and is minimal on P 1 (Q 2 ), each point has either 0 or 2 pre-images since φ can not have critical point in P 1 (Q 2 ). This contradicts to the fact that φ is isometric.

For a rational map φ of degree 3 is defined as (1.1), the good reduction property and minimality give the reduction φ(z) = (z-1)f (z) zg(z) , where f, g ∈ F 2 [z] are two quadratic irreducible polynomials. However, 1+z+z 2 is the unique quadratic irreducible polynomial over F 2 . Hence, f = g, which contradicts that φ has good reduction.

For a rational map φ of degree 4 which is defined as (1.1), the good reduction property and minimality give the reduction φ(z) = (z-1)f (z) zg(z) , with f, g ∈ F 2 [z] being two different cubic irreducible polynomials. It is known that 1 + z + z 3 and 1 + z 2 + z 3 are the only two cubic irreducible polynomials over F 2 . Hence, φ(z) can only have two cases:

φ(z) = (z -1)(1 + z + z 3 ) z(1 + z 2 + z 3 ) or φ(z) = (z -1)(1 + z 2 + z 3 ) z(1 + z + z 3 ) .
By simple calculations, in both cases, a 3 -b 3 ≡ 0 (mod 2), which contradicts the seventh equation in (1.2).

5.2.

Minimal rational maps of degree 4. This section is devoted to investigate the minimal rational maps of degree 4 which are 1-Lipschitz continuous and their induced orbits on B 3 . For a rational map

φ(z) = a 0 + a 1 z + • • • + a d-1 z d-1 + a d z d b 1 z + • • • + b d-1 z d-1 + b d z d of degree d with a i , b j ∈ Q 2 and a d = b d = 1. Assume that φ 1-Lipschitz continuous and minimal on P 1 (Q 2 ). The orbit of φ 1 on B 1 is 0 → ∞ → 1 → 0.
For the convenience of investigating the induced orbit on B n with n ≥ 2, we choose the local coordinate around ∞ by f (z) = 1/z. Denote by B 1 (0) the image of B 1 (∞) under f . we have the following communicating graph.

d d d d d d B 1 (0) B 1 (0) ψ φ q φ B 1 (∞) c I φ f B 1 (1) ϕ '
Actually, the two balls B 1 (0) and B 1 (0) have no difference. To avoid confusion, the elements in B 1 (0) are denoted by z, such as 1, 2.

Let P 1 (Q 2 ) be the disjoint union of B 1 (0), B 1 (0) and B 1 (1), i.e.

P 1 (Q 2 ) = B 1 (0) B 1 (0) B 1 (1).
Then we can define a map φ from P 1 (Q 2 ) to itself as follows:

φ(z) :=    ψ(z), if z ∈ B 1 (0); ϕ(z), if z ∈ B 1 (0); φ(z), if z ∈ B 1 (1).
Instead of investigating the dynamical system (P 1 (Q 2 ), φ), we study the dynamical system ( P 1 (Q 2 ), φ), so that we can do modulo calculation by computer. We obtain all rational maps of degree 4 with coefficients in {0, 1, 2, 3} which have good reduction and are minimal on P 1 (Q p ). We associate each of the 12 vertices at level 3 a label i,

where i ∈ {0, 1, 2 • • • 7}, if the vertex corresponds to the ball B n (i), i,
where i ∈ {0, 2, 4, if the vertex corresponds to the ball B n (1/i).

The rational maps and their induced orbits at level 3 are showed in the following table.

Coefficients a 0 , a 1 , a 2 , a 3 , b 1 , b 2 , b 3 Induced periodic orbits at level 3 1, 0, 1, 3, 3, 1, 0 1, 1, 1, 2, 3, 2, 3 1, 2, 1, 1, 3, 3, 2 1, 2, 3, 3, 3, 3, 0 0 → 0 → 1 → 6 → 6 → 3 1, 3, 1, 0, 3, 0, 1 → 4 → 4 → 5 → 2 → 2 → 7 1, 3, 3, 2, 3, 0, 3 3, 2, 1, 3, 1, 1, 0 3, 3, 1, 2, 1, 2, 3 1, 0, 1, 3, 1, 1, 2 1, 1, 1, 2, 1, 2, 1 1, 2, 1, 1, 1, 3, 0 1, 2, 3, 3, 1, 3, 2 0 → 0 → 1 → 6 → 2 → 3 1, 3, 1, 0, 1, 0, 3 → 4 → 4 → 5 → 2 → 6 → 7 1, 3, 3, 2, 1, 0, 1 3, 2, 1, 3, 3, 1, 2 3, 3, 1, 2, 3, 2, 1 1, 0, 1, 3, 1, 1, 0 1, 1, 1, 2, 1, 2, 3 1, 2, 1, 1, 1, 3, 2 1, 2, 3, 3, 1, 3, 0 0 → 0 → 1 → 2 → 6 → 3 1, 3, 1, 0, 1, 0, 1 → 4 → 4 → 5 → 6 → 2 → 7 1, 3, 3, 2, 1, 0, 3 3, 2, 1, 3, 3, 1, 0 3, 3, 1, 2, 3, 2, 3

Coefficients a 0 , a 1 , a 2 , a 3 , b 1 , b 2 , b 3 Induced periodic orbits at level 3 1, 0, 1, 3, 3, 1, 2 1, 1, 1, 2, 3, 2, 1 1, 2, 1, 1, 3, 3, 0 1, 2, 3, 3, 3, 3, 2 0 → 0 → 1 → 2 → 2 → 3 1, 3, 1, 0, 3, 0, 3 → 4 → 4 → 5 → 6 → 6 → 7 1, 3, 3, 2, 3, 0, 1 3, 2, 1, 3, 1, 1, 2 3, 3, 1, 2, 1, 2, 1 1, 0, 3, 1, 3, 1, 0 1, 1, 3, 0, 3, 2, 3 3, 0, 1, 1, 1, 3, 0 3, 0, 3, 3, 1, 3, 2 0 → 0 → 1 → 6 → 6 → 7 3, 1, 1, 0, 1, 0, 3 → 4 → 4 → 5 → 2 → 2 → 3 3, 1, 3, 2, 1, 0, 1 3, 2, 3, 1, 1, 1, 0 3, 3, 3, 0, 1, 2, 3 1, 0, 3, 1, 1, 1, 2 1, 1, 3, 0, 1, 2, 1 3, 0, 1, 1, 3, 3, 2 3, 0, 3, 3, 3, 3, 0 0 → 0 → 1 → 6 → 2 → 7 3, 1, 1, 0, 3, 0, 1 → 4 → 4 → 5 → 2 → 6 → 3 3, 1, 3, 2, 3, 0, 3 3, 2, 3, 1, 3, 1, 2 3, 3, 3, 0, 3, 2, 1 1, 0, 3, 1, 1, 1, 0 1, 1, 3, 0, 1, 2, 3 3, 0, 1, 1, 3, 3, 0 3, 0, 3, 3, 3, 3, 2 0 → 0 → 1 → 2 → 6 → 7 3, 1, 1, 0, 3, 0, 3 → 4 → 4 → 5 → 6 → 2 → 3 3, 1, 3, 2, 3, 0, 1 3, 2, 3, 1, 3, 1, 0 3, 3, 3, 0, 3, 2, 3 1, 0, 3, 1, 3, 1, 2 1, 1, 3, 0, 3, 2, 1 3, 0, 1, 1, 1, 3, 2 3, 0, 3, 3, 1, 3, 0 0 → 0 → 1 → 2 → 2 → 7 3, 1, 1, 0, 1, 0, 1 → 4 → 4 → 5 → 6 → 6 → 3 3, 1, 3, 2, 1, 0, 3 3, 2, 3, 1, 1, 1, 2 3, 3, 3, 0, 1, 2, 1 6. SOME EXAMPLES Example 6.1. Let p = 3 and φ(x) = -2z 2 +2z+1 z 3 -3z 2 +z+1 . The dynamical system (P 1 (Q p ), φ) is minimal.

The reduction mod 3 of φ is φ(z) = z 2 + z + 2 z 3 + z + 1 .

So φ has good reduction. Consider the map φ : P 1 (F 3 ) → P 1 (F 3 ). By a simple calculation, we get φ(1) = ∞, φ(∞) = 0, φ(0) = 2, φ(2) = 1. Thus φ is transitive on P 1 (F 3 ). Further, using the software Mathmatica, we can check (φ 4 ) (1)) ≡ (φ 4 ) (0) ≡ 1 (mod 3).

Let σ = (1, ∞, 0, -1) be the cycle at level 1. Since φ 4 (1) ≡ 7 (mod 3 3 ), σ grows. Let σ be the lift of σ at level 2. By calculating φ 12 (1) ≡ 10 (mod 3 3 ), we deduce that σ grows which then implies that σ grows forever. Therefore, the system (P 1 (Q p ), φ) is minimal. Example 6.2. Let p = 3 and φ(z) = 2z+3 (z-1)(z-2) . The dynamical system (P 1 (Q p ), φ) is not minimal and we decompose P 1 (Q p ) as

P 1 (Q p ) = B 1 (0) P 1 (Q p ) \ B 1 (0) ,
where B 1 (0) is a minimal component of φ and the points in P 1 (Q p ) \ B 1 (0) are attracted into B 1 (0).

One can check that φ(z) = 2z z 2 +2 . Then deg φ = deg φ, which implies that φ has good reduction. First, we consider the map φ : P 1 (F 3 ) → P 1 (F 3 ). Since φ(1) = ∞, φ(2) = ∞, φ(∞) = 0 and φ(0) = 0, it follows that φ(B 1 (1))) ⊂ B 1 (∞), φ(B 1 (2)) ⊂ B 1 (∞), φ(B 1 (∞)) ⊂ B 1 (0) and φ(B 1 (0)) ⊂ B 1 (0). So P 1 (Q p ) \ B 1 (0) lies in the attracting basin of B 1 (0). It suffices to study the subsystem (B 1 (0), φ). We shall show that the system (B 1 (0), φ) is minimal. In fact, the derivative of φ at 0 is 13/4. Thus φ (0) ≡ 1 (mod 3). Let σ = (0) be the cycle at level 1. Then σ grows or splits. A simple calculation shows that

  obtained by canceling common factors in the reductions f (z), g(z). If deg φ = deg φ, we say φ has good reduction. If deg φ < deg φ, we say φ has bad reduction.

Proposition 3 . 3 (

 33 [START_REF] Fan | On minimal decomposition of p-adic polynomial dynamical systems[END_REF] Proposition 1, and[START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF] Proposition 4.4). Let σ be a cycle of ϕ n .

  Corollary 1] for the third assertion. Remark that the results in Proposition 3.4 are presented only for polynomials in Z p [z]. Actually, these results also hold for convergent series in Z p [[z]] (see[START_REF] Fan | Dynamics of convergent power series on the integral ring of a finite extension of Qp[END_REF] Propositions 4.7, 4.8, 4.10, 4.11] for a more general setting).It is now well known that the subsystem (E, ϕ) is minimal if and only if the induced map ϕ n : E/p n Z p → E/p n Z p is minimal for any n ≥ 1 (see[1, p. 111], [2, Theorem 1.2] and[START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF] Corollary 4]

Let E ⊂ pZ p be a ϕ-invariant compact open set. Let E/p n Z p := {x ∈ Z p /p n Z p : ∃ y ∈ E such that x ≡ y (mod p n )}.

φ(0) ≡ 15 (mod 3 3 ), φ(15) ≡ 3 (mod 3 3 ), φ(3) ≡ 18 (mod 3 3 ).

So σ grows, by Proposition 3.4, the lift of σ grows too. It then follows that σ grows forever. Hence, the system (B 1 (0), φ) is minimal.

A. H. FAN was supported by NSF of China (Grant No. 11471132); S. L. FAN was supported by NSF of China (Grant No.s 11401236 and 11611530542); L. M. LIAO was supported by 12R03191A-MUTADIS; Y. F. WANG was supported by NSF of China (Grant No. 11231009).

Now we assume that the coefficients of φ satisfy the condition (5.3). By (5.3), we can check directly that (B 1 , φ 1 ) is minimal.

By Lemma 3.2, the conditions A φ ≡ 0 (mod 2), and B φ ≡ 1 (mod 2) imply that φ(z + 1) is 1-Lipschitz from pZ p to pZ p and can be written as the form of (3.1). Recall that φ 3 = φ • ϕ • ψ and note that ψ(0) = 0 and ϕ(0) = 1. It suffices to prove that ψ and ϕ -1 are both 1-Lipschitz from pZ p to itself and have the form (3.1). These can be obtained by direct calculations and by applying Lemma 3.2.

In the following lemma, we will calculate the derivative of φ 3 at 0. Using the chain rule, we obtain

For simplicity, we denote A φ := i≥1 ia i and B φ := i≥1 ib i . By calculation, we have

Lemma 5.2. Assume that the rational map φ has good reduction. Then the system

Proof. The rational map φ has good reduction and (B 2 , φ 2 ) is minimal implies that (B 1 , φ 1 ) is also minimal. So pZ p is φ 3 -invariant. By the classification of the lifts of the cycles, (B 2 , φ 2 ) is minimal if and only if

) is minimal, (φ 3 ) (0) ≡ 1 (mod 2), and φ 3 (0)φ(1) ≡ 2 (mod 4) (5.5)

under the condition that φ has good reduction. By Lemma 5.1, (B 1 , φ 1 ) is minimal if and only if a 0 ≡ 1 (mod 2), B φ ≡ 1 (mod 2) and A φ ≡ 0 (mod 2).

Recall that

The second condition (φ 3 ) (0) ≡ 1 (mod 2) of (5.5) implies that


Abderrahim Okba

Alexandru Takacs

Hervé Aubert

Samuel Charlot

Pierre-François Calmon

Multi

Multiband Rectenna for Microwave Applications Rectenna

Keywords: electromagnetic energy harvesting, satellite health monitoring, multiband rectenna, cross dipole antenna array. Mots-clés-récupération d'énergie électromagnétique, rectenna multi-bandes, antenne, satellite géostationnaire

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The last decade has known a significant development of a wide range of wireless electronic devices such as RFID systems and wireless sensors networks. For many critical applications (e.g. space, aeronautic and medical applications) the sensors used to monitor physical and structural quantities should be wireless and energetically autonomous. The energy, under its different forms (kinetic, thermal, solar, radio waves…) can be used to power these sensors. In this work, the use of the radio waves as energy source for autonomous power sensors is addressed. The autonomy of the wireless sensors can be achieved mainly by two techniques: (i) wireless power transfer and (ii) energy harvesting. These are two important concepts of "energy in Radiosciences", the first one consists of transmitting an electromagnetic energy generated by a dedicated/intentional transmitter and convert this energy into DC on the wireless sensor node by using a rectenna (an acronym coming from rectifier antenna), and the second one consists of harvesting the energy of the existing/surrounding radio waves by using a rectenna. In geostationary telecommunication satellites, high gain microwave antennas are mounted on the external panels of the satellite for communication broadcasting. The radiofrequency (RF) power radiated by such antennas is in the range of 100W. Consequently, an important amount of electromagnetic energy is available around the satellite and illuminates the (antenna) panels. The electric field generated by the spill-over loss of these microwave antennas can reach the following highest levels (effective values): 40 V/m in C-band, 49.5 V/m in X-band, 106 V/m in Ku-band, and 127 V/m in K-band [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. Thus, the harvesting of the electromagnetic energy was identified as a realistic solution to power autonomous wireless sensors deployed along the surface of the panels for structure health monitoring [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. The rectenna (antenna + rectifier circuit) plays a crucial role in receiving and converting RF power into DC power. It consists of two main blocks: the first block is the antenna used to capture the RF energy and converts it into an RF signal, the second block is the rectifier that converts this RF signal into a DC voltage used to power an autonomous wireless sensor. This paper presents recent results obtained in Ku and K bands from using an innovative rectenna topology reported in [START_REF] Okba | Cross dipole rectenna for satellite health monitoring[END_REF] and developed in [START_REF] Okba | Multiband Rectenna for microwave applications[END_REF] and [START_REF] Okba | Cross dipoles rectenna for microwave applications[END_REF], is composed of four printed cross dipoles antenna array and uses a single low cost Schottky diode [5]. Rectennas using a cross dipole antenna were also reported in [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF], [START_REF] Sun | Design of a High-Efficiency 2.45-GHz Rectenna for Low-Input-Power Energy Harvesting[END_REF] while a rectenna using a two cross dipoles array was reported in [START_REF] Takacs | Rectenna Design for K Band Application[END_REF]. The rectenna topology and the design methodology are presented in Section II. The experimental results are reported in Section III and show that very good performances can be achieved in three frequency bands by using a single rectenna. The proposed compact topology is an extension of the rectenna using two cross dipoles antenna array reported in [START_REF] Takacs | Rectenna Design for K Band Application[END_REF].

II. RECTENNA TOPOLOGY AND DESIGN

In order to implement a multiband rectenna, two topologies can be considered. They are shown in Fig. 1. They consist of a multiband antenna which captures the ambient electromagnetic energy and converts it into a RF signal. This RF signal is then transferred to the rectifier via an impedance matching circuit/bandpass filter for maximizing the transfer of power and minimizing the return loss at the operating frequencies. The rectifier (e.g., a Schottky diode) is another key element of a rectenna and must be chosen carefully. Its role is to rectify the RF signal and a low-pass filter is then used in order to suppress the fundamental frequency component and its harmonics. The sensing device (typically a resistor) is modeled as a load impedance to be powered.

As shown in Fig. 1 a) the first possible topology uses only one rectifier for addressing all the targeted frequencies. As a drawback this topology generally requires a multiband/broadband matching network that can be very challenging to design/implement in practice. A rectenna based on this topology is reported in [START_REF] Kuhn | A Multi-Band Stacked RF Energy Harvester with RF-to-DC Efficiency Up to 84%[END_REF], [START_REF] Niotaki | A Compact Dual-Band Rectenna Using Slot-Loaded Dual Band Folded Dipole Antenna[END_REF]. Another solution is shown in the Fig. 1 b). A specific rectifier is used with its own impedance matching circuit at each targeted frequency. Then all the obtained DC powers are combined for powering the load (i.e. the sensing device). The rectenna proposed in [START_REF] Song | A High-Efficiency Broadband Rectenna for Ambient Wireless Energy Harvesting[END_REF] is based on this topology. For compactness purposes the topology shown in Fig. 1 a) is adopted here. Moreover, by adopting this topology, a non-resonant matching technique is used. The matching is carried out by controlling the impedances of the antenna and the rectifier. From a theoretically point of view, another topology can be proposed: it consists of using a specific rectenna (with a narrow band/single band antenna) for each frequency band and then combining the harvested DC powers. Nevertheless this solution leads to very large design. The proposed topology is composed of an array of four cross dipoles antenna array (4CDAA) in parallel and a silicon Schottky diode -Skyworks SMS7630 [5] mounted in shunt configuration. In order to select the topology of our antenna several requirements were taken into account: (i) the rectenna should be as compact as possible (we note that the size of our design is determined mainly from the size of its antenna), (ii) the gain should be as higher as possible in order to increase the rectenna efficiency, (iii) the antenna should have a multiband behavior. As we will show later the 4CDAA topology represents a good trade-off between the previously mentioned criteria. The best way to feed the 4CDAA is to use Coplanar Strip Lines (CPS). Thus the use of a single diode in a shunt configuration seems to be the best solution to implement the rectifier. Then the rectifier is composed by (i) a silicon SMS7630 diode mounted in shunt configuration, (ii) a shunt capacitor to implement the DC low-pass filter, (iii) the CPS lines and via-holes used to interconnect the lumped components (diode and capacitance). From an electrical point of view the rectifier views the input impedance of the CDAA as shunt impedance.

The PCB layout and a photo of the manufactured rectenna are shown in Fig. 2. The antenna array and the diode are implemented on the top of the PCB while the RF shunt capacitor is mounted on the bottom side of the PCB. A metallic plate is added on the bottom side in order to enhance the antenna gain. This compact topology (its size is fixed by the size of the antenna array) is an extension of the rectenna using two cross dipoles antenna array 2CDAA reported in [START_REF] Takacs | Rectenna Design for K Band Application[END_REF]. Moreover, a non-resonant matching technique is used here. Instead of using a dedicated matching circuit that acts as a (resonant) band-pass filter we try here to implement directly the (conjugate) matching requirements by properly controlling the input impedances of the 4CDAA and of the rectifier. Then, the matching circuit is virtually suppressed and leads to a more compact structure. The main goal is to achieve the best impedance matching at the resonant frequencies of the 4CDAA.

The rectifier consists of the diode and the low pass RC filter (composed of the shunt capacitor and the load itself). The 4CDAA can be viewed as an array formed by four cross dipole arrays printed on the top of the PCB. Each cross dipole is an array with two dipoles in cross configuration as shown in Fig. 2. The substrate used is the Rogers RT/Duroïd 6002 (substrate thickness: 0.508 mm, relative dielectric permittivity: 2.94 and dielectric loss tangent: 0.0012) [START_REF]RT/duroid®6002 high frequency laminates[END_REF]. III. DESIGN OF THE RECTENNA Some specific technical requirements for space applications must be fulfilled for designing an efficient rectenna: (i) the rectenna topology have to be as simple as possible; (ii) all the components in the rectenna (dielectric substrate, diode, capacitors, etc.) must be qualified for space applications, ready to be qualified or at least with performances suitable for space requirements; (iii) the operating frequencies must be chosen from the electromagnetic environment existing on board of satellite antenna panels. An analysis of the electromagnetic energy available on the panels at different frequencies is reported in [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF].

Design of the antenna

Intensive full-wave electromagnetic simulations were performed by using Ansys HFSS software in order to design the 4CDAA structure. The diode was replaced by a voltage port while the shunt capacitor and the load were modeled as lumped RLC boundaries as shown in Fig. 3. The antenna was initially simulated and optimized for 12 GHz, but it also displays a good impedance matching at two other frequencies that corresponds to the higher resonant modes of the 4CDAA. The simulated return loss shows a multi-band behavior (12 GHz, 17.6 GHz and 20.2 GHz). Fig. 4 displays the simulated results regarding the return loss and the input impedance versus frequency. where n is the number of dipoles (𝑛𝑛 =8 in our case) and 𝐺𝐺𝐺𝐺 is the gain of each (lossless) dipole. This gain can be increased when a reflector is properly positioned below the antenna surface. The expected maximum gain is 14.2 dBi when half-wavelength dipoles (𝐺𝐺𝐺𝐺 =2.15 dBi) are used. This gain should be considered as an upper limit for any practical 4CDAA design. There are no accurate closed-form expressions for designing such arrays but some rules and methodology can be applied in order to start the design. The length of each dipole (Ld in Fig 2 ) should be approximately half wavelength and the distance between the center of two opposite cross dipoles (L2 in Fig. 2) should be an integer multiple of half wavelength. The angle between the dipoles (α in Fig 2) impacts the polarization of the radiated field and the maximum achievable gain. CoPlanar Strip (CPS) lines are used to feed each cross dipole.

We target here a linear polarization but other polarizations (e.g. circular polarization) can be obtained by controlling the angle α between the crossed dipoles and the relative phase at the input of each cross dipole. Depending on the system requirements, the microwave antennas of the broadcasting satellites may operate using the circular or linear polarization [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. The linear polarization is adopted here because the (in house) experimental setup used for rectenna characterization has been built up by using a linearly polarized horn antenna. The diode is mounted at the center of the array as shown in Fig. 2. Six metallic via holes were used to connect the top and the bottom side of the PCB. A shunt capacitor is mounted on the bottom side of the PCB. The main dimensions of the fabricated (optimized) rectenna are: L1= 4 mm, L2= 15.3 mm, Ld= 10.6 mm, Lx=Ly= 19.5 mm.

At the first operating frequency (f 0 =12GHz), the length of each dipole (Ld) is close to the half wavelength (Ld~λ/2) and the distance between the center of the opposite dipoles (L2) is close to wavelength (L2~λ). The second operating/resonant frequency is f 1 ~1.5f 0 . At this frequency, Ld~0.7λ and L2~1.5λ. The third operating/resonant frequency is f 2 ~2f 0 . At this frequency Ld~λ and L2~2λ. Fig. 5 shows the simulated radiation patterns (gain) of 4CDAA at the 3 resonant (operating) frequencies 12 GHz, 17.6 GHz and 20.2 GHz and Fig. 6 shows the surface current density on the rectenna. From a theoretical point of view, at the third frequency (20.2 GHz), the radiation pattern is expected to be null along the Oz axis. This is not the case because Ld is not exactly equal to λ and also because an electromagnetic coupling exists between the cross dipoles. The designed 4CDAA exhibits the following simulated gain (values obtained for θ=φ=0°): 11.7 dBi at 12 GHz, 11.7 dBi at 17.6 GHz and 12.7 dBi at 20.2 GHz. As expected the simulated gains for the 4CDAA are much higher than the theoretical gain of a single dipole (2.15 dBi for an ideal half-wavelength dipole).

Rectifier

The diode is key element of the rectifier. The choice of an appropriate diode is crucial and must be done carefully in the design of a high-efficient rectifying circuit. This is because the diode is the main source of losses and its performance impacts significantly the performances of rectenna. The equivalent circuit of the diode/rectifier is shown in Fig. 7. It consists of the series resistance (RS), a nonlinear junction resistance (Rj) derived from the I-V characteristic of the diode and a nonlinear junction capacitance (𝐶𝐶 𝑗𝑗 ). Due to the self-biasing nature of the shunt connected diode, 𝐶𝐶 𝑗𝑗 depends on the bias voltage across the diode as follows [START_REF] Yoo | Theoretical and Experimental Development of 10 and 35 GHz Rectennas[END_REF], [START_REF] Guo | An Improved Analytical Model for RF-DC Conversion Efficiency in Microwave Rectifiers[END_REF].

𝐶𝐶 𝑗𝑗 = 𝐶𝐶 𝑗𝑗0 � 𝑉𝑉 𝑓𝑓 𝑉𝑉 0 + 𝑉𝑉 𝑓𝑓
where 𝐶𝐶 𝑗𝑗0 is zero bias junction capacitance of the diode and 𝑉𝑉 0 denotes the DC output voltage. Following [START_REF] Yoo | Theoretical and Experimental Development of 10 and 35 GHz Rectennas[END_REF], an ideal DC passband filter is assumed here. Three key parameters of the diode impact the RF-to-DC conversion efficiency. The series resistance (Rs) limits the efficiency through the dissipation losses, the zero-bias junction capacitance (𝐶𝐶 𝑗𝑗0 ) affects the harmonic currents through the diode and the breakdown voltage (𝑉𝑉 𝐵𝐵𝐵𝐵 ) limits the power handling capability of the rectifier. The diode used in this design is the silicon Schottky diode SMS7630 from Skyworks. Its main parameters are reported in the Table 1 (see [14], [START_REF]Skyworks application notes 'Mixer and detector diodes[END_REF])

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The experimental setup used in order to characterize the rectenna is shown in the Fig. 8. A microwave signal provided by an Anritsu MG3694B generator is injected at the input of a horn antenna using a coaxial cable. During the measurement, the rectenna was aligned in order to match the (linear) polarization of the transmitting horn antenna. The horn antenna is placed at a distance d above the rectenna and allows illuminating the rectenna with a linearly polarized E-field. An automatic acquisition routine is implemented in Labview software from National Instruments to speed-up the acquisition process. The harvested DC voltage measured by using a DC multimeter. The DC power can be derived from the measured DC voltage as long as the load is known. The measured losses due to the coaxial cable and connectors placed between the signal generator and the horn antenna are around 3 dB at the operating frequency band (12 GHz). Two different transmitting horn antennas are used here: the first one covers the 12GHz -16 GHz frequency band and the second one covers the 16GHz -22 GHz frequency band. The efficiency η (in %) of the rectenna can be computed as follows [START_REF] Popovic | Low-Power Far-Field Wireless Powering for Wireless Sensors[END_REF]:

𝜂𝜂 (%) = 𝑃𝑃 𝐷𝐷𝐷𝐷 𝑆𝑆. 𝐴𝐴 𝑒𝑒𝑓𝑓𝑓𝑓 • 100 = 4. 𝜋𝜋. 𝑃𝑃 𝐷𝐷𝐷𝐷 𝑆𝑆. 𝐺𝐺 𝐵𝐵 . 𝜆𝜆² • 100 ( 3
)
where 𝑃𝑃 𝐷𝐷𝐷𝐷 is the harvested DC power, 𝑆𝑆 is the incident electromagnetic power density, 𝐴𝐴 𝑒𝑒𝑓𝑓𝑓𝑓 is the antenna effective area, 𝐺𝐺 𝐵𝐵 is the simulated gain of the (rectenna's) antenna and 𝜆𝜆 is the free-space wavelength of the illuminating electromagnetic wave at a given frequency.

From equation (3) The power density (µW/cm 2 ) can be computed as a function of the E-field effective value 𝐸𝐸 (V/m) on the antenna surface or as a function of the RF power P t injected at the input of the transmitting horn antenna (gain 𝐺𝐺 𝑡𝑡 ) and positioned at a distance 𝐺𝐺 from the rectenna, as follows:

𝑆𝑆 = 𝐸𝐸 2 120. 𝜋𝜋 • 100 = 30. 𝑃𝑃 𝑡𝑡 . 𝐺𝐺 𝑡𝑡 𝐺𝐺 2 . 120. 𝜋𝜋 • 100 (4)

TABLE 2. Comparison of rectenna performances *Non Communicated

The E-field and the power density illuminating the rectenna can be computed as function of 𝐺𝐺 𝑡𝑡 and d by using eq. ( 3). Fig. 9 shows the measured harvested DC power at the input port of a resistive load (RL=300Ω) in the frequency band 10GHz -16GHz demonstrate also a multi-band behavior. The proposed rectenna harvests efficiently the RF power at 17 GHz and 20 GHz. The optimal load was also determined and Fig. 11 shows the DC power as a function of the load for an E-field of 48 V/m at 12GHz. A standard potentiometer (0-10kΩ) was used as a load impedance. The results shows that the DC power decreases (i.e. the efficiency decreases) when the load impedance increases. The highest DC power is obtained when R = 300 Ω. The harvested DC power for RL=300Ω as function of E-field illuminating the 4CDAA at the frequency of 12 GHz is reported on Fig. 12. From a practical point a view at least 1 mW of DC power is required to supply an autonomous wireless sensor for satellite health application. Typically such a wireless sensor is composed of a sensor (DC power consumption in the range of few µW), a DC-to-DC convertor, a power management unit and a wireless low-power transceiver (DC power consumption in the range of the mW). However the electromagnetic energy illuminating the antenna panels is almost constant and the DC circuitry (including the power management unit) should be minimal [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. As shown in Fig. 12, the rectenna can provide a DC power of 1mW or higher when it is illuminated by electric field amplitude higher than 38 V/m. The efficiency of 4CDAA rectenna is 1.8 1 0 0 3 0 0 5 0 0 7 0 0 9 0 0 2 0 0 0 5 0 0 0 1 0 0 0 0 DC power (mW) R(ohm) computed using eq. ( 2). Fig. 13 represents the RF-to-DC conversion efficiency as a function of the E-field illuminating the rectenna under test. The highest efficiency is approximately 41% at 12 GHz for illuminating E-field amplitude ranging between 51 V/m and 83 V/m. At the second resonant frequency (f1=17.6 GHz) the highest efficiency is approximately 12% while an efficiency of about 20% is obtained at 20 GHz. The main sources of power loss are: (i) the ohmic resistance of the selected low-cost silicon diode and (ii) the eventual mismatch between the antenna and the rectifier (due to the adopted non-resonant matching technique and to the lack of accurate simulation models of the selected diode). The overall performances of the 4CDAA rectenna can be further improved by optimizing PCB layout on the bottom side which has a major impact on the impedance matching between the rectifier and the antenna located on the top side of the PCB. Efficiencies greater than 25% can be obtained at the first resonant frequency by using a low cost silicon Schottky diode (SMS7630) when the E-field amplitude exceeds 19 V/m. The rectenna is very compact as shown in Fig. 2. Its size is 2.5 cm x 2.5 cm. Moreover this rectenna exhibits a tri-band feature which is obtained from the multiple resonant frequencies of the cross dipoles antenna array.

Ref

V. CONCLUSION

A compact rectenna based on the use of an original four cross dipoles antenna array was designed, manufactured and measured. The experimental results demonstrate that a DC power greater than 1 mW can be harvested in the Ku band (12 GHz) for electric field amplitude higher than 38 V/m. This DC power is sufficient to power a wireless sensor (including the sensor, the sensing device, the power management unit and the wireless transceiver) for satellite health monitoring application. A maximum RF-to-DC conversion efficiency of 41% was obtained for a load of 300Ω and an incident electric field of 51 V/m. Moreover this rectenna exhibits a multi-band behavior at frequencies that corresponds to the available multiple resonant frequencies of the cross dipoles antenna array.

Fig. 1 .

 1 Fig.1. Topologies for multiband rectennas: (a) rectenna with one RF band-pass filter for a broadband/multiband RF matching and (b) rectenna combining the harvested DC powers at each frequency.

Fig. 2 .

 2 Fig.2. (a) Top view (not to scale) of the layout of the 4CDAA rectenna and (b) photos of top and bottom side of the manufactured rectenna. The wires shown here were used to connect a variable resistance in order to reach the optimal load impedance.

Fig. 3 .

 3 Fig.3. The antenna with its voltage port and the lumped RLC boundaries

Fig. 4 .

 4 Fig.4. (a) S11 viewed at the input port of the diode vs frequency; (b) the input impedance viewed at the input port of the diode vs frequency

Fig. 5 . 2 GHzFig. 6 .

 526 Fig.5. Simulated (HFSS) radiation pattern (gain) in the xOz plane (phi=0°, continuous line) and in the yOz plane (phi=90, dashed line) of the 4CDAA at the frequencies of (a) 12 GHz, (b) 17.6GHz and (c) 20.2 GHz

Fig. 7 .

 7 Fig.7. Equivalent circuit of the diode

Fig. 8 .

 8 Fig.8. Experimental setup used to characterize the mutiband rectenna

Fig. 9 .

 9 Fig.9. Measured DC power at the input port of a resistive load (RL=300Ω) as a function of frequency and for various E-field amplitudes

Fig. 11 .

 11 Fig.11. Measured DC power as a function of the resistance of the load for E-field=48V/m.

Fig. 12 .Fig. 13 .

 1213 Fig.12. Measured DC power at the input of a load (RL=300Ω) as a function of the illuminating E-field amplitude at 12 GHz

TABLE 1 :

 1 Schottky diode parameters

	Parameter	Value
	Ohmic resistance (Rs)	20 Ω
	Junction capacitance (C j0 )	0.14 pF
	Saturation current (I s )	5 µA
	Forward voltage (V f )	0.34V
	Reverse breakdown voltage (B v )	2 V

microwave generator DC multimeter transmitting antenna rectenna under test powermeter for

  various amplitudes of the (the gain 𝐺𝐺 𝑡𝑡 the first transmitting horn antenna at 12 GHz is 19 dBi). The maximum DC power was measured at 12 GHz as shown in Fig.9.

		Operating	Diode	Incident power density	Dimensions	Maximum
		Frequency		/ RF power at the		Conversion
				rectifier input		Efficiency
	[12]	35GHz	Skyworks	20.79 dBm, RL=400Ω	NC*	39%
			DMK6606	17 dBm, RL=100Ω
	[17]	2.45GHz-5.8GHz	Macom	2.38mW/cm² -	NC*	84.4% -82.7%
			MA4E1317	8.77mW/cm²
	[18]	24GHz	Macom	10 mW/cm²	Length : 55 mm	24%
	[19]	35GHz	MA4E1317 Macom	30 mW/cm²	1.9𝜆𝜆 0 Length : 23.55 mm	35%
	[20]	25.7GHz	MA4E1317 Macom	8 dBm	2.7𝜆𝜆 0 21.9 x 15 mm²	16%
	[21]	5.8GHz	MA4E2502L Macom	12 mW/cm²	2.4𝜆𝜆 0 ² NC*	76%
			MA4E1317
	This	12GHz-17GHz-	Skyworks	1.8 mW/cm²	25 x 25 mm²	42%
	work	20GHz	SMS7630		𝜆𝜆 0 ² at 12 GHz

ACKNOWLEDGEMENT

The authors acknowledge the support of French Space Agency within the framework of several past R&T contracts.


Abderrahim Okba

Samuel Charlot

Pierre-François Calmon

Alexandru Takacs

Hervé Aubert

P-F Calmon

Multiband rectenna for microwave applications

Keywords: electromagnetic energy harvesting, wireless power transfer, multi-band rectennas, printed dipoles array, satellite application

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The electromagnetic harvesting of the spill-over losses of microwave antennas was identified as a realistic solution to power autonomous wireless sensors for health monitoring of the antenna panels of the broadcasting satellites [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. This paper presents recent results in Ku and K band obtained by using an innovative rectenna topology [START_REF] Okba | Cross dipoles rectenna for microwave applications[END_REF] composed of four printed cross dipoles antenna array and using a low cost Schottky diode [3]. Rectennas using a cross dipole antenna were also reported in [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF], [START_REF] Sun | Design of a High-Efficiency 2.45-GHz Rectenna for Low-Input-Power Energy Harvesting[END_REF] while a rectenna using a two cross dipoles array was reported in [START_REF] Takacs | Rectenna Design for K Band Application[END_REF]. The rectenna topology and the design methodology are presented in Section II. The experimental results reported in Section III and obtained by using the manufactured rectenna show that very good performances can be achieved in three frequencies bands.

II. RECTENNA TOPOLOGY AND DESIGN

The proposed topology is composed of an array of four cross dipoles (4CDAA) and a silicon Schottky diode -Skyworks SMS7630 [3]. The PCB layout and a photo of the manufactured rectenna are shown in Fig. 1.The antenna array and the diode are implemented on the top of the PCB while the RF shunt capacitor is mounted on the bottom side of PCB. A reflector metallic plate is added on the bottom side in order to improve the antenna gain. This compact topology (its size is fixed by the size of the antenna array) is an extension of a rectenna using two cross dipoles antenna array 2CDAA reported in [START_REF] Takacs | Rectenna Design for K Band Application[END_REF] and a non-resonant matching technique was used. This non-resonant matching is implemented by properly controlling (i) the input impedance of the antenna array and (ii) the input impedance of the rectifier itself. The rectifier consists of the diode and the low pass RC filter (composed of the shunt capacitor and the load itself). The 4CDAA can be viewed as an array formed by four cross dipoles printed on the PCB. Each cross dipole consists of two dipoles in cross configuration as shown in Fig. 1. The substrate used is Rogers 6002 (substrate thickness: 0.508 mm, relative dielectric permittivity: 2.94, dielectric loss tangent: 0.0012). The maximum theoretical gain (Gmax in dBi) of such array can be predicted by using the following expression:

𝐺𝐺𝐺𝐺 = 10 • 𝑙𝑙𝑙(8) + 𝐺𝐺 = 11.18 𝐺𝑑𝑑 (1)
where Gd is the gain of each lossless dipole. This gain can be increased by 3 dB when a metallic reflector is properly positioned below the antenna surface. The expected maximum There are no accurate analytical formulas for designing such arrays but some rules can be used in order to start the design. The length of each dipole (Ld in Fig 1) should be approximately half wavelength and the distance between the center of two opposite cross dipoles (L2 in Fig. 1) should be an integer multiple of half wavelength. The angle between the dipoles (α in Fig 1) impacts the polarization of the radiated field and the maximum achievable gain. Linear or circular polarization can be obtained by using this topology as function of the relative phase at the access ports of the crossed dipoles.

Coplanar strip lines are used to feed each cross dipole. We target here a linear polarization. Thus the diode are mounted in the center of the array as represented in Fig. 1. The distance between the mounting position of the diode and each cross dipole should be in the range of half wavelength. Six metallic via holes were used to connect the top and the bottom side of the PCB. A shunt capacitor is mounted on the bottom side of the PCB. The main dimensions of the fabricated (optimized) rectenna are: L1= 4 mm, L2= 15.3 mm, Ld= 10.6 mm, Lx=Ly= 19.5 mm. Intensive full-wave electromagnetic simulations were performed by using Ansys HFSS software in order to optimize the 4CDAA. The proposed 4CDAA has a multi-band behavior. At the first operating frequency (f 0 ) the length of each dipole (Ld) is in the range of the half wavelength (Ld~λ/2) and the distance between the center of the opposite dipoles (L2) is in the range of wavelength (L2~λ). Another operating/resonant frequency is f 2 ~2f 0 . At this frequency Ld~λ and L2~2λ. There is another resonant frequency at approximately f 1 ~1.5f 0 . At this frequency Ld~0.7λ and L2~1.5λ. Note that the relationships f 1 ~1.5f 0 and f 2 ~2f 0 are not exact because the 4CDAA is very compact and a non-negligible electromagnetic coupling between the four cross dipoles exists. Fig. 2 shows the simulated results for: (i) the radiation pattern of 4CDAA at the 3 resonant frequencies (12 GHz, 17 GHz and 20 GHz), (ii) the simulated gain (over Oz axis) and input impedance (viewed at the diode input port) as a function of the frequency for the same 4CDAA. The simulation results in the frequency band of 13-16 GHz were not yet available where this paper was prepared was submitted. The optimized 4CDAA exhibits the following simulated gain: G=11 dBi at 12 GHz, G=11.9 dBi at 17 GHz and G=12.4 dBi at 20 GHz as it can be observed from Fig. 2 c). The simulated input impedance is shown in the Fig. 2 d).

A generic electrical model for the diode exists in the ADS software. It was customized with the following parameters [6][7]: ohmic resistance R s = 20Ω, junction capacitance C j0 = 0.14 pF, saturation current I s = 5 µA, forward voltage V f = 0.34V, reverse breakdown voltage B v = 2V and the current at reverse breakdown voltage I bv = 0.1mA. The ADS simulation model is represented in Fig. 3. The PCB located on the bottom side of the PCB was simulated by using standard printed PCB lines and via holes founded in the ADS library. The harmonic balance port was charged by the input impedance viewed at the input port of the diode. The first simulations results obtained from this model demonstrates that the maximum DC power is obtained at the frequency of 12 GHz . Nevertheless we observed that ADS harmonic balance engine exhibited numerical convergence problem in this case and an important sensitivity of the DC power delivered to the load (RL) as function of the length of the printed PCB line. Thus we decided to manufacture a prototype and to keep as compact as possible the PCB layout of the bottom side in order to avoid any uncontrolled coupling between the 4CDAA (bottom side) and the PCB layout of the bottom side.

III. EXPERIMENTAL RESULTS AND DISCUSSION

The experimental setup shown in the Fig. 4 was used to characterize the rectenna. A microwave signal generated from an Anritsu MG3694B generator was injected at the input of a horn antenna which illuminated rectenna under test with a linearly polarized E-field. An automatic acquisition routine was implemented in Labview software from National Instruments to speed-up the acquisition process. The harvested DC voltage was measured by using a DC multimeter. The DC power can be derived from the measured DC voltage as long as the load is known. The measured loss due to the coaxial cable and connectors between the antenna and the signal generator was close to 3 dB in the operating frequency band. Two different transmitting horn antennas were used: one for covering the 12GHz -16 GHz frequency band and another one to cover the 16GHz -22 GHz frequency band.

The efficiency η (in %) of the rectenna can be computed by using the following expression [8]:

(2) where P DC is the harvested DC power, S is the incident electromagnetic power density, A eff is the antenna effective area, G R is the gain of the (rectenna's) antenna and λ is the wavelength of the illuminating electromagnetic wave. The power density (µW/cm 2 ) can be computed as a function of E-field effective value E (V/m) on the antenna surface or as a function of the RF power P t injected to the input of the transmitting horn antenna of gain G t and positioned at the distance d from the rectenna, as follows:

(3) The E-field and the power density illuminating the rectenna can be computed as function of the G t and d by using eq. (3). Fig. 5 shows the measured harvested DC power at the input port of a resitive load (RL=300Ω) in the frequency band 12GHz -16GHz for various amplitudes of the E-field (the first transmitting horn antenna with G t =19 dBi at 12 GHz was used). The E-field values were computed by using eq. (3). The maximum DC power was measured at 12 GHz as represented in Fig. 5.

Fig. 6 displays the measured DC power as function of frequency on a load of RL=300Ω in the frequency band 16GHz -22 GHz as function of E-field (the second transmitting horn antenna with G t =19.5 dBi positioned at the distance d=22 cm was used). The E-field amplitudes were computed by using eq. ( 3). The measurement results shown in Fig. 6 demonstrate also a multi-band behavior. Beside of its main operating frequency (12 GHz) the proposed rectenna harvests efficiently the RF power at 20 GHz. One can see also an important amount of the DC power for the frequencies of 17 GHz and 16.6 GHz. The optimal load was also determined (results not shown here). The maximum DC power was obtained for a load impedance RL of 300Ω. The harvested DC power for RL=300Ω as function of Efield illuminating the 4CDAA at the frequency of 12 GHz is reported on Fig. 7. From a practical point a view at least 1 mW of DC power is required to supply an autonomous wireless sensors for satellite health application. Typically such a wireless sensor is composed of a sensor (DC power comsumption in the range of few µW), a DC-to-DC convertor, a power management unit and a low-power wireless tranceiver (DC power consumption in the range of the mW). However the electromagnetic energy illuminating the antenna panels is almost constant and the DC circuitry (including the power management unit) should be minimal [START_REF] Takacs | Microwave power harvesting for satellite health monitoring[END_REF]. As represented in Fig. 7 the rectenna can provide a DC power of 1mW or higer when it is illuminated by an electric field amplitude higher than 38 V/m. The efficiency of 4CDAA rectenna is computed by using eq. ( 2). Fig. 8 represents the RF-to-DC conversion efficiency as function of the E-field illuminating the rectenna under test. The maximun achieved efficiency is approximately 41% at 12 GHz (first resonant frequency for 4CDAA) for an illuminating E-field amplitude ranging between 51 V/m and 58 V/m. The maximum achieved efficiency at the second resonant frequency (f=17 GHz) is approximately 12% while at 20 GHz (the third resonant frequency) the maximum efficiency is 16.3 %. The reported efficiencies are limited by: (i) the ohmic resistance of the selected low-cost silicon diode, (ii) the adopted non-resonant matching technique, (iii) the lack of accurate simulation models of the selected diode. The overall performances of the 4CDAA rectenna can be further improved by optimizing PCB layout on the bottom side which has a major impact on the impedance matching between the rectifier and the antenna located on the top side of the PCB. Works are under run and more results will be presented at the conference. Despite of the adopted non-resonant matching technique and of the inherent inaccuracy of the simulation model an efficiency greater than 25% can be obtained at the first resonant frequency by using a low cost silicon Schottky diode (SMS7630) when the E-field amplitude exceeds 19 V/m. The rectenna is very compact as shown in Fig. 1. Its size is 2.5 cm x 2.5 cm that is square wavelength at 12 GHz. Moreover this rectenna exibits an tri-band behaviour explained mainly by the multiple resonant frequencies of the cross dipoles antenna array.

IV. CONCLUSION

A compact rectenna (overall size is approximately square wavelength at 12 GHz) based on the use of an original four cross dipoles antenna array was designed, manufactured and measured. The experimental results demonstrate that a DC power greater than 1 mW can be harvested in the Ku band (12 GHz) for electric field amplitude higher than 38 V/m. This DC power is sufficient to power a wireless sensor (including the sensor, the power management unit and the wireless tranceiver) for satellite health monitoring application. A maximum RF-to-DC conversion efficiency of 41% was obtained for a load of 300Ω and an incident electric field of 51 V/m. Moreover this rectenna exibitis a multi-band behavior at frequencies that corresponds to the multiple resonant frequencies of the cross dipoles antenna array.

Fig. 1 .

 1 Fig. 1. (a) Top view (not to scale) of the layout of the 4CDAA rectenna, its main geometrical dimensions and (b) photos of top and bottom side of the manufactured rectenna. The wires shown here was used to connect a variable resistance in order to reach the optimal load.

Fig. 2 .

 2 Fig. 2. (a) Simulated (HFSS) gain in the xOz plane (phi=0°) -continous line -and in the xOy plane (phi=90°) -dashed-line of the 4CDAA at the frequency of 12 GHz, (b) simulated gain at 17 GHz and 20 GHz (the insets show the 3D radiation patterns for 12 GHz, 17 GHz and 20 GHz), (c) Simulated (HFSS) maximum gain of the 4CDAA vs the frequency, (d) the input impedance viewed at the input port of the diode vs frequency.

Fig

  Fig. 3. ADS electrical circuit model.

Fig. 4 .

 4 Fig. 4. Experimental setup used to characterize the tri-band rectenna

Fig. 5 .

 5 Fig. 5. Measured DC power at the input port of a resistive load (RL=300Ω) as function of frequency and for various E-field amplitudes

Fig. 7 .

 7 Fig. 7. Measured DC power at the input of a load (RL=300Ω) as a function of the illuminating E-field amplitude at 12 GHz

[ 6 ]

 6 http://www.skyworksinc.com/uploads/documents//www.skyworksinc.com/uploads/documents/200826A.pdf [8] Z. Popovic; E.A. Falkenstein, D. Costinett, R. Zane, "Low-Power Far-Field Wireless Powering for Wireless Sensors", Proceedings of the IEEE, Vol. 101, No. 6, pp.1397 -1407, June 2013.

ACKNOWLEDGEMENT

The authors acknowledge the support of French Space Agency within the framework of several past R&T contracts.