ISSN 1064-5624, Doklady Mathematics, 2012, Vol. 86, No. 3, pp. 843–845. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © V.S. Anashin, A.Yu. Khrennikov, E.I. Yurova, 2012, published in Doklady Akademii Nauk, 2012, Vol. 447, No. 6, pp. 595–598.
843 The theory of p-adic dynamical systems is under intense development [1, 2, 6], and more new tions are emerging. In addition to “traditional applica-tions” in areas, such as theoretical p-adic physics [7], population dynamics [8], psychology [9], and cryp-tography [1, 3, 5], we note applications to genetic code analysis [10]. In particular, the ergodicity of dynamics plays an important role in many applied problems. One of the first problems concerning the ergodicity of p-adic dynamics was the study of dynamical systems on p-adic spheres [2, 11, 12]. First, the simplest dynamical systems were considered, namely, itera-tions of monomial mappings x → xm, m = 2, 3, …, sending spheres into themselves. It was shown that even this dynamics is rather complicated: for the sys-tems to be ergodic, the positive integer m > 1 has to be related to the prime number p > 1 in a nontrivial man-ner. In [11] the problem was posed of the stability of the ergodicity condition with respect to a perturbation of monomial dynamics by “small polynomial map-pings x→xs +
v(x), where v(x) is a “small polyno-mial.” This problem was solved in 2006 [4].
In the last years, interest has been aroused in the study of (discrete) p-adic dynamical systems x→f(x) with nonpolynomial functions. Moreover, for many applications, especially, for cryptography [1], the study of dynamics for nonanalytical and even nons-mooth functions is of great interest. Specifically, in cryptography, this is associated with the fact that many natural pseudorandom number generators con-structed using ergodic 2-adic dynamical systems are based on nonsmooth mappings x → f(x), which include, for example, bitwise logical operations. We have recently developed a new approach to the study of the ergodicity of p-adic dynamical systems based on using the van der Put basis in the space of continuous
functions f: ⺪p→ ⺪p, where ⺪p is the ring of p-adic integers (topologically, it is the unit closed ball U1(0) =
{x∈⺡p: |x|p≤ 1} in the field ⺡p of p-adic numbers). The elements of the van der Put basis are constructed using the characteristic functions of p-adic balls, which are locally constant functions. (Note that, in the p-adic case, such functions are continuous; more-over, the set of piecewise constant functions is dense in the space of continuous functions. Thus, it is not sur-prising that piecewise constant functions can be used to construct bases with different properties.) By using the van der Put basis, we can examine the ergodicity of
p-adic dynamical systems with continuous functions, i.e., go beyond the traditionally used classes of polyno-mial, analytical, and smooth functions. In [13] ergod-icity conditions were found for compatible mappings f:
⺪p→⺪p (i.e., mappings satisfying the p-adic Lipschitz condition with constant 1). The strongest results were obtained in the 2-adic case (which is of greatest inter-est for cryptographic applications), but some results were also obtained for odd p.
In this paper, a technique based on using the expansion coefficients with respect to the van der Put basis is applied to study dynamical systems on 2-adic spheres, f: Sr(a) →Sr(a), where Sr(a) = {x∈⺡p: |x – a|p = r}, a∈⺪p, r = , k = 1, 2, …. Specifically, we solve the problem of the stability of the ergodicity con-dition for nonsmooth perturbations of monomial dynamical systems, i.e., for mappings of the form x→ xs +
v(x), where v is a “small perturbation.” Thus, posed in [11], the problem of deriving ergodicity con-ditions for small perturbations of monomial dynami-cal systems on spheres is completely solved in the 2-adic case. The transition to the case of odd p is non-trivial. In this case, the above problem for nonsmooth functions remains open.
Note that the study of ergodicity of dynamical sys-tems on spheres (rather than on the whole ring ⺪p of p-adic integers) is important for the design of congru-ent pseudorandom generators for quasi-Monte Carlo numerical methods. Namely, if the composition of the recurrence law for a generator involves a partially
1 2k
----Ergodicity of Dynamical Systems on 2-Adic Spheres
V. S. Anashina, A. Yu. Khrennikovb
, and E. I. Yurovab
Presented by Academician V.S. Vladimirov May 19, 2011
Received August 19, 2011
DOI: 10.1134/S1064562412060312
a Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991 Russia
e-mail: vladimir.anashin@u-picardie.fr
b International Center for Mathematical Modeling, Linnaeus University, S-35195 Växjö, Sweden
e-mail: andrei.khrennikov@lnu.se, ekaterina.yurova@lnu.se
844
DOKLADY MATHEMATICS Vol. 86 No. 3 2012
ANASHIN et al.
defined operation (e.g., taking an inverse element) or an operation whose domain is smaller than the ring of p-adic integers (e.g., exponentiation), then such a generator reaches a maximum period only if its recur-rence law is an ergodic transformation of a p-adic sphere (for more detail, see [1, Subsection 9.2.2]). Generators using such operations are rather wide-spread (see, for example, [14, 15]). It should also be noted that the performance of the generator is an important characteristic for these applications. That is why not only “slow” operations (such as multiplica-tion, taking an inverse, or exponentiation) but also “fast” operations, such as bitwise logical ones, are use-ful in the composition of the recurrence law. However, in the latter case, the recurrence law, though remain-ing continuous with respect to the p-adic metric, is no longer a smooth p-adic function. Thus, the results of this paper surpass the boundaries of purely mathemat-ical interest.
Let a function f: ⺪p→⺪p be defined (and take val-ues) on the set ⺪p of all p-adic integers (p is prime), and let f satisfy the Lipschitz condition with constant 1 with respect to the p-adic metric |·|p: |f(x) – f(y)|p≤
|x– y|p for all x, y ∈⺪p. Recall that a mapping of an algebraic system A to itself is called compatible if it preserves all the congruences of A. Since ⺪p is a com-mutative ring with respect to addition and multiplica-tion, a function f: ⺪p→⺪p satisfies the Lipschitz con-dition with constant 1 if and only if it is compatible; i.e., f(x) ≡f(y)(modpk) whenever x≡y (modpk); x, y∈ smallest nonnegative residue of z modulo pk.
The space ⺪p is equipped with the natural probabil-ity measure, namely, the Haar measure μp normalized so that μp(⺪p) = 1. Elementary μp-measurable sets are p-adic balls. Recall that the p-adic ball of radius p–k is the set a + pk⺪
p of all p-adic integers that are congruent to a modulo pk. The volume of this ball is defined as
μp(a + pk⺪p) = p–k.
Recall that a measurable mapping f : ⺣→⺣ of a measurable space S with a probability measure μ is said to preserve the measure μ (in what follows, measure-preserving) if μ(f–1(S)) = μ(S) for each measurable
subset S ⊂S. A μ-preserving mapping f : ⺣→ ⺣ is called ergodic if it has no proper invariant subsets, i.e., if f–1(S) = S for a measurable subset S⊂⺣ implies μ(S)
= 0 or μ(S) = 1.
We say that a compatible function f: ⺪p → ⺪p is bijective (transitive) modulo pk if the induced mapping x哫f(x)modpk is a permutation (a permutation that is a single cycle, respectively) on ⺪/pk⺪. A compatible
function f: ⺪p→⺪p is measure-preserving (ergodic) if and only if it is bijective (transitive, respectively) mod-ulo pk for all k = 1, 2, … (see [1, Theorem 4.23]).
It is well known that any continuous mapping f:
⺪p→⺪p of the space ⺪p to the space ⺪p of p-adic inte-gers can be defined using a van der Put series. Namely, given a mapping f, there is a unique sequence {Bf(0), it is equal to the number of digits in the base-p expan-sion of m ∈ ⺞0. Namely, given m ∈⺞, if
denotes the largest rational integer not greater than , then is the length of the base-p expan-of the ball expan-of radius centered at the point m∈⺞0:
χ(m, x) =
=
CRITERION FOR THE ERGODICITY OF A COMPATIBLE MAPPING
OF A SPHERE IN TERMS OF THE VAN DER PUT SERIES
DOKLADY MATHEMATICS Vol. 86 No. 3 2012
ERGODICITY OF DYNAMICAL SYSTEMS 845
sphere (a) coincides with the ball (a + 2r) of radius 2–r– 1 centered at the point a + 2r: (a) = {a+ 2r + 2r+ 1x: x∈⺪
2} = (a + 2r). Note that we then
have f(a + 2r + 2r+ 1x) = f(a + 2r) +2r+ 1g(x), where g: ⺪ 2 →⺪2 is a compatible function.
Theorem 1.The function f is ergodic on the sphere
(a) if and only if and
g(x) is ergodic on⺪2.
It was shown [13] that a function f: ⺪2→⺪2
repre-sented as a van der Put series f(x) = (m)χ(m, x)
is ergodic on ⺪2 if and only if Bf(m) ≡ 0 (mod ) for all m = 0, 1, 2, …; i.e., if and only if Bf(m) = for suitable p-adic integers bf(m), m = 0, 1, 2, … .
Theorem 2.A compatible function f: ⺪2→⺪2 repre-sented as a van der Put series
wherebf(m) ∈⺪2andm = 0, 1, 2, …, is ergodic on the sphere (a) if and only if the following conditions are satisfied:
(i) f(a + 2r) ≡a + 2r + 2r+ 1(mod2r+ 1);
(ii) |bf(a + 2r + m · 2r+ 1)|
2 = 1 form> 0;
(iii) ≡ 1 (mod 4);
(iv) + ≡ 2
(mod4);
(v) ≡ 0 (mod4) forn≥ 3.
CRITERION FOR THE ERGODICITY OF SMALL PERTURBATIONS OF A MONOMIAL MAPPING OF A SPHERE
Consider the mapping f(x) = xs + 2r+ 1u(x) of
the sphere (1) = {1 + 2r + 2r+ 1x: x∈⺪
2}, s, r∈ {1, 2,
3, …}.
Theorem 3. Letu: ⺪2→⺪2be an arbitrary compat-ible function. The function f(x) = xs + 2r+ 1u(x) is ergodic on the sphere (1) = {1 + 2r+ 1x: x∈ ⺪
2} if and only ifs≡ 1 (mod 4) andu(1) ≡ 1 (mod 2).
ACKNOWLEDGMENTS
Anashin acknowledges the support of the Russian Foundation for Basic Research (project no. 12-01-00680-a) and the Science Foundation of the People’s Republic of China (visiting professorship for senior international scientists, grant no. 2009G2-11).
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S
2–r U2–r–1
S
2–r U
2–r–1
S
2–r f a 2
r
+
( ) ≡(a+2r)(mod2r+1)
Bf m=0
∞
∑
2 log2m
2 log2m bf( )m
f x( ) Bf( )χm (m x, )
m=0 ∞
∑
2 log2mbf( )χm (m x, ), m=0
∞
∑
= =
S
2–r
bf a 2 r
2r+1
+ +
( )
bf a 2r 2r+2
+ +
( ) bf a 2r 3×2r+1
+ +
( )
bf(a+2r+m⋅2r+1) m=2n–1
2n–1
∑
S
2–r
S
