Ergodicity of dynamical systems on 2 adi
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.
MATHEMATICS
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
The theory of p-adic dynamical systems is under
intense development [1, 2, 6], and more new applications are emerging. In addition to “traditional applications” in areas, such as theoretical p-adic physics [7],
population dynamics [8], psychology [9], and cryptography [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, iterations of monomial mappings x → xm, m = 2, 3, …,
sending spheres into themselves. It was shown that
even this dynamics is rather complicated: for the systems to be ergodic, the positive integer m > 1 has to be
related to the prime number p > 1 in a nontrivial manner. In [11] the problem was posed of the stability of
the ergodicity condition with respect to a perturbation
of monomial dynamics by “small polynomial mappings x → xs + v(x), where v(x) is a “small polynomial.” 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 nonsmooth functions is of great interest. Specifically, in
cryptography, this is associated with the fact that many
natural pseudorandom number generators constructed 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
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
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; moreover, the set of piecewise constant functions is dense in
the space of continuous functions. Thus, it is not surprising 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 polynomial, analytical, and smooth functions. In [13] ergodicity 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 interest 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 –
1
a|p = r}, a ∈ ⺪p, r = ---k , k = 1, 2, …. Specifically, we
2
solve the problem of the stability of the ergodicity condition 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 conditions for small perturbations of monomial dynamical systems on spheres is completely solved in the 2adic case. The transition to the case of odd p is nontrivial. In this case, the above problem for nonsmooth
functions remains open.
Note that the study of ergodicity of dynamical systems on spheres (rather than on the whole ring ⺪p of
p-adic integers) is important for the design of congruent pseudorandom generators for quasi-Monte Carlo
numerical methods. Namely, if the composition of the
recurrence law for a generator involves a partially
843
844
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 recurrence 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 widespread (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 multiplication, taking an inverse, or exponentiation) but also
“fast” operations, such as bitwise logical ones, are useful in the composition of the recurrence law. However,
in the latter case, the recurrence law, though remaining 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 mathematical interest.
Let a function f : ⺪p → ⺪p be defined (and take values) 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 commutative ring with respect to addition and multiplication, a function f : ⺪p → ⺪p satisfies the Lipschitz condition with constant 1 if and only if it is compatible;
i.e., f (x) ≡ f (y)(mod pk) whenever x ≡ y (modpk); x, y ∈
⺪p. In this case, a reduction modulo pk is an epimorphism modpk: ⺪p → ⺪p/pk⺪ (with the kernel pk⺪p) of
the ring ⺪p to the ring ⺪/pk⺪ of residues modulo pk.
The elements of ⺪/pk⺪ are identified with 0, 1, …, pk – 1
and, throughout this paper, for z ∈ ⺪p, by zmodpk we
denote the corresponding element of ⺪/pk⺪, i.e., the
smallest nonnegative residue of z modulo pk.
The space ⺪p is equipped with the natural probability 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, measurepreserving) 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) modulo 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 integers can be defined using a van der Put series. Namely,
given a mapping f, there is a unique sequence {Bf(0),
Bf (1), Bf (2), …} of p-adic integers such that
∞
∑ B ( m ) ⋅ χ ( m, x )
f(x) =
(1)
f
for all x ∈ ⺞0; here,
m=0
1 if x – m p ≤ p –n ,
χ ( m, x ) = ⎨
⎩ 0 otherwise,
where n = 1 if m = 0 and n is uniquely determined by
the inequality pn – 1 ≤ m ≤ pn – 1 if m ≥ 1. The number
n in the definition of χ(m, x) has a clear interpretation:
it is equal to the number of digits in the base-p expansion of m ∈ ⺞0. Namely, given m ∈ ⺞, if log pm
denotes the largest rational integer not greater than
log pm , then log pm is the length of the base-p expansion of m minus 1; i.e., n = log pm + 1 for m ∈ ⺞0 (by
definition, we set log p0 = 0). The coefficients Bf(m)
are related to the values of f as follows. Let m = m0 +
m1p + … + mn – 2pn – 2 + mn – 1pn – 1 be the base-p expansion of m, i.e., mj ∈ {0, 1, …, p – 1}, j = 0, 1, …, n – 1
and mn – 1 ≠ 0. Then
f ( m ) – f ( m – mn – 1 pn – 1 )
Bf ( m ) = ⎨
⎩ f ( m ) otherwise.
if
m ≥ p,
Note also that χ(m, x) is the characteristic function
of the ball of radius p
m ∈ ⺞0:
1
χ(m, x) =
– log pm – 1
if
centered at the point
x ≡ m ( mod p
log pm + 1
);
0 otherwise,
1
if
x∈U
=
p
– log m – 1
p
( m );
0 otherwise.
CRITERION FOR THE ERGODICITY
OF A COMPATIBLE MAPPING
OF A SPHERE IN TERMS
OF THE VAN DER PUT SERIES
Let p = 2, f : ⺪2 → ⺪2 be a compatible function, and
S 2–r (a) be the sphere of radius 2–r centered at the point
a ∈ {0, 1, …, 2r – 1}. Suppose that S 2–r (a) is invariant
under f; i.e., f : S 2–r (a) → S 2–r (a). In the case p = 2, the
DOKLADY MATHEMATICS
Vol. 86
No. 3
2012
ERGODICITY OF DYNAMICAL SYSTEMS
845
2r + 2r + 1x: x ∈ ⺪2} = U 2– r – 1 (a + 2r). Note that we then
Theorem 3. Let u: ⺪2 → ⺪2 be an arbitrary compatible function. The function f (x) = xs + 2r + 1u(x) is
ergodic on the sphere S 2–r (1) = {1 + 2r + 1x: x ∈ ⺪2} if and
only if s ≡ 1 (mod 4) and u(1) ≡ 1 (mod 2).
have f(a + 2r + 2r + 1x) = f(a + 2r) + 2r + 1g(x), where g: ⺪2
→ ⺪2 is a compatible function.
ACKNOWLEDGMENTS
sphere S 2–r (a) coincides with the ball U 2– r – 1 (a + 2r) of
radius 2–r – 1 centered at the point a + 2r: S 2–r (a) = {a +
Theorem 1. The function f is ergodic on the sphere
r
r
S 2–r (a) if and only if f ( a + 2 ) ≡ ( a + 2 ) ( mod2
r+1
) and
g(x) is ergodic on ⺪2.
It was shown [13] that a function f: ⺪2 → ⺪2 repre-
Anashin acknowledges the support of the Russian
Foundation for Basic Research (project no. 12-0100680-a) and the Science Foundation of the People’s
Republic of China (visiting professorship for senior
international scientists, grant no. 2009G2-11).
∞
sented as a van der Put series f (x) =
∑ B (m)χ(m, x)
f
m=0
log m
is ergodic on ⺪2 if and only if Bf(m) ≡ 0 (mod 2 2 )
for all m = 0, 1, 2, …; i.e., if and only if Bf(m) =
log 2m
2
b f ( m ) for suitable p-adic integers bf(m), m = 0,
1, 2, … .
Theorem 2. A compatible function f : ⺪2 → ⺪2 represented as a van der Put series
∞
∞
f(x) =
∑ B ( m )χ ( m, x ) = ∑ 2
log 2m
f
b f ( m )χ ( m, x ),
m=0
m=0
where bf(m) ∈ ⺪2 and m = 0, 1, 2, …, is ergodic on the
sphere S 2–r (a) if and only if the following conditions are
satisfied:
(i) f(a + 2r) ≡ a + 2r + 2r + 1(mod2r + 1);
r
r
r+1
(iv) b f ( a + 2 + 2
(mod4);
r
2. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological
Models (Kluwer, Dordrecht, 1997).
3. V. S. Anashin, Math. Notes 55, 109–133 (1994).
4. V. Anashin, Proceedings of 2nd International Conference on p-Adic Mathematical Physics, Belgrade, September 15–21, 2005, AIP Conference Proc. Melville (Am.
Inst. Phys, New York, 2006), Vol. 826, pp. 3–24.
5. V. Anashin, Proceedings of Advanced Study Institute
Boolean Functions in Cryptology and Information Security, NATO Sci. Peace Secur. Ser. D. Inf. Commun.
Secur. (IOS, Amsterdam, 2008), pp. 33–57.
6. S. De Smedt and A. Khrennikov, Comment. Math.
Univ. St. Pauli 4 (2), 117–132 (1997).
8. A. Yu. Khrennikov, Dokl. Math. 58, 151–153 (1998).
) ≡ 1 (mod 4);
r+2
1. V. S. Anashin and A. Yu. Khrennikov, Applied Algebraic
Dynamics (Walter de Gruyter, Berlin, 2009).
7. B. Dragovich, A. Khrennikov, S. V. Kozyrev, and
I. V. Volovich, p-Adic Numbers Ultrametric Anal.
Appl. 1 (1), 1–17 (2009).
(ii) |bf(a + 2r + m · 2r + 1)|2 = 1 for m > 0;
(iii) b f ( a + 2 + 2
REFERENCES
) + bf ( a + 2 + 3 × 2
r+1
) ≡ 2
9. A. Y. Khrennikov, p-Adic Numbers, Ultrametric Anal.,
Appl. 2 (1), 1–20 (2010).
10. A. Khrennikov and S. V. Kozyrev, Phys. A: Stat. Mech.
Appl. 381, 265–272 (2007).
n
2 –1
(v)
∑
m=2
r
bf ( a + 2 + m ⋅ 2
r+1
) ≡ 0 (mod4) for n ≥ 3.
n–1
12. M. Gundlach, A. Khrennikov, and K.-O. Lindahl, Infinite Dimen. Anal., Quantum Prob. Related Fields 4,
569–577 (2001).
CRITERION FOR THE ERGODICITY
OF SMALL PERTURBATIONS
OF A MONOMIAL MAPPING OF A SPHERE
xs
2r + 1u(x)
Consider the mapping f (x) =
+
of
r
r
+
1
the sphere S 2–r (1) = {1 + 2 + 2 x: x ∈ ⺪2}, s, r ∈ {1, 2,
3, …}.
DOKLADY MATHEMATICS
11. M. Gundlach, A. Khrennikov, and K.-O. Lindahl,
Lect. Notes Pure Appl. Math. 222, 127–132 (2001).
Vol. 86
No. 3
2012
13. V. S. Anashin, A. Yu. Khrennikov, and E. I. Yurova,
Dokl. Math. 83, 306–308 (2011).
14. J. Eichenauer-Herrmann, E. Herrmann, and S. Wegenkittl, Lect. Notes Stat. 127, 66–97 (1996).
15. H. Niederreiter, Random Number Generation and
Quasi-Monte Carlo Methods (SIAM, Philadelphia, PA,
1992).
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.
MATHEMATICS
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
The theory of p-adic dynamical systems is under
intense development [1, 2, 6], and more new applications are emerging. In addition to “traditional applications” in areas, such as theoretical p-adic physics [7],
population dynamics [8], psychology [9], and cryptography [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, iterations of monomial mappings x → xm, m = 2, 3, …,
sending spheres into themselves. It was shown that
even this dynamics is rather complicated: for the systems to be ergodic, the positive integer m > 1 has to be
related to the prime number p > 1 in a nontrivial manner. In [11] the problem was posed of the stability of
the ergodicity condition with respect to a perturbation
of monomial dynamics by “small polynomial mappings x → xs + v(x), where v(x) is a “small polynomial.” 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 nonsmooth functions is of great interest. Specifically, in
cryptography, this is associated with the fact that many
natural pseudorandom number generators constructed 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
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
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; moreover, the set of piecewise constant functions is dense in
the space of continuous functions. Thus, it is not surprising 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 polynomial, analytical, and smooth functions. In [13] ergodicity 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 interest 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 –
1
a|p = r}, a ∈ ⺪p, r = ---k , k = 1, 2, …. Specifically, we
2
solve the problem of the stability of the ergodicity condition 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 conditions for small perturbations of monomial dynamical systems on spheres is completely solved in the 2adic case. The transition to the case of odd p is nontrivial. In this case, the above problem for nonsmooth
functions remains open.
Note that the study of ergodicity of dynamical systems on spheres (rather than on the whole ring ⺪p of
p-adic integers) is important for the design of congruent pseudorandom generators for quasi-Monte Carlo
numerical methods. Namely, if the composition of the
recurrence law for a generator involves a partially
843
844
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 recurrence 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 widespread (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 multiplication, taking an inverse, or exponentiation) but also
“fast” operations, such as bitwise logical ones, are useful in the composition of the recurrence law. However,
in the latter case, the recurrence law, though remaining 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 mathematical interest.
Let a function f : ⺪p → ⺪p be defined (and take values) 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 commutative ring with respect to addition and multiplication, a function f : ⺪p → ⺪p satisfies the Lipschitz condition with constant 1 if and only if it is compatible;
i.e., f (x) ≡ f (y)(mod pk) whenever x ≡ y (modpk); x, y ∈
⺪p. In this case, a reduction modulo pk is an epimorphism modpk: ⺪p → ⺪p/pk⺪ (with the kernel pk⺪p) of
the ring ⺪p to the ring ⺪/pk⺪ of residues modulo pk.
The elements of ⺪/pk⺪ are identified with 0, 1, …, pk – 1
and, throughout this paper, for z ∈ ⺪p, by zmodpk we
denote the corresponding element of ⺪/pk⺪, i.e., the
smallest nonnegative residue of z modulo pk.
The space ⺪p is equipped with the natural probability 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, measurepreserving) 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) modulo 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 integers can be defined using a van der Put series. Namely,
given a mapping f, there is a unique sequence {Bf(0),
Bf (1), Bf (2), …} of p-adic integers such that
∞
∑ B ( m ) ⋅ χ ( m, x )
f(x) =
(1)
f
for all x ∈ ⺞0; here,
m=0
1 if x – m p ≤ p –n ,
χ ( m, x ) = ⎨
⎩ 0 otherwise,
where n = 1 if m = 0 and n is uniquely determined by
the inequality pn – 1 ≤ m ≤ pn – 1 if m ≥ 1. The number
n in the definition of χ(m, x) has a clear interpretation:
it is equal to the number of digits in the base-p expansion of m ∈ ⺞0. Namely, given m ∈ ⺞, if log pm
denotes the largest rational integer not greater than
log pm , then log pm is the length of the base-p expansion of m minus 1; i.e., n = log pm + 1 for m ∈ ⺞0 (by
definition, we set log p0 = 0). The coefficients Bf(m)
are related to the values of f as follows. Let m = m0 +
m1p + … + mn – 2pn – 2 + mn – 1pn – 1 be the base-p expansion of m, i.e., mj ∈ {0, 1, …, p – 1}, j = 0, 1, …, n – 1
and mn – 1 ≠ 0. Then
f ( m ) – f ( m – mn – 1 pn – 1 )
Bf ( m ) = ⎨
⎩ f ( m ) otherwise.
if
m ≥ p,
Note also that χ(m, x) is the characteristic function
of the ball of radius p
m ∈ ⺞0:
1
χ(m, x) =
– log pm – 1
if
centered at the point
x ≡ m ( mod p
log pm + 1
);
0 otherwise,
1
if
x∈U
=
p
– log m – 1
p
( m );
0 otherwise.
CRITERION FOR THE ERGODICITY
OF A COMPATIBLE MAPPING
OF A SPHERE IN TERMS
OF THE VAN DER PUT SERIES
Let p = 2, f : ⺪2 → ⺪2 be a compatible function, and
S 2–r (a) be the sphere of radius 2–r centered at the point
a ∈ {0, 1, …, 2r – 1}. Suppose that S 2–r (a) is invariant
under f; i.e., f : S 2–r (a) → S 2–r (a). In the case p = 2, the
DOKLADY MATHEMATICS
Vol. 86
No. 3
2012
ERGODICITY OF DYNAMICAL SYSTEMS
845
2r + 2r + 1x: x ∈ ⺪2} = U 2– r – 1 (a + 2r). Note that we then
Theorem 3. Let u: ⺪2 → ⺪2 be an arbitrary compatible function. The function f (x) = xs + 2r + 1u(x) is
ergodic on the sphere S 2–r (1) = {1 + 2r + 1x: x ∈ ⺪2} if and
only if s ≡ 1 (mod 4) and u(1) ≡ 1 (mod 2).
have f(a + 2r + 2r + 1x) = f(a + 2r) + 2r + 1g(x), where g: ⺪2
→ ⺪2 is a compatible function.
ACKNOWLEDGMENTS
sphere S 2–r (a) coincides with the ball U 2– r – 1 (a + 2r) of
radius 2–r – 1 centered at the point a + 2r: S 2–r (a) = {a +
Theorem 1. The function f is ergodic on the sphere
r
r
S 2–r (a) if and only if f ( a + 2 ) ≡ ( a + 2 ) ( mod2
r+1
) and
g(x) is ergodic on ⺪2.
It was shown [13] that a function f: ⺪2 → ⺪2 repre-
Anashin acknowledges the support of the Russian
Foundation for Basic Research (project no. 12-0100680-a) and the Science Foundation of the People’s
Republic of China (visiting professorship for senior
international scientists, grant no. 2009G2-11).
∞
sented as a van der Put series f (x) =
∑ B (m)χ(m, x)
f
m=0
log m
is ergodic on ⺪2 if and only if Bf(m) ≡ 0 (mod 2 2 )
for all m = 0, 1, 2, …; i.e., if and only if Bf(m) =
log 2m
2
b f ( m ) for suitable p-adic integers bf(m), m = 0,
1, 2, … .
Theorem 2. A compatible function f : ⺪2 → ⺪2 represented as a van der Put series
∞
∞
f(x) =
∑ B ( m )χ ( m, x ) = ∑ 2
log 2m
f
b f ( m )χ ( m, x ),
m=0
m=0
where bf(m) ∈ ⺪2 and m = 0, 1, 2, …, is ergodic on the
sphere S 2–r (a) if and only if the following conditions are
satisfied:
(i) f(a + 2r) ≡ a + 2r + 2r + 1(mod2r + 1);
r
r
r+1
(iv) b f ( a + 2 + 2
(mod4);
r
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8. A. Yu. Khrennikov, Dokl. Math. 58, 151–153 (1998).
) ≡ 1 (mod 4);
r+2
1. V. S. Anashin and A. Yu. Khrennikov, Applied Algebraic
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(iii) b f ( a + 2 + 2
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