Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems
We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always...
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| Zitieren: | Origination of the singular continuous spectrum in the conflict dynamical systems / T. Karataieva, V. Koshmanenko // Methods of Functional Analysis and Topology. — 2009. — 15, N 1. — С. 15-30. — Бібліогр.: 25 назв. — англ. |
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| citation_txt | Origination of the singular continuous spectrum in the conflict dynamical systems / T. Karataieva, V. Koshmanenko // Methods of Functional Analysis and Topology. — 2009. — 15, N 1. — С. 15-30. — Бібліогр.: 25 назв. — англ. |
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| description | We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous.
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 15–30
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM IN
THE CONFLICT DYNAMICAL SYSTEMS
T. KARATAIEVA AND V. KOSHMANENKO
Dedicated to the memory of wise Ukrainian mathematician A. Ya. Povzner
Abstract. We study the spectral properties of the limiting measures in the conflict
dynamical systems modeling the alternative interaction between opponents. It has
been established that typical trajectories of such systems converge to the invariant
mutually singular measures. We show that ”almost always” the limiting measures are
purely singular continuous. Besides we find the conditions under which the limiting
measures belong to one of the spectral type: pure singular continuous, pure point,
or pure absolutely continuous.
1. Introduction
We study the spectral properties of the limiting invariant measures of the dynami-
cal systems with composition of the alternative conflict interaction. This composition
generates the evolution of occupations of the living positions (regions) for a couple of
opponents. A starting state of the system is fixed by a pair of probability measures µ and
ν corresponding to opponents which exist on the common space Ω (the space of the living
resources). The fight between opponents for the control in regions of Ω is described by
the non-commutative and non-linear operation (composition) which is defined in terms
of µ and ν. This transformation we call the conflict composition and denote by � (see
below Section 4).
Actually the transformation � provides a certain redistribution between opponents of
the occupation probabilities in regions at various moments of the conflict. Iteration of
this transformation generates the evolution of the probability redistribution in terms of
the changed measures at a sequence of the discrete moments of time t = 1, 2, . . . , N, . . .
So, the dynamical system of conflict appears
{µN−1, νN−1} �−→ {µN , νN}, µ0 = µ, ν0 = ν, N = 1, 2, . . .
In papers [15, 16] it has been proved (for finite or countable spaces Ω) that the limiting
states
µ∞ = lim
N→∞
µN , ν∞ = lim
N→∞
νN
exist and are invariant with respect to the conflict composition �.
In the present paper we extend the above result for the case of measures having the
similar structure on the segment [0, 1] (c.f. with [17, 1]).
The significant fact is that an every similar structure measure (for the exact definition
see Section 3) belongs to the pure spectral type is the following sense. Namely, it means
2000 Mathematics Subject Classification. 91A05, 91A10, 90A15, 90D05, 37L30, 28A80.
Key words and phrases. Conflict dynamical system, similar structure probability measure, conflict
composition, local priority, directed priority, pure point, pure absolutely continuous, pure singular con-
tinuous measures.
15
16 T. KARATAIEVA AND V. KOSHMANENKO
that each such measure has a single component in the Lebesgue decomposition: either
purely point, or purely absolutely continuous, or purely singular continuous.
A key problem is to find the conditions which ensure that one or both limiting measures
µ∞, ν∞ belong to the before chosen spectral type. Some results in this direction have
been already obtained in [17, 1].
In this paper we stress that the limiting measures µ∞, ν∞ are singular continuous
almost always. Only in exceptional cases they might become point or absolutely continu-
ous. This fact exposes the mostly distinctive property of the considered conflict dynami-
cal systems. In particular, starting with absolutely continuous measures µ, ν ∈ Mac and
applying an infinite sequence of conflict composition �, we usually obtain in the limit
the singular continuous measures µ∞, ν∞ ∈ Msc of the Cantor type. The supports of
µ∞, ν∞ ∈ Msc are nowhere dense sets of zero Lebesgue measure. We will produce the
conditions ensuring this result.
At the same time in the exceptional cases we obtain the criterion guaranteeing that
one of limiting measures will be purely point or purely absolutely continuous. Of course,
the appearance of the such type of limiting measures is rather exotic. We illustrate it by
simple examples.
Finally we remark that the growing interest in study of singular continuous mea-
sures and their supports (=spectra) takes place not only in the fractal geometry but in
mathematical physics too. In particular, in the connection with the problem of physical
interpretation of the fine effects from presence of the singular spectrum (see, for example,
[3, 8, 18, 24] ). We also refer reader to the papers [6, 23] where conflict composition � has
been used in the applications for construction of the migration and population models .
2. The similar structure measures
In the present paper we shall use a specific class of measures, the similar structure
measures, having a certain structural similarity of its supports. This class of measures are
exactly appropriate for construction of the conflict dynamical systems. The set of similar
structure measures are considerably wider than the well-known class of the self-similar
measures introduced by Hutchinson [13] (see also [12, 24]).
Let us describe shortly the notion of probability similar structure measure which are
supported on the segment ∆0 = [0, 1].
Let T = {Tik}n
i=1, k = 1, 2, . . . , 2 ≤ n < ∞ be a family of contractive similarities on
R1 having the following properties. For all k
(a) 0 < c ≤ cik < 1,
where cik is the contraction coefficient for Tik,
(b) ∆0 =
n
⋃
i=1
Tik∆0,
(c) λ(Tik∆0
⋂
Ti′k∆0) = 0, i �= i′,
where λ denotes Lebesgue measure.
Put
Ui1···ik,i′
1
···i′
k
:= Ti11 · · ·Tikk(Ti′
1
1 · · ·Ti′
k
k)−1, 1 ≤ ik, i′k ≤ n.
We recall that each Tikk is a bijection and hence the inverse transformation T−1
ikk is well
defined.
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 17
We say that set S ⊆ ∆0 has a structural similarity, if for some fixed family of contrac-
tive similarities T with above properties, S admits the representation
(2.1) S =
n
⋃
i1=1
si1 , si1 =
n
⋃
i2=1
si1i2 , · · · , si1···ik−1
=
n
⋃
ik=1
si1i2···ik
, · · · ,
where si1···ik
⊂ Ti11 · · ·Tikk∆0 and all non-empty si1···ik
of every fixed rank k = 1, 2, . . .
are similar one to other
(2.2) si1···ik
= Ui1···ik,i′
1
···i′
k
si′
1
···i′
k
.
We observe that
(2.3) diam(si1···ik
) → 0, k → ∞,
and
(2.4) λ
(
scl
i1···ik
⋂
scl
i′
1
···i′
k
)
= 0, if il �= i′l
at least for single 1 ≤ l ≤ k, where cl stands for closure.
Emphasize that in contrast to the definition of a self-similar set (see [13, 24]), subsets
of different ranks si1 , si1i2 , · · · , si1···ik
, · · · are in general not similar. In particular, they
are not similar in general to the whole set S. Roughly speaking, each similar structure
set on any ε-level (ε > 0) admits the decomposition into a finite amount of ”cells” with
diameters smaller or equal to ε, which are similar one to other but are not necessarily
similar to whole set and to ”cells” of another level of decomposition.
A Borel measure µ on ∆0 is called the similar structure measure if its support S =
suppµ ≡ Sµ has the structural similarity (see (2.1)–(2.4)) and besides,
(2.5) µ
(
si1···ik
⋂
si′
1
···i′
k
)
= 0, if il �= i′l
at least for single 1 ≤ l ≤ k, and the defined for non-empty sets si1···ik
and si1···ik−1
the
ratios
(2.6)
µ(si1···ik−1ik
)
µ(si1···ik−1
)
=: pikk > 0 (si0 ≡ ∆0)
are independent on i1, . . . , ik−1. We put pikk = 0 if si1···ik
is empty. The family of such
measures will be denoted by Mss(∆0) ≡ Mss (ss stands for similar structure).
Every measure µ ∈ Mss may be fixed in the following way.
Let us consider an infinite sequence of stochastic vectors qk, k = 1, 2, . . . , from Rn, 1 <
n < ∞ with strictly positive coordinates
qk = (q1k, q2k, . . . , qnk), q1k, . . . , qnk > 0, q1k + · · · + qnk = 1.
We assume that
(2.7) qinf := inf
i,k
qik > 0.
Then
(2.8) Q = {qk}∞k=1 = {qik}n, ∞
i=1, k=1
denotes the stochastic matrix whose columns are formed with coordinates of vectors qk.
Each matrix Q fixes on the segment ∆0 the so-call (see [22, 2]) Q∗-representation, which
we call here simply as the Q-representation on ∆0. Let us shortly recall this construction.
For each k = 1, 2, . . . let us decompose the segment ∆0 from the left to the right in
the family of closed intervals of rank k
∆0 =
n
⋃
i1=1
∆i1 =
n
⋃
i1,i2=1
∆i1i2 = · · · =
n
⋃
i1,...,ik=1
∆i1i2···ik
= · · ·
18 T. KARATAIEVA AND V. KOSHMANENKO
We suppose that different intervals of the same rank overlap at most only in the extreme
points. That is, the lengths of these intervals are fixed by elements of the matrix Q
(2.9) λ(∆i1 ) = qi11, λ(∆i1i2) = qi11qi22, · · · , λ(∆i1i2···ik
) = qi11qi22 · · · qikk, · · ·
Of course, these decompositions are consistent
(2.10)
∆0 = [0, 1] =
n
⋃
i1=1
∆i1 , ∆i1 =
n
⋃
i2=1
∆i1i2 , · · · , ∆i1i2···ik−1 =
n
⋃
ik=1
∆i1i2···ik
, · · ·
It is not hard to see that relations
∆i1i2···ik
= Ti11 · · ·Tikk∆0
define the one-to-one correspondence between the family of all Q-representations on ∆0
and the family of above mentioned contractive similarities T .
Clearly that under a given Q-representation the σ-algebra generated by the family of
subsets {∆i1···ik
}∞k=0 coincides with the usual Borel σ-algebra on [0, 1]. Moreover due to
(2.7) for every point x ∈ [0, 1] there exists a sequence of embedded segments ∆i1i2···ik
containing this point and such that x =
⋂∞
k=1 ∆i1i2···ik
. This fact can be written in the
following form:
(2.11) x = ∆i1i2···ik··· ,
where the sequence of indexes i1, i2, . . . , ik, . . . defines the point x uniquely. Thus,
i1, i2, . . . , ik, . . . are coordinates of x in the fixed Q-representation.
We remark that each point of a view (2.11) admits the representation in the terms of
the corresponding family of similarities T :
x = lim
k→∞
yk, yk = Ti11 · · ·Tikky, ∀y ∈ R1.
In what follows we will fix some Q-representation on ∆0 or, that is the same, a family
of contractive similarities T .
From (2.6) it follows that each measure µ ∈ Mss is uniquely associated with the
stochastic matrix
(2.12) P ≡ {pk}∞k=1 = {pik}n, ∞
i=1, k=1 ,
whose columns are formed by coordinates of stochastic vectors pk ∈ Rn, k = 1, 2, . . .
pk = (p1k, . . . , pnk), p1k, . . . , pnk ≥ 0, p1k + · · · + pnk = 1.
In order to point the dependence µ from P we write sometimes µ = µP .
The construction of the measure µP starting of P may be realized in the following
way.
Using the first k columns of matrices Q and P we define the Borel measure µk on ∆0
by the formula
(2.13) µk :=
n
∑
i1,i2,...,ik=1
ci1i2···ik
λi1i2···ik
, ci1i2···ik
:=
pi11pi22 · · · pikk
qi11qi22 · · · qikk
,
where λi1i2···ik
:= λ|∆i1i2···ik
denotes the restriction of Lebesgue measure on segment
∆i1···ik
. By (2.13) it follows that
(2.14) µ1(∆i1) = pii1, . . . , µk(∆i1...ik
) = pi11pi22 · · · pikk, · · ·
Hence, µk, k = 1, 2, . . . is a sequence of probability measures uniformly distributed on
∆i1...ik
. From (2.13), (2.14) it also follows that the plot for the distribution function
fk(x) = µk{(−∞, x)} for the measure µk is a piece-wise linear non-decreasing line. The
values of the function fk(x), k > 1 in end-points of each segment ∆i1···ik−1
of the rank
k − 1 are the same as for the function fk−1(x). Obviously for the sequence {fk(x)}∞k=1
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 19
we have fk(x) → f(x), k → ∞ in the sense of the uniform convergence, where f(x) is
a left continuous non-decreasing function. Thus, f(x) is the distribution function for a
fixed probability measure µ on [0, 1]. So we can write
µ = lim
k→∞
µk.
The above constructed measure µ has the similar structure on all segments ∆i1···ik
under each fixed k ≥ 1. Besides, by condition (2.7) λ(∆i1...ik
) → 0, k → ∞. Therefore,
this measure belongs to the class Mss.
Let us discuss in more details the structure of the support for µ, which will be often
called the spectrum of µ. This set can be written as follows:
(2.15) Sµ ≡ suppµ =
⋂
k
Sµk
, Sµk
= suppµk.
The set Sµ admits another representation
Sµ ≡ suppµ = ∆0\S̄0(µ), S0(µ) =
⋃
{ik:pikk=0}
∆int
i1···ik
where the union takes place along open intervals ∆int
i1···ik
(int means interior) for which
pikk = 0 and the bar stands for adding to S0(µ) the isolated end-points if they appear.
We note that all intervals ∆int
i1···ik
such that pill = 0 with some l < k has also empty
intersection with Sµ, because these intervals are included in ∆int
i1···il
.
We are able now to define
(2.16) si1···ik
:= Sµ
⋂
∆#
i1···ik
,
where # denotes the possible removing from ∆i1···ik
one of the end-points. Namely,
we have to remove the right end-point, if PL
k :=
∏∞
s=1 p1,k+s > 0, or the left one, if
PR
k :=
∏∞
s=1 pn,k+s > 0. The set ∆i1···ik
in (2.16) remains without any changes, if
PL
k = PR
k = 0. This procedure is caused by condition (2.5). Indeed, if one of the
conditions PL
k > 0 or PR
k > 0 holds (they can not be fulfilled simultaneously!) the
corresponding end-points of the segment ∆i1···ik
become atoms of the point spectrum
for the measure µ (see below Theorem 3.1). Thus the above describing construction of
si1···ik
ensures the validity of condition (2.5). Of course, it may occurs that si1···ik
is
empty.
It is evident now that the non-empty subsets si1···ik
, si′
1
···i′
k
defined by (2.16) are
similar one to other for each fixed k ≥ 1. It is true since all segments ∆i1···ik
, ∆i′
1
···i′
k
are similar, and on every step described above, only similar sets from these segments has
been removed. In fact, by virtue of the previous observations subsets si1···ik
, si′
1
···i′
k
are
connected by a certain kind of similarity of the form (2.2).
Further, due to (2.14) we obtain the important relations
(2.17) µ(si1···ik
) = µ(∆#
i1···ik
) = µk(∆i1···ik
) = pi11 · · · pikk.
We observe that si1···ik
is non-empty iff
µ(si1···ik
) = pi11 · · · pikk �= 0.
Thus (2.5) and (2.6) is fulfilled.
By the way, from (2.11) and (2.17) it follows that
(2.18) µ(x) =
∏
k
pikk,
where ik, k = 1, 2, . . . denote coordinates of a point x in Q-representation (2.11).
Conversely, one can reconstruct a certain stochastic P -matrix by a given similar struc-
ture measure µ ∈ Mss using meanings µ(si1···ik
) (see (2.17)).
20 T. KARATAIEVA AND V. KOSHMANENKO
We shall use below the following well-known result [22].
Lemma 2.1. The constructed above measure µP = µ is singular with necessity if for
infinite set of values k at least one coordinate of pk equals zero.
Proof. Without loss of generality we assume
0 = pi
′
1
1 = · · · = pi
′
k
k = · · ·
By (2) we have
S0(µ) ⊃
⋃
{pi′
k
k=0}
∆int
i1···ik−1i′
k
.
So we can calculate Lebesgue measure of the set S0(µ)
λ(S0(µ)) ≥ qi
′
1
1 +
∑
i1 �=i′
1
qi11qi′
2
2 +
∑
i1 �=i′
1
qi11
∑
i2 �=i′
2
qi22qi′
3
3 + · · ·
+
∑
i1 �=i′
1
qi11
∑
i2 �=i′
2
qi22 · · ·
∑
ik−1 �=i′
k−1
qik−1k−1qi′
k
k + · · ·
= qi′
1
1 + qi′
2
2(1 − qi′
1
1) + qi′
3
3(1 − qi′
1
1)(1 − qi′
2
2) + · · ·
+ qi′
k
k(1 − qi′
1
1) · · · (1 − qi′
k−1
k−1) + · · ·
Obviously λ(S0(µ)) = 0 because
(1 − qi′
1
1)(1 − qi′
2
2) · · · (1 − qi′
k
k) · · · =
∞
∏
k=1
(1 − qi′
k
k) = 0,
where we take into account (2.7). �
3. Spectral purity of the similar structure measures
One of the characteristic property of measures µ ∈ Mss(∆0) is that each such measure
has only a single component in the Lebesgue decomposition: either purely point, or
purely absolutely continuous, or purely singular continuous (see [2]). Here we produce
this fact about purity of a spectrum (=support) of the similar structure measure as a
relevant version of the famous Jessen-Wintner theorem for probability distributions (see
[10, 14, 22]). The proof in essence belongs to G. Torbin and actually is the same as in
[2].
Let us introduce some notations. For a measure µ = µP of class Mss(∆0) we write
µ ∈ Mss
pp,Mss
ac,Mss
sc, if this measure is purely point (µ = µpp), purely absolutely contin-
uous (µ = µac), or purely singular continuous (µ = µsc), respectively. Given stochastic
matrices P and Q, we define two values
Pmax(µ) :=
∞
∏
k=1
pmax,k and ρ(µ, λ) :=
∞
∏
k=1
ρk,
where pmax,k := maxn
i=1{pik}, ρk := Σn
i=1
√
pikqik.
Theorem 3.1. Each similar structure measure µ = µP ∈ Mss(∆0) associated with the
stochastic matrix P under the fixed Q-representation on ∆0 = [0, 1] has a pure spectral
type. Namely,
(a) µ ∈ Mss
pp
, if and only if Pmax(µ) > 0,
(b) µ ∈ Mss
ac
, if and only if ρ(µ, λ) > 0,
(c) µ ∈ Mss
sc
, if and only if Pmax(µ) = 0 and ρ(µ, λ) = 0.
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 21
Proof. Let us denote by
(Ω,A, µ∗) =
∞
∏
k=1
(Ωk,Ak, mk)
the infinite direct product of discrete probability spaces
(Ωk,Ak, mk), Ωk = {ωik
}n
ik=1, mk(ωik
) = pikk,
where space Ωk and σ-algebra Ak depend on k only formally (in fact they are the same
for all k). But pikk is changed with ik and k in a correspondence with the matrix P .
Thus the measure µ∗ is uniquely connected with the matrix P . In particular, for the
cylindrical sets Ωi1···ik
:= ωi1 × · · · × ωik
× ∏∞
l=1 Ωk+l ∈ A,
(3.1) µ∗(Ωi1···ik
) =
k
∏
s=1
piss.
In what follows we use the measurable mapping from Ω = Ω1 × Ω2 · · · × Ωk · · · to [0, 1],
π : Ω ∋ ω∗ = {ωi1 × ωi2 × · · · × ωik
× · · · } → x =
∞
⋂
k=1
∆i1i2···ik··· ∈ ∆0.
We note that Ω possibly is replaced by Ω�Ω0, where the set Ω0 is not empty only if
there exists k0 such that one of the following inequalities occurs:
PL
k0
=
∏
s≥k0
p1s > 0, PR
k0
=
∏
s≥k0
pns > 0.
Namely,
Ω0 = {ω∗ ∈ Ω | ωik
= ω1k
, ∀k ≥ k0},
or
Ω0 = {ω∗ ∈ Ω | ωik
= ωnk
, ∀k ≥ k0},
respectively to the first or to the second case. In any case the mapping π preserves the
meanings of the measure:
µ∗(Ωi1···ik
) = µ(∆#
i1···ik
) = pi11 · · · pikk,
where ∆i1···ik
= π(Ωi1···ik
)) (see (2.16), (2.17) (3.1)). By this reason µ is often called the
image-measure for µ∗ with respect to the mapping π. We will use this property of π to
preserve the meanings of measure without reminding.
Let us prove (a). If Pmax(µ) = 0, then µ∗(ω∗) = 0 for each point ω∗ ∈ Ω. Indeed,
µ∗(ω∗) =
∞
∏
k=1
mk(ωik
) =
∞
∏
k=1
pikk ≤
∞
∏
k=1
pmax,k = 0.
In this case µ∗ is continuous and so is µ. Thus, we proved the necessity of the condition
Pmax(µ) > 0 in the statement (a).
To prove the sufficiency let us introduce the set
A+ := {ω∗ ∈ Ω | µ∗(ω∗) > 0}.
A+ is not empty, if Pmax(µ) > 0, because this set contains at least the point ω∗
max =
ωi′
1
× · · · ×ωi′
k
× · · · with coordinates for which pi′
k
k = pmax,k. Besides, A+ contains also
all points ω∗ ∈ Ω such that ωik
�= ωi′
k
for a finite amount of coordinates under condition
pikk > 0. The set A+ does not contain another points. It follows from the condition
Pmax(µ) > 0. Indeed, since vectors pk are stochastic the sequence pmax,k is an unique
one (up to changing of a finite amount of coordinates) which converges to 1. In other
words, ω∗ ∈ A+ means that coordinates ω∗ differ from coordinates of point ω∗
max only
in finite number of places. Thus, the set A+ is countable. Applying now Kolmogorov’s
principle of zero and unit we conclude: either µ∗(A+) = 0 or µ∗(A+) = 1. However
22 T. KARATAIEVA AND V. KOSHMANENKO
µ∗(A+) ≥ µ∗(ω∗
max) > 0. Therefore the statement µ∗(A+) = 1 is true. This means that
the measure µ∗ is concentrated on a not more than countable number of atoms. Thus
µ∗ = µ∗
pp and hence for its image-measure under the mapping π, the equality µ = µpp is
established too.
The statements (b), (c) are direct consequences of the below presented Kakutani’s
theorem (see [14, 7] and also [1, 2] ). �
Theorem. ([14]). Let
(Ω,A, µ∗) =
∞
∏
k=1
(Ωk,Ak, mk) and (Ω,A, λ∗) =
∞
∏
k=1
(Ωk,Ak, λk)
be a pair of the infinite direct products of abstract probability spaces. Suppose that each
measure mk is equivalent to λk (notation, mk ∼ λk). Define
ρ(µ∗, λ∗) :=
∞
∏
k=1
ρk, ρk =
∫
Ωk
√
ϕk(ω)dλk(ω),
where ϕk(ω) = dmk(ω)
dλk(ω) is the Radon-Nikodim derivative. Then
(a) µ∗ ∼ λ∗, only if ρ(µ∗, λ∗) > 0;
(b) µ∗ ⊥ λ∗, only if ρ(µ∗, λ∗) = 0, where ⊥ denotes the mutual singularity.
For our consideration we may put Ωk = {ω1, . . . , ωn}, n ≥ 2 for all k ≥ 1, assign
λk(ωik
) = qikk, mk(ωik
) = pikk, and take into account that µP coincides with Lebesgue
measure on ∆0, if P = Q.
Finally we remark that µ = µpp has the discrete spectrum which is concentrated in a
finite set of atoms only if there exists some k0 ≥ 1 such that pmax,k = 1 for all k ≥ k0. In
this case the number of points in A+ is the same as the amount of atoms in the spectrum
of µpp. Otherwise, each point of the spectrum is accumulating.
4. The conflict dynamical system
Let us consider for a fixed k ≥ 1 a pair of discrete probability measures mk, vk de-
fined on a common space Ωk = {ω1k
, . . . , ωnk
}, nk > 1. These measures are naturally
associated with the stochastic vectors pk = (p1k, . . . , pnk), rk = (r1k, . . . , rnk)
mk(ωik
) := pik, vk(ωik
) := rik, ik = 1, . . . , n.
At first we define the conflict dynamical system for a couple of measures mk, vk
(4.1) {mN−1
k , vN−1
k } �−→ {mN
k , vN
k }, N = 1, 2, . . . , m0
k = mk, v0
k = vk
using the non-commutative and non-linear transformation (the conflict composition �)
in terms of the stochastic vectors pk, rk (for more details see [15, 16])
pN
k = pn−1
k � rN−1
k , rN
k = rN−1
k � pN−1
k , p0
k = pk, r0
k = rk.
Here the coordinates of vectors pN
k , rN
k are calculated by the formulae
(4.2) pN
ik :=
pN−1
ik (1 − rN−1
ik )
zN−1
k
, rN
ik :=
rN−1
ik (1 − pN−1
ik )
zN−1
k
,
with zN−1
k = 1 − (pN−1
k , rN−1
k ), where (·, ·) stands for the inner product in Rn. The
formulae (4.2) are well-defined if we suppose the requirement that (pk, rk) �= 1. Then the
condition (pN−1
k , rN−1
k ) �= 1 automatically occurs for all N . For coming to the measures
in (4.1) we put mN
k (ωik
) =: pN
ik, vN
k (ωik
) =: rN
ik , i = 1, . . . , n.
One can interpret the formulas (4.2) as follows (see [16] and cf. with [11, 20]). Let the
measures mk,vk correspond to some opponents A, B. Then the value pN
ik (rN
ik) means the
probability for A (B) to occupy the position ωik
after N steps of the conflict actions. This
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 23
value equals to the conditional probability to occupy the position ωik
by the opponent
A (B) under the assumption that A could not meet B at any of n positions.
Let us consider now a pair of similar structure measures µ = µP , ν = νR ∈ Mss(∆0)
associated with stochastic matrices P and R respectively. The matrix R has a form
similar to (2.12) and it is composed with a sequence of the stochastic vectors rk ∈
Rn
+, k = 1, 2, . . . We suppose that the Q-representation of the interval ∆0 = [0, 1] (or
that is the same, the family of contractive similarities T on R1) is fixed.
By a pair of similar structure measures µP , νR the conflict dynamical system is defined
as follows:
(4.3) {µN−1, νN−1} �−→ {µN , νN}, N = 1, 2, . . . , µ0 = µP , ν0 = νR,
where the measures µN , νN , N = 1, 2, . . . are associated with the stochastic matrices
PN , RN respectively. In turn, the matrices PN , RN is constructed using the stochastic
vectors pN
k and rN
k defined by (4.2).
The next theorem follows from the so-called conflict Theorem (see [15, 16]) and The-
orem 2 and Theorem 3 from [4].
Theorem 4.1. Let µ = µP , ν = νR ∈ Mss(∆0) be a pair of similar structure measures.
Assume that stochastic matrices P, R satisfy the following condition,
(4.4) (pk, rk) �= 1, ∀k.
Then there exist two limiting measures
µ∞ = lim
N→∞
µN , ν∞ = lim
N→∞
νN ,
in the sense of the uniform convergence, which are invariant with respect to the compo-
sition � defined in (4.3).
Further, the elements of the stochastic matrices P∞, R∞ associated with the limiting
measures µ∞, ν∞ have the following explicit form:
if pk �= rk, then
(4.5) p∞ik =
{
dik/Dk, i ∈ N+,k
0, i /∈ N+,k
, r∞ik =
{
−dik/Dk, i ∈ N−,k
0, i /∈ N−,k
where
dik = pik − rik, N+,k := {i : dik > 0}, N−,k := {i : dik < 0},
Dk =
∑
i∈N+,k
dik =
∑
i∈N−,k
−dik;
if pk = rk , then the non-zero coordinates of vectors p∞
k = r∞k have a form
(4.6) p∞ik = r∞ik = 1/ck,
where ck denotes an amount of non-zero initial coordinates pik = rik �= 0.
Thus, one can easy calculate the values of measures µ∞, µ∞ on subsets ∆#
i1···ik
:
µ∞(∆#
i1···ik
) =
∏
s≤k
diss/Ds,
under assumption that is ∈ N+,s for all s ≤ k. In particular, if a coordinate is of the
vector ps has some priority with respect to the corresponding coordinate of the vector
rs, s ≤ k: piss > riss, then due to (4.5) we obtain that p∞iss > 0, but r∞iss = 0,
and therefore µ∞(∆#
i1···ik
) > 0, but ν∞(∆#
i1···ik
) = 0. Nevertheless, it is possible that
µ∞(∆#
i1···ik
) = ν∞(∆#
i1···ik
) = 0 even in the case ps �= rs, s ≤ k. It happens due to (4.5)
for all coordinates such that piss = riss > 0 since then diss = p∞iss = r∞iss = 0.
24 T. KARATAIEVA AND V. KOSHMANENKO
5. Origination of the singular continuous spectrum
In this main section of the paper we show that origination of the singular continuous
spectrum is generic for the limiting measures µ∞ and ν∞.
We say that similar structure measures µ = µP , ν = νR are essentially different, write
simply µ �= ν, if pk �= rk almost for all k (this means that pk = rk at most for a finite
amount of indices k). We say that the measure µ has a local priority with respect to the
measure ν at a position ik, if pikk > rikk. If the measure µ has a local priority with respect
to the measure ν almost for all k from some sequence (= a direction) i1, i2, . . . , ik, . . .,
then we say that µ has the directed priority with respect to ν, write (µ > ν)i1i2···ik···
Finally, we say that µ has the shock directed priority with respect to the essentially
different measure ν, if the the total value of local priorities equals to infinity
(5.1)
∑
k
dikk = ∞, dikk = pikk − rikk
and moreover, the normalized meanings of the local priorities
dik
:=
dikk
Dk
, Dk = 1/2
∞
∑
ik=1
|dikk|
converge to 1 so quickly that
(5.2) d(µ, ν) :=
∏
k
dik
> 0.
Clearly that condition (5.2) is highly specific and in general does not fulfilled.
The main result of the paper is formulated in the next theorem. It assert that for
any couple of starting essentially different measures µ, ν, the generic type of the limiting
measures µ∞ and ν∞ in the conflict dynamic system (4.3) is singular continuous.
Theorem 5.1. Let µ = µP , ν = νR ∈ Mss be a couple of similar structure measures
on [0, 1] associated with stochastic matrices P = {pk}∞k=1, R = {rk}∞k=1 respectively.
Assume µ, ν are essentially different and neither µ nor ν has a shock directed priority
one respect to other, in particular, d(µ, ν) = d(ν, µ) = 0 for any sequence i1, i2, . . . , ik, . . .
Then both µ∞, ν∞ ∈ Msc.
Proof. If µ �= ν then due to (4.5) all limiting vectors p∞
k , r∞k for almost all k contain
at least one non-zero coordinate. Therefore, the measures µ∞, ν∞ are singular due
to Lemma 2.1. We have only to check that these measures are continuous, i.e., that
µ∞(x) = ν∞(x) = 0, ∀x ∈ ∆0. Indeed, by (2.11) and (2.18) a value of the measure µ∞
at a point x with coordinates ik is defined by the formula µ∞(x) =
∏
k p∞ikk. The latter
product is possible strictly positive, only if x belongs to the point spectrum of µ∞ (see
Theorem 3.1), i.e., if the following condition
(5.3) Pmax(µ
∞) =
∏
k
p∞max,k > 0, p∞max,k := maxi{p∞ik} = max
ik
dikk
Dk
is carried (see Theorem 4.1). Condition (5.3) means that the convergence di′
k
→ 1, k →
∞ is so quick that
∏
k di′
k
> 0, where di′
k
:= maxi dik/Dk (see (5.2)). But then the
sequence i′k (on this sequence the maximal values p∞max,k are realized) constitutes the
shock directed priority of the measure µ with respect to ν (see (5.2)). Thus we get the
contradiction with assumptions of the theorem. Thus µ∞(x) = 0 for each point x ∈ [0, 1]
and therefore µ∞ ∈ Msc. The similar arguments is true also for ν. �
In [21] (see also [9, 18, 19]) it has been shown that operators with singular continuous
spectrum are generic. The similar fact is valid for the class of singular continuous mea-
sures in the space Mss. Here we’ll construct on the space of similar structure measures
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 25
Mss the non-trivial ”global” probability measure m and prove that the sub-class Mss
sc is
a set of full m-measure (see Theorem 5.2 below). Thus, the statement that in the dy-
namical conflict system limiting measures µ∞, ν∞ are almost always singular continuous,
has a more precise sense.
To this end we firstly construct in Mss a family of cylinder subsets defined by induction
as follows.
On the first step we put
I
βi1
i1
:= {µ ∈ Mss| µ(∆i1) ∈ βi1 , µ(∆i1 ) = max
j1=1,...,n
{µ(∆j1)}},
where βi1 stands for a usual Borel set from the σ-algebra B of subsets from [0, 1].
Further, for every k ≥ 1 we define
I
βi1
···βik
i1···ik
:= {µ ∈ I
βi1
···βik−1
i1···ik−1
| µ(∆i1···ik
)
µ(∆i1···ik−1
)
∈ βik
, µ(∆i1···ik
) = max
jk=1,...,n
{µ(∆i1···ik−1jk
)}},
where µ(∆i0 ) ≡ µ(∆0) = 1.
It is clear that
Mss =
n
⋃
i1,...,ik=1
Ii1···ik
, Ii1···ik
:=
⋃
βi1
,...,βik
∈B
I
βi1
···βik
i1···ik
, k = 1, 2, . . .
Starting of the family
{Iβi1
···βik
i1···ik
, βik
∈ B, ik ∈ {1, . . . , n}, k = 1, 2, . . .}
and using the standard procedure (see, for example, [5]) we generate the rink of subsets
which is denoted as R.
Now we introduce on R the probability measure m. At first we define it on cylinder
sets:
m(I
βi1
···βik
i1···ik
) := qi11 · · · qikk ·
k
∏
l=1
λ′(βil
),
where λ′(βi1) := cnλ(βi1), cn = n
n−1 taking into account that λ([1/n, 1]) = n−1
n . It is
easy to check that
n
∑
i1,...,ik=1
m(Ii1···ik
) = m
(
n
⋃
i1,...,ik=1
(
⋃
βi1
,··· ,βik
∈B
I
βi1
···βik
i1···ik
))
= 1,
since
∑n
i1,...,ik=1 qi1 · · · qik
= 1 and λ′([1/n, 1]) = 1.
Let m∗ be the outer measure for m. Denote by J ss the minimal σ-algebra generated
by the family of the cylinder subsets {Iβi1
···βik
i1···ik
} in Mss. The standard procedure of re-
striction of m∗ on σ−algebra J ss concludes the construction of the σ-additive probability
measure, which we denote by m.
Let us recall that Mss
pp, Mss
ac, Mss
sc denote the classes of purely point, purely absolutely
continuous and purely singularly continuous measures respectively.
Theorem 5.2. Under a fixed Q-representation of the segment [0, 1] the class of purely
singular continuous similar structure measures Mss
sc([0, 1]) is a set of full measure for m:
m(Mss
sc) = 1.
Proof. At first we show that
m∗(Mss
pp) = m∗(Mss
ac) = 0.
26 T. KARATAIEVA AND V. KOSHMANENKO
To this end we fix a sequence εk → 0, k → ∞ such that
∑
k εk = ∞. For example one
can put εk = 1/k. By εk we introduce the sequence of subsets
Ik :=
n
⋃
i1,...,ik=1
I
βi1
···βik
i1···ik
, βi1 = [a1, 1], . . . , βik
= [ak, 1],
where ak are chosen in such a case that λ′(βik
) = εk. Let us denote
Ipp,k := {µ ∈ Mss
pp|µ ∈ Ik} = Ik
⋂
Mss
pp.
It is clear that
m∗(Ipp,k) ≤ m(Ik) ≤
n
∑
i1,...,ik=1
m(I
βi1
···βik
i1···ik
)
=
n
∑
i1,...,ik=1
qi11 · · · qikk
k
∏
l=1
λ′(βil
) ≤
n
∑
i1,...,ik=1
qi11 · · · qikkλ′(βik
) = εk,
since
∑n
i1,...,ik=1 qi11 · · · qikk = 1.
It is easy to see that for each measure µ ∈ Mss
pp there exists k0 = k0(µ) such that
µ ∈ Ik for all k ≥ k0 (see condition (a) in Theorem 3.1, and also (5.3)). Therefore
µ ∈ Ipp,k beginning with some k which is defined by µ. Hence
(5.4) δk := m
(
Ipp,k\
k−1
⋃
l=1
Ipp,l
)
→ 0, k → ∞
Let us denote I ′
pp,k :=
⋃k
l=1 Ipp,l. Clearly I ′
pp,k ⊂ I′
pp,k+1 and also Mss
pp =
⋃∞
k=1 I ′
pp,k.
Thus, by (5.4) and due to the σ-additive property of the outer measure, m∗(Mss
pp) = 0.
Indeed by virtue of the theorem on continuity for a union of subsets (see. [5], Theorem
6.2) we have
m∗(Mss
pp) = lim
k→∞
m∗
(
k
⋃
l=1
I ′
pp,l
)
= lim
k→∞
m∗(Ipp,k) ≤ lim
k→∞
m(Ik) = lim
k→∞
εk = 0.
The equality m(Mss
ac) = 0 we prove on a similar way. With this aim we introduce
another sequence of subsets with the same notations
Ik := {µ ∈ Mss|µ ∈ I
βi1
...βik
i1...ik
, βil
= [qill − εl/2, qill + εl/2], l = 1, ..., k}
and
Iac,k := {µ ∈ Mss
ac|µ ∈ Ik} = Ik
⋂
Mss
ac.
We note that in the case where qikk = 1/n one can put βik
= [1/n, 1/n + εk]. It is not
hard to understand that for every measure µ ∈ Mss
ac there exists k0 = k0(µ) such that
µ ∈ Iac,k for all k ≥ k0. Therefore
(5.5) δk := m
(
Iac,k\
k−1
⋃
l=1
Iac,l
)
→ 0, k → ∞.
Let us denote I ′
ac,k :=
⋃k
l=1 Iac,l. Clearly that I ′
ac,k ⊂ I′
ac,k+1 and also Mss
ac =
⋃∞
k=1 I ′
ac,k. Hence m∗(Mss
ac) = 0 since
m∗(Mss
ac) = lim
k→∞
m∗
(
k
⋃
l=1
I ′
ac,l
)
= lim
k→∞
m∗(Iac,k) ≤ lim
k→∞
m(Ik) = lim
k→∞
εk = 0,
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 27
where we used
m(Ik) =
n
∑
i1,...,ik=1
qi11 · · · qikkλ′(βik
) = εk.
By m∗(Mss
pp) = m∗(Mss
ac) = 0 we conclude that Mss
pp, Mss
ac ∈ J ss (see, for instance,
Theorem 6.1 from [5]). Thus m(Mss
pp) = m(Mss
ac) = 0. It means by Theorem 3.1 that
m(Mss
sc) = 1. �
6. When µ∞ ∈ Mpp ?
We want to find the conditions ensuring the pure point type at least for one of the
limiting measures.
The next theorem follows from assertion (a) of Theorem 3.1, formulae (4.5), and proof
of Theorem 5.1 (especially the formula (5.3)).
Theorem 6.1. The limiting measure µ∞ is pure point, µ∞ ∈ Mpp, if and only if the
initial measure µ possesses a directed shock priority with respect to ν, i.e., if (5.2) is
valid.
Proof. By Theorem 3.1 condition (5.3) is necessary and sufficient for µ∞ ∈ Mpp. This
condition is obviously equivalent to (5.2). �
We note, it might happen that each of measures µ �= ν could have their own directed
priorities one with respect to other (even not one), i.e., the relations (µ > ν)i1i2···ik···,
(ν > µ)i′
1
i′
2
···i′
k
··· may fulfilled simultaneously. Moreover, measures µ, ν may have simul-
taneously shock directed priorities one with respect to other. So, this case is not excluded
that both limiting measures are pure point. The next result is rather unexpected.
Theorem 6.2. Let µ, ν be a couple of different similar structure measures. The limiting
measure µ∞ has a pure point type, µ∞ ∈ Mpp, if and only if in the set of all directed
priorities there is unique one (µ > ν)i′
1
i′
2
···i′
k
···, up to replacing of a finite amount of
indices, which satisfies the condition
(6.1)
∞
∑
k=1
di′
k
= ∞, di′
k
=
di′
k
k
Dk
.
Proof. Let µ∞ ∈ Mpp. Then due to Theorem 6.1 conditions (5.2) and (5.3) are held.
This means that there exists the directed priority (µ > ν)i′
1
,i′
2
,...,i′
k
,... such that for any
other direction the differences dikk = pikk − rikk, ik �= i′k converge to zero when k → ∞.
Moreover, this convergence is so quick that the total sum of the above differences is finite,
i.e., the following series is convergent,
(6.2)
∞
∑
k=1
∑
ik �=i′
k
dik
< ∞.
Now we observe that conditions (5.3) and (6.2) are equivalent since p∞max,k = di′
k
. There-
fore the unique non-convergent series consist of di′
k
.
Conversely, let condition (6.1) is fulfilled for the unique directed priority fixed by
the direction i′1, . . . , i
′
k, . . . By Theorem 4.1 all non-zero coordinates of the matrix P∞
associated with µ∞ have a form p∞ik = dik
Dk
= dik
. Therefore, by condition (6.1) we
have that
∑
k p∞i′
k
k =
∑
k di′
k
= ∞ just for the direction i′1, i
′
2, . . . , i
′
k, . . . If we assume
that µ∞ /∈ Mpp, then
∏
k p∞ikk = 0 for any (other) directions i1, . . . , ik, . . . And then,
28 T. KARATAIEVA AND V. KOSHMANENKO
instead (6.2) we, in particular, have:
∑
k(
∑
ik �=i′
k
dikk/Dk) = ∞. Moreover, the series
constituted from the maximal values among dikk/Dk, ik �= i′k is divergent too
∑
k
max
ik �=i′
k
(dikk/Dk) = ∞.
So we get the contradiction with the assumption about of uniqueness of the direction
i′1, . . . , i
′
k, . . . for which (6.1) is fulfilled. Thus, µ∞ ∈ Mpp. �
As a consequence we get the fruitful criterion of continuity of the similar structure
measures.
Theorem 6.3. A measure µ = µP ∈ Mss(∆0) associated with the stochastic matrix P
is pure continuous, i.e., µ /∈ Mpp, if and only if one can pick at least two sequences of
elements of the matrix P , which are different, pikk �= pi′
k
k, k = 1, 2, . . ., and such that
(6.3)
∑
k
pikk =
∑
k
pi′
k
k = ∞.
Proof. Let (6.3) hold. Of course, it is possible that pikk = pi′
k
k for finite amount of
meanings of the index k. Assume in addition that µ ∈ Mpp. Then by Theorem 3.1,
∏∞
k=1 pmax,k > 0. Clearly, that in the such case
∑
k pmax,k =
∑
k pikk = ∞, where we
consider that pmax,k is reached just on ik. We emphasize that latter series is unique (up
to replacing of a finite number of coordinates) divergent one formed from elements of
the matrix P . This follows from the fact that the condition
∏
k pmax,k > 0 is equivalent
to the convergent of the series
∑
k(
∑
jk �=ik
pjkk). However then
∑
k pi′
k
k < ∞ for any
sequence i′k �= ik, that contradicts to (6.3). So, we prove that µ /∈ Mpp under condition
(6.3).
Let us prove the necessity of (6.3). Let µ /∈ Mpp and therefore the series
∑
k
∑
{ik:pikk �=pmax,k}
pikk
converges. Define pi′
k
k := max{ik:pikk �=pmax,k}{pikk}. It is clear that
∑
k pi′
k
k = ∞. But
∑
k pmax,k = ∞ too, since pmax,k ≥ 1/n (n ≥ 2) due to vectors pk are stochastic. Thus
we constructed two divergent series. The theorem is proved. �
We note that if |N+,k| = 1 (see Theorem 4.1) for almost all k (|N+,k| denotes the
cardinality of the set N+,k), then the measure µ∞ has a discrete spectrum and its
support consists of the finite number of points. This number is equal to the product
1 ≤
∏
k |N+,k| < ∞. But if |N+,k| ≥ 2 for an infinite amount of values k, then σ(µ∞) is
a countable nowhere dense set including only accumulating points.
7. Examples
In the last section we build the examples of the limiting measures which are not
necessarily singular continuous.
We write qik ∼ 1/n, if
∏
k(
∑
i
√
qik/n) > 0.
Example 7.1. Let measures µ, ν ∈ Mss be associated with the stochastic matrices
formed with vectors pk, rk ∈ Rn
+ n = 2. Then µ∞, ν∞ ∈ Mac, if qik ∼ 1/2 and
pk = rk almost for all k. Indeed, in this case p∞ik = r∞ik = 1/2 almost for all k. Therefore
ρ(µ, ν) = ρ(ν, µ) > 0 due to qik ∼ 1/2. Thus, by Theorem 3.1 both measures µ∞, ν∞ are
pure absolutely continuous.
But if µ, ν are different, then both limiting measures are pure point, µ∞, ν∞ ∈ Mpp
without any additional condition on qik.
ORIGINATION OF THE SINGULAR CONTINUOUS SPECTRUM . . . 29
Statement 7.1. Let n = 2 and pk �= rk for almost all k (excluded finite set is denoted
by K). Then both limiting measures µ∞, ν∞ are pure point.
Proof. If pk �= rk, k /∈ K and pk, rk ∈ R2 then one of coordinates of the vectors p∞
k , r∞k
is equal to unit, and other one is equal to zero (see (4.5)). Therefore
∏
k/∈K pmax,k =
∏
k/∈K rmax,k > 0 and µ∞, ν∞ ∈ Mpp due to Theorem 3.1. In such the case each of
considered measure is concentrated at 2|K| atoms, where |K| is an amount of point in
the set K. �
Example 7.2. Let the measures µ, ν ∈ Mss be associated with matrices formed with
the stochastic vectors pk, rk ∈ Rn
+, n = 3. Then similar to Example 7.1, µ∞, ν∞ ∈ Mac
only if qik ∼ 1/3 and pk = rk for almost all k. Indeed, in this case almost all elements
p∞ik = r∞ik = 1/3 and therefore ρ(µ, ν) = ρ(ν, µ) > 0. But if pk �= rk for almost all k then,
in general, both limiting measures are pure singular continuous. However, if one of initial
measures, say µ, has a local priority, pikk > rikk, only at a single position for almost all
k, then µ∞ ∈ Mpp, and ν∞ ∈ Msc. It is true since almost all values p∞max,k = 1 and the
corresponding coordinates r∞ikk = 0.
In general case (n > 2) the similar result occurs.
Statement 7.2. Let pk, rk ∈ Rn
+, n ≥ 2 Then µ∞ ∈ Mac if and only if qik ∼ 1/n, n ≥ 3
and µ = ν in the sense that pik = rik for almost all k. But if dikk > 0 only at a single
position ik for each k, then the measure µ possesses the shock directed priority with
respect to the measure ν, d(µ, ν) > 0, and therefore µ∞ ∈ Mpp.
Proof. If µ = ν then by Theorem 4.1 the elements of the limiting matrices P∞, R∞ have
the following values p∞ik = r∞ik = 1/n for almost all k. Therefore by Theorem 3.1 we have:
ρ(µ∞, λ) > 0. This ensures that µ∞ ∈ Mac. Let us show the necessity of the conditions.
Due to Theorem 5.1, µ∞ /∈ Mac, if µ �= ν in the sense that pik �= rik for an infinite
amount of meanings of k. Therefore the condition pik = rik is necessary almost for all k.
Moreover, since p∞ik = 1/n (see (4.6)), the value ρ(µ∞, λ) is strongly positive only under
the condition qik ∼ 1/n. Finally, taken into account that the measure µ has only a single
priority position ik for each k we observe that p∞ikk = 1. Thus d(µ, ν) > 0 and µ∞ is the
pure point measure. �
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Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka,
Kyiv, 01601, Ukraine
E-mail address: karat@imath.kiev.ua
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka,
Kyiv, 01601, Ukraine
E-mail address: kosh@imath.kiev.ua
Received 15/02/2008
|
| id | nasplib_isofts_kiev_ua-123456789-5703 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-12-07T18:05:56Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Karataieva, T. Koshmanenko, V. 2010-02-02T13:00:43Z 2010-02-02T13:00:43Z 2009 Origination of the singular continuous spectrum in the conflict dynamical systems / T. Karataieva, V. Koshmanenko // Methods of Functional Analysis and Topology. — 2009. — 15, N 1. — С. 15-30. — Бібліогр.: 25 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5703 We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous. en Інститут математики НАН України Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems Article published earlier |
| spellingShingle | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems Karataieva, T. Koshmanenko, V. |
| title | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems |
| title_full | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems |
| title_fullStr | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems |
| title_full_unstemmed | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems |
| title_short | Origination of the Singular Continuous Spectrum in the Conflict Dynamical Systems |
| title_sort | origination of the singular continuous spectrum in the conflict dynamical systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5703 |
| work_keys_str_mv | AT karataievat originationofthesingularcontinuousspectrumintheconflictdynamicalsystems AT koshmanenkov originationofthesingularcontinuousspectrumintheconflictdynamicalsystems |