# On One 2-Valued Transformation: Its Invariant Measure and Application to Masked Dynamical Systems

Authors: P. I. Troshin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We consider one familyS of 2-valued transformations on the interval [0, 1] with measure μ, endowed with a set of weight functions. We construct invariant measure μ
S
=
μ for this multivalued dynamical system with weights and show the interplay between such systems and masked dynamical systems, which leads to image processing.


---

## Body

## 1. Introduction

LetX be a space with finite measure μ on σ-field B of subsets of X, N
∈
N an integer, I
=
{
1
,
…
,
N
}, and S
i
:
X
→
X—some measurable transformations. Consider a set of measurable functions (endowment):
(1)
α
i
:
X
⟶
[
0,1
]
,
i
∈
I
∣
∑
i
∈
I
‍
α
i
≡
1
.
A collection
(2)
X
;
B
;
μ
;
S
1
,
…
,
S
N
;
α
1
,
…
,
α
N
is calledmultivalued dynamical system with weights, and the map S
=
∪
i
∈
I
S
i with fixed pairs {
(
S
i
,
α
i
)
}
i
∈
I—endowed
N
-transformation (see [1]). Regarding this, we can establish a new measure on B:
(3)
μ
S
(
B
)
=
∑
i
∈
I
‍
∫
S
i
-
1
(
B
)
‍
α
i
(
x
)
d
μ
.
One of the important questions of dynamical system theory is finding an invariant measure μ
S
=
μ.The endowmentα plays a role of a parameter which controls measure μ
S. On the other hand, α
i
(
x
) could be considered as a probability of choosing and applying the transformation S
i (out of S) to a point x
∈
X in stochastic dynamical system. Finally, as we show further, this parameter can uniquely define some single-valued dynamical system connected to S.In this paper we continue (after [2]) studying the following 2-transformation S
=
S
1
∪
S
2 of the interval [
0,1
] (see Figure 1):
(4)
S
1
(
x
)
=
1
1
-
a
x
,
x
∈
[
0,1
-
a
)
;
1
1
-
a
x
-
a
1
-
a
,
x
∈
[
1
-
a
,
1
]
,
S
2
x
=
1
1
-
a
x
,
x
∈
0
,
a
;
1
1
-
a
x
-
a
1
-
a
,
x
∈
a
,
1
,
with a shift a
∈
0
,
1
/
2 as its parameter. Dynamical system (
[
0,1
]
,
S
) is tightly connected to the theory of β-decompositions (see [3–6]).Figure 1
The design of2-transformation S.As a motivation for this paper in introduction we examine two points: invariance of measure for this endowed2-transformation and masked dynamical system associated with it.


### 1.1. Invariance of Measure

Letλ be the Lebesgue measure on [
0,1
] and B the Borel σ-field on [
0,1
]. Let also μ
(
B
)
=
∫
B
‍
p
(
x
)
d
λ be a measure, absolutely continuous with respect to the Lebesgue measure (μ
≪
λ), with density p
(
x
)
∈
L
1
(
[
0,1
]
,
B
,
λ
) and p
(
x
)
≥
0.According to [1], we endow 2-transformation S with a set of weight functions α
=
{
α
1
(
x
)
,
α
2
(
x
)
}, α
1
(
x
)
,
α
2
(
x
)
∈
L
1
(
[
0,1
]
,
B
,
λ
) such that α
1
(
x
)
+
α
2
(
x
)
=
1 and α
1
(
x
)
,
α
2
(
x
)
≥
0. Then we can introduce a new measure μ
S on B:
(5)
μ
S
(
B
)
=
∫
S
1
-
1
(
B
)
α
1
(
x
)
p
(
x
)
d
λ
+
∫
S
2
-
1
(
B
)
α
2
(
x
)
p
(
x
)
d
λ
.There are three independent parameters in the abovementioned construction: density functionp
(
x
), shift number a, and endowment α
=
{
α
1
(
x
)
,
α
2
(
x
)
}. Whether we search for endowed transformation for a given measure μ or a measure μ
S
=
μ for a given transformation S, there is a certain relation between these parameters, defined by equality μ
S
=
μ.Further on, we fix three parameters:a
∈
0
,
1
/
2, {
α
1
(
x
)
,
α
2
(
x
)
}, and p
(
x
), and let n
∈
N be such that
(6)
1
n
+
1
<
a
≤
1
n
(
n
≥
2
)
.Here we cite the following criterion of existence of invariant measure.Theorem 1 (see [2]).
μ
S
=
μ if and only if the following conditions hold true:
(7)
∑
k
=
-
1
n
-
1
‍
p
x
+
k
a
=
1
1
-
a
∑
k
=
-
1
n
-
2
‍
p
x
+
k
a
1
-
a
,
∀
x
∈
a
,
1
-
n
-
1
a
;
(8)
∑
k
=
-
1
n
-
2
‍
p
x
~
+
k
a
=
1
1
-
a
∑
k
=
-
1
n
-
3
‍
p
x
~
+
k
a
1
-
a
,
∀
x
~
∈
1
-
n
-
1
a
,
2
a
;
(9)
α
1
x
+
m
a
p
x
+
m
a
=
∑
k
=
-
1
m
‍
p
(
x
+
k
a
)
-
1
1
-
a
∑
k
=
-
1
m
-
1
‍
p
x
+
k
a
1
-
a
,
where for n
=
2, m
=
0, x
∈
[
a
,
1
-
a
), for n
≥
3, m
=
0, x
∈
[
a
,
2
a
), for n
≥
3, m
=
1,2
,
…
,
x
∈
[
a
,
2
a
), and x
+
m
a
∈
[
2
a
,
1
-
a
).
There is no restriction on functionα
1
(
x
) on the sets [
0
,
a
) and [
1
-
a
,
1
].Equations (7)-(8) define function p
(
x
) on the interval [
0,1
], and (9) defines endowment α. We can revise (9) into more compact and constructive formula:
(10)
α
1
x
p
x
=
∑
k
=
0
s
‍
p
x
-
k
a
-
1
1
-
a
∑
k
=
1
s
‍
p
x
-
k
a
1
-
a
,
x
∈
a
,
1
-
a
,
where s
=
[
x
/
a
] ([
x
] is an integer part of x).To clarify the meaning of the theorem we give two corollaries from it.Corollary 2 (see [2]).
Given measureμ
≪
λ there exists endowed 2-transformation S
(
a
) preserving measure μ if and only if p
(
x
), a, and {
α
1
(
x
)
,
α
2
(
x
)
} satisfy conditions (7)–(9).Corollary 3 (see [2]).
Given endowed2-transformation S
(
a
) there exists measure μ
≪
λ which is preserved by transformation S if and only if p
(
x
), a, and {
α
1
(
x
)
,
α
2
(
x
)
} satisfy conditions (7)–(9).There is a convenient graphical scheme of summation intervals placement on the interval[
0,1
] for (7)-(8); see Figures 2 and 3.Figure 2
Scheme of summation intervals placement for (7) (upper) and (8) (lower), here b
=
1
-
(
n
-
1
)
a, even n.Figure 3
Scheme of summation intervals placement for (7) (upper) and (8) (lower), here b
=
1
-
(
n
-
1
)
a, odd n.Informally, we can depict (7)-(8) as follows:
(11)
∑
‍
p
(
•
)
=
1
1
-
a
∑
‍
p
(
▪
)
.Regarding Theorem1, the following question arises.Question 1.
Are there functions satisfying (7)-(8)?One trivial solution isp
≡
0.Slightly less trivial example of constant densityp
≡
c, c
∈
R, c
>
0, is presented in the following corollary.Corollary 4 (see [2]).
(
c
·
λ
)
S
=
c
·
λ if and only if a
=
1
/
n, n
=
2,3
,
….However, this2-valued dynamical system allows even more sophisticated density: (7)–(9) hold true for some nonconstant p
(
x
), as shown in the next theorem.Letχ
B
(
x
)
=
0
,
x
∉
B
,
1
,
x
∈
B
, be a characteristic function for a subset B
⊂
[
0,1
].Theorem 5 (see [2]).
Givenn
=
2,3
,
… there exist a shift a (1
/
(
n
+
1
)
<
a
<
1
/
n), piecewise constant density p
(
x
), and endowment {
α
1
(
x
)
,
α
2
(
x
)
}, such that μ
S
=
μ. Namely,
(12)
p
x
=
β
χ
0
,
δ
x
+
β
+
γ
1
-
δ
χ
δ
,
1
-
δ
x
+
γ
χ
[
1
-
δ
,
1
]
(
x
)
,
a
=
n
+
1
-
n
2
+
1
n
,
δ
=
a
n
2
,
β
,
γ
>
0
,
n
i
s
e
v
e
n
,
p
x
=
β
χ
0,1
-
δ
x
+
β
+
γ
1
-
δ
χ
1
-
δ
,
δ
x
+
γ
χ
[
δ
,
1
]
(
x
)
,
a
=
n
+
1
-
n
2
-
1
n
+
1
,
δ
=
a
n
+
1
2
,
β
,
γ
>
0
,
n
i
s
o
d
d
.Remark 6.
Theorem5 yields a family of densities with two parameters β
,
γ
>
0.For computational simplicity in this theorema is chosen in such a way that the middle intervals in the graphical scheme touch each other; see Figure 4 for even n.Figure 4
Special choice of a shifta: (
n
/
2
)
a
=
1
-
(
n
/
2
)
(
a
/
1
-
a
), even n.The resulting piecewise density consists of three domains; see Figure5.Figure 5
Typical view of a piecewise constant density from Theorem5.However, the same question arises again: are there other nontrivial (nonconstant) densities satisfying (7)-(8)?In Section2 we present a scheme to construct nontrivial densities in case of a
=
(
3
-
5
)
/
2, n
=
2, and study some properties of the functions we obtain there. In Section 3 there is a scheme to construct such densities for arbitrary a
∈
(
0
,
1
/
2
] (n
≥
2).Finally, in this subsection we cite the following lemma which implies “mirror twoness” of invariant measures densities (see Corollary8): if p
(
x
) is such a density, then the function g
(
x
)
=
p
(
1
-
x
) is again a density of invariant measure.Lemma 7 (see [2]).
LetA
i
(
x
)
=
α
i
(
x
)
p
(
x
), i
=
1,2. Then μ
S
=
μ if and only if λ-almost everywhere on [
0,1
]
(13)
A
1
1
-
a
x
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
1
1
-
a
x
+
a
+
A
2
(
(
1
-
a
)
x
+
a
)
+
χ
0
,
a
/
1
-
a
(
x
)
A
2
(
(
1
-
a
)
x
)
=
p
x
1
-
a
.Corollary 8.
Ifp
(
x
) is invariant measure density, then the function g
(
x
)
=
p
(
1
-
x
) with endowment β
i
(
x
)
=
α
3
-
i
(
1
-
x
), i
=
1,2, is also invariant measure density.Proof.
Letp
(
x
) be invariant measure density, g
(
x
)
=
p
(
1
-
x
), and B
i
(
x
)
=
β
i
(
x
)
g
(
x
)
=
α
3
-
i
(
1
-
x
)
p
(
1
-
x
)
=
A
3
-
i
(
1
-
x
). Substituting 1
-
x instead of x in equality (13) yields
(14)
g
x
1
-
a
=
p
1
-
x
1
-
a
=
A
1
1
-
a
1
-
x
+
χ
0
,
a
/
1
-
a
x
A
1
1
-
a
1
-
x
+
a
+
A
2
1
-
a
1
-
x
+
a
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
2
1
-
a
1
-
x
=
A
1
1
-
1
-
a
x
+
a
+
χ
0
,
a
/
(
1
-
a
)
x
A
1
1
-
1
-
a
x
+
A
2
1
-
1
-
a
x
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
2
1
-
1
-
a
x
+
a
=
B
2
1
-
a
x
+
a
+
χ
0
,
a
/
(
1
-
a
)
x
B
2
1
-
a
x
+
B
1
(
(
1
-
a
)
x
)
+
χ
(
(
1
-
2
a
)
/
(
1
-
a
)
,
1
]
(
x
)
B
1
(
(
1
-
a
)
x
+
a
)
.
Thus equality (13) holds true for g
(
x
) almost everywhere.


### 1.2. Masked Dynamical System

As an extra motivation we consider here the following argument: endowmentα of dynamical system S can be connected with mask endowment of some iterated functions system F (see below).Consider some disjoint coverM
=
{
M
i
}
i
∈
I of the set X: M
i
∈
B, i
∈
I, M
i
∩
M
j
=
∅, i
,
j
∈
I, i
≠
j, ∪
i
∈
I
M
i
=
X. Let α
i
=
χ
M
i, i
∈
I, be characteristic functions of the subsets M
i
⊂
X.We may say that, regarding the contribution ofS
i
-
1
(
B
)
∩
M
i to the measure
(15)
μ
S
(
B
)
=
∑
i
∈
I
‍
μ
(
S
i
-
1
(
B
)
∩
M
i
)
=
μ
∪
i
∈
I
(
S
i
-
1
(
B
)
∩
M
i
)
,
N-valued transformation turns into the following single-valued one:
(16)
S
~
(
x
)
=
S
1
(
x
)
,
x
∈
M
1
,
⋮
S
N
(
x
)
,
x
∈
M
N
.In the case of arbitrary endowmentα we may consider single-valued stochastic dynamical system:
(17)
S
~
~
(
x
)
=
S
1
(
x
)
with
probability
α
1
(
x
)
,
⋮
S
N
(
x
)
with
probability
α
N
(
x
)
.Such an approach that turns multivalued dynamical system into single-valued one is implemented in [7] for mappings S, connected with iterated function systems (IFS). It lets us establish and control fractal transformations between IFS attractors. Such transformations have direct practical value (see below). Here we introduce main points from [7] (relevant to this paper).LetX
≠
∅ be a compact Hausdorff space and K
(
X
) a set of nonempty compact subsets of X. Let I
=
{
1
,
…
,
N
} be a finite set of positive integers, I
∞ a set of infinite sequences of numbers from I, and f
i
:
X
→
X
,
i
∈
I, continuous mappings. Then F
=
(
X
;
f
1
,
…
,
f
N
) is callediterated function system (IFS).Due to decreasing monotone inclusion of corresponding compact subsets one can correctly define the mapping(18)
Π
:
I
∞
⟶
K
X
,
σ
=
σ
1
σ
2
⋯
⟼
⋂
k
=
1
∞
‍
f
σ
1
∘
f
σ
2
∘
⋯
∘
f
σ
k
X
.
If, for all σ
∈
I
∞, Π
(
σ
) is a singleton, then the IFS is calledpoint-fibred. In this case a mapping
(19)
π
:
I
∞
⟶
A
=
π
(
I
∞
)
⊂
X
,
{
π
(
σ
)
}
=
Π
(
σ
)
,
is called thecoding map of F, I
∞ thecode space of F, and σ
∈
I
∞ theaddress of the point π
(
σ
)
∈
A.For point-fibred IFS on a compact Hausdorff space there exists a unique setA
∈
K
(
X
) such that
(20)
A
=
⋃
i
∈
I
‍
f
i
(
A
)
,
and A
=
π
(
I
∞
) (see [7]). This set is called theattractor of the given IFS.IFS attractor often happens to be a fractal set or even self-similar one, which is usually of huge interest.Henceforth, we constrain ourselves to point-fibred IFS on some compact Hausdorff space only (however, this is rather typical, cf. Remark2.5 in [7]).A pointx
∈
A may have more than one address (even uncountably many). The following definition will be useful to make the choice of address unique. A subset Ω
⊂
I
∞ is called theaddress space of the IFS F if π
|
Ω
:
Ω
→
A is bijective. Then the inverse mapping
(21)
τ
:
A
⟶
Ω
,
x
⟼
π
Ω
-
1
(
x
)
,
is called thesection of
π.If there are two point-fibred IFSF
=
{
X
;
f
1
,
…
,
f
N
} and G
=
{
Y
;
g
1
,
…
,
g
N
} (with common I
∞) on compact Hausdorff spaces X and Y, A
F and A
G being their attractors, π
G the coding mapping of G, and τ
F the section of π
F, then we can define thefractal transformation (under this transformation the fractal dimension of a set could be changed) between attractors of F and G:
(22)
T
F
G
:
A
F
⟶
A
F
,
x
⟼
π
G
∘
τ
F
(
x
)
.The paper [7] gives a continuity criteria for T
F
G and also describes some applications of fractal transformations for conversion and filtering images and steganography (hidden data transmission, e.g., packing several images into one).The choice of the address spaceΩ
F of F defines a fractal transformation. In [7] two methods for construction of Ω
F are proposed, and they lead to sections τ
F with good properties.One of the methods is to usetop addresses: sequences from I
∞ may be put in lexicographic order, which lets us choose a unique (“top”) element from π
-
1
(
x
) for all x
∈
A (see [5, 8]). This method is computationally simple and can be easily implemented on computer. However, only a few certain sections can be obtained in this way.Let us consider the second method in more detail. LetF be a point-fibred IFS with injective maps f
i
,
i
∈
I. A collection of subsets M
=
{
M
i
⊂
A
,
i
∈
I
} is called themask of F if(1)
M
i
⊂
f
i
(
A
)
,
i
∈
I;
(2)
M
i
∩
M
j
=
∅
,
i
,
j
∈
I
,
i
≠
j;
(3)
∪
i
∈
I
M
i
=
A.For allx
∈
A, there exists a unique i
∈
A such that x
∈
M
i
⊂
f
i
(
A
). The mapping
(23)
T
:
A
⟶
A
,
x
⟼
f
1
-
1
(
x
)
,
x
∈
M
1
⋮
f
N
-
1
(
x
)
,
x
∈
M
N
,
is called themasked dynamical system for F.This system is used to construct a sectionτ
:
A
→
τ
(
A
)
⊂
I
∞ by following the orbit T
n
(
x
)
=
T
∘
⋯
∘
T
︸
n
times
(
x
) of point x; namely,
(24)
τ
x
=
σ
x
=
σ
1
x
σ
2
x
…
,
where
x
∈
T
k
-
1
-
1
M
σ
k
x
,
k
=
1,2
,
…
.
In this case π
(
σ
(
x
)
)
=
x (see [7]).Thus the maskM of dynamical system connected with IFS is a special case of endowment α, when α
i
=
χ
M
i, i
∈
I. We can also consider stochastic mask defined by endowment weight functions: if supp
⁡
α
i
⊂
f
i
(
A
), i
∈
I, then
(25)
T
~
~
(
x
)
=
f
1
-
1
(
x
)
with
probability
α
1
(
x
)
,
⋮
f
N
-
1
(
x
)
with
probability
α
N
(
x
)
.Let us describe the connection between this mask construction and2-transformation S. Consider the following IFS (see Figure 6):
(26)
X
=
0,1
;
f
1
x
=
1
-
a
x
,
f
2
x
=
1
-
a
x
+
a
.IFS equation (26) (a) and multivalued (without mask) dynamical system (coincides with S) connected with it (b).
(a)
(b)This is point-fibred IFS with injective functionsf
1
,
f
2, and its attractor is the interval A
=
X
=
[
0,1
]. Consider f
1
-
1
,
f
2
-
1 for construction of masked dynamical system T. As might be seen on Figure 6, this dynamical system is the object of this paper. Let M
=
{
M
1
,
M
2
} be a mask of this IFS. Then obviously, [
0
,
a
)
⊂
M
1 and (
1
-
a
,
1
]
⊂
M
2. Define M
1
∩
[
a
,
1
-
a
] and M
2
∩
[
a
,
1
-
a
] arbitrarily (M
1
∩
M
2
=
∅, M
1
,
M
2
∈
B). The example of a mask and the process of finding masked address of a point x
∈
A are illustrated on Figure 7.Figure 7
Example of masked dynamical system for IFS equation (26), M
1
=
[
0,0.5
), M
2
=
[
0.5,1
], and τ
(
x
)
=
222211
….As we have already mentioned, mask endowmentM of F in this case coincides with endowment α
=
{
α
1
(
x
)
=
χ
M
1
(
x
)
,
α
2
(
x
)
=
χ
M
2
(
x
)
} of S.Then the following question arises.Question 2.
Is there an invariant measure for this masked dynamical system?We give an example of such a measure in Section2.


## 1.1. Invariance of Measure

Letλ be the Lebesgue measure on [
0,1
] and B the Borel σ-field on [
0,1
]. Let also μ
(
B
)
=
∫
B
‍
p
(
x
)
d
λ be a measure, absolutely continuous with respect to the Lebesgue measure (μ
≪
λ), with density p
(
x
)
∈
L
1
(
[
0,1
]
,
B
,
λ
) and p
(
x
)
≥
0.According to [1], we endow 2-transformation S with a set of weight functions α
=
{
α
1
(
x
)
,
α
2
(
x
)
}, α
1
(
x
)
,
α
2
(
x
)
∈
L
1
(
[
0,1
]
,
B
,
λ
) such that α
1
(
x
)
+
α
2
(
x
)
=
1 and α
1
(
x
)
,
α
2
(
x
)
≥
0. Then we can introduce a new measure μ
S on B:
(5)
μ
S
(
B
)
=
∫
S
1
-
1
(
B
)
α
1
(
x
)
p
(
x
)
d
λ
+
∫
S
2
-
1
(
B
)
α
2
(
x
)
p
(
x
)
d
λ
.There are three independent parameters in the abovementioned construction: density functionp
(
x
), shift number a, and endowment α
=
{
α
1
(
x
)
,
α
2
(
x
)
}. Whether we search for endowed transformation for a given measure μ or a measure μ
S
=
μ for a given transformation S, there is a certain relation between these parameters, defined by equality μ
S
=
μ.Further on, we fix three parameters:a
∈
0
,
1
/
2, {
α
1
(
x
)
,
α
2
(
x
)
}, and p
(
x
), and let n
∈
N be such that
(6)
1
n
+
1
<
a
≤
1
n
(
n
≥
2
)
.Here we cite the following criterion of existence of invariant measure.Theorem 1 (see [2]).
μ
S
=
μ if and only if the following conditions hold true:
(7)
∑
k
=
-
1
n
-
1
‍
p
x
+
k
a
=
1
1
-
a
∑
k
=
-
1
n
-
2
‍
p
x
+
k
a
1
-
a
,
∀
x
∈
a
,
1
-
n
-
1
a
;
(8)
∑
k
=
-
1
n
-
2
‍
p
x
~
+
k
a
=
1
1
-
a
∑
k
=
-
1
n
-
3
‍
p
x
~
+
k
a
1
-
a
,
∀
x
~
∈
1
-
n
-
1
a
,
2
a
;
(9)
α
1
x
+
m
a
p
x
+
m
a
=
∑
k
=
-
1
m
‍
p
(
x
+
k
a
)
-
1
1
-
a
∑
k
=
-
1
m
-
1
‍
p
x
+
k
a
1
-
a
,
where for n
=
2, m
=
0, x
∈
[
a
,
1
-
a
), for n
≥
3, m
=
0, x
∈
[
a
,
2
a
), for n
≥
3, m
=
1,2
,
…
,
x
∈
[
a
,
2
a
), and x
+
m
a
∈
[
2
a
,
1
-
a
).
There is no restriction on functionα
1
(
x
) on the sets [
0
,
a
) and [
1
-
a
,
1
].Equations (7)-(8) define function p
(
x
) on the interval [
0,1
], and (9) defines endowment α. We can revise (9) into more compact and constructive formula:
(10)
α
1
x
p
x
=
∑
k
=
0
s
‍
p
x
-
k
a
-
1
1
-
a
∑
k
=
1
s
‍
p
x
-
k
a
1
-
a
,
x
∈
a
,
1
-
a
,
where s
=
[
x
/
a
] ([
x
] is an integer part of x).To clarify the meaning of the theorem we give two corollaries from it.Corollary 2 (see [2]).
Given measureμ
≪
λ there exists endowed 2-transformation S
(
a
) preserving measure μ if and only if p
(
x
), a, and {
α
1
(
x
)
,
α
2
(
x
)
} satisfy conditions (7)–(9).Corollary 3 (see [2]).
Given endowed2-transformation S
(
a
) there exists measure μ
≪
λ which is preserved by transformation S if and only if p
(
x
), a, and {
α
1
(
x
)
,
α
2
(
x
)
} satisfy conditions (7)–(9).There is a convenient graphical scheme of summation intervals placement on the interval[
0,1
] for (7)-(8); see Figures 2 and 3.Figure 2
Scheme of summation intervals placement for (7) (upper) and (8) (lower), here b
=
1
-
(
n
-
1
)
a, even n.Figure 3
Scheme of summation intervals placement for (7) (upper) and (8) (lower), here b
=
1
-
(
n
-
1
)
a, odd n.Informally, we can depict (7)-(8) as follows:
(11)
∑
‍
p
(
•
)
=
1
1
-
a
∑
‍
p
(
▪
)
.Regarding Theorem1, the following question arises.Question 1.
Are there functions satisfying (7)-(8)?One trivial solution isp
≡
0.Slightly less trivial example of constant densityp
≡
c, c
∈
R, c
>
0, is presented in the following corollary.Corollary 4 (see [2]).
(
c
·
λ
)
S
=
c
·
λ if and only if a
=
1
/
n, n
=
2,3
,
….However, this2-valued dynamical system allows even more sophisticated density: (7)–(9) hold true for some nonconstant p
(
x
), as shown in the next theorem.Letχ
B
(
x
)
=
0
,
x
∉
B
,
1
,
x
∈
B
, be a characteristic function for a subset B
⊂
[
0,1
].Theorem 5 (see [2]).
Givenn
=
2,3
,
… there exist a shift a (1
/
(
n
+
1
)
<
a
<
1
/
n), piecewise constant density p
(
x
), and endowment {
α
1
(
x
)
,
α
2
(
x
)
}, such that μ
S
=
μ. Namely,
(12)
p
x
=
β
χ
0
,
δ
x
+
β
+
γ
1
-
δ
χ
δ
,
1
-
δ
x
+
γ
χ
[
1
-
δ
,
1
]
(
x
)
,
a
=
n
+
1
-
n
2
+
1
n
,
δ
=
a
n
2
,
β
,
γ
>
0
,
n
i
s
e
v
e
n
,
p
x
=
β
χ
0,1
-
δ
x
+
β
+
γ
1
-
δ
χ
1
-
δ
,
δ
x
+
γ
χ
[
δ
,
1
]
(
x
)
,
a
=
n
+
1
-
n
2
-
1
n
+
1
,
δ
=
a
n
+
1
2
,
β
,
γ
>
0
,
n
i
s
o
d
d
.Remark 6.
Theorem5 yields a family of densities with two parameters β
,
γ
>
0.For computational simplicity in this theorema is chosen in such a way that the middle intervals in the graphical scheme touch each other; see Figure 4 for even n.Figure 4
Special choice of a shifta: (
n
/
2
)
a
=
1
-
(
n
/
2
)
(
a
/
1
-
a
), even n.The resulting piecewise density consists of three domains; see Figure5.Figure 5
Typical view of a piecewise constant density from Theorem5.However, the same question arises again: are there other nontrivial (nonconstant) densities satisfying (7)-(8)?In Section2 we present a scheme to construct nontrivial densities in case of a
=
(
3
-
5
)
/
2, n
=
2, and study some properties of the functions we obtain there. In Section 3 there is a scheme to construct such densities for arbitrary a
∈
(
0
,
1
/
2
] (n
≥
2).Finally, in this subsection we cite the following lemma which implies “mirror twoness” of invariant measures densities (see Corollary8): if p
(
x
) is such a density, then the function g
(
x
)
=
p
(
1
-
x
) is again a density of invariant measure.Lemma 7 (see [2]).
LetA
i
(
x
)
=
α
i
(
x
)
p
(
x
), i
=
1,2. Then μ
S
=
μ if and only if λ-almost everywhere on [
0,1
]
(13)
A
1
1
-
a
x
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
1
1
-
a
x
+
a
+
A
2
(
(
1
-
a
)
x
+
a
)
+
χ
0
,
a
/
1
-
a
(
x
)
A
2
(
(
1
-
a
)
x
)
=
p
x
1
-
a
.Corollary 8.
Ifp
(
x
) is invariant measure density, then the function g
(
x
)
=
p
(
1
-
x
) with endowment β
i
(
x
)
=
α
3
-
i
(
1
-
x
), i
=
1,2, is also invariant measure density.Proof.
Letp
(
x
) be invariant measure density, g
(
x
)
=
p
(
1
-
x
), and B
i
(
x
)
=
β
i
(
x
)
g
(
x
)
=
α
3
-
i
(
1
-
x
)
p
(
1
-
x
)
=
A
3
-
i
(
1
-
x
). Substituting 1
-
x instead of x in equality (13) yields
(14)
g
x
1
-
a
=
p
1
-
x
1
-
a
=
A
1
1
-
a
1
-
x
+
χ
0
,
a
/
1
-
a
x
A
1
1
-
a
1
-
x
+
a
+
A
2
1
-
a
1
-
x
+
a
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
2
1
-
a
1
-
x
=
A
1
1
-
1
-
a
x
+
a
+
χ
0
,
a
/
(
1
-
a
)
x
A
1
1
-
1
-
a
x
+
A
2
1
-
1
-
a
x
+
χ
(
1
-
2
a
)
/
(
1
-
a
)
,
1
x
A
2
1
-
1
-
a
x
+
a
=
B
2
1
-
a
x
+
a
+
χ
0
,
a
/
(
1
-
a
)
x
B
2
1
-
a
x
+
B
1
(
(
1
-
a
)
x
)
+
χ
(
(
1
-
2
a
)
/
(
1
-
a
)
,
1
]
(
x
)
B
1
(
(
1
-
a
)
x
+
a
)
.
Thus equality (13) holds true for g
(
x
) almost everywhere.


## 1.2. Masked Dynamical System

As an extra motivation we consider here the following argument: endowmentα of dynamical system S can be connected with mask endowment of some iterated functions system F (see below).Consider some disjoint coverM
=
{
M
i
}
i
∈
I of the set X: M
i
∈
B, i
∈
I, M
i
∩
M
j
=
∅, i
,
j
∈
I, i
≠
j, ∪
i
∈
I
M
i
=
X. Let α
i
=
χ
M
i, i
∈
I, be characteristic functions of the subsets M
i
⊂
X.We may say that, regarding the contribution ofS
i
-
1
(
B
)
∩
M
i to the measure
(15)
μ
S
(
B
)
=
∑
i
∈
I
‍
μ
(
S
i
-
1
(
B
)
∩
M
i
)
=
μ
∪
i
∈
I
(
S
i
-
1
(
B
)
∩
M
i
)
,
N-valued transformation turns into the following single-valued one:
(16)
S
~
(
x
)
=
S
1
(
x
)
,
x
∈
M
1
,
⋮
S
N
(
x
)
,
x
∈
M
N
.In the case of arbitrary endowmentα we may consider single-valued stochastic dynamical system:
(17)
S
~
~
(
x
)
=
S
1
(
x
)
with
probability
α
1
(
x
)
,
⋮
S
N
(
x
)
with
probability
α
N
(
x
)
.Such an approach that turns multivalued dynamical system into single-valued one is implemented in [7] for mappings S, connected with iterated function systems (IFS). It lets us establish and control fractal transformations between IFS attractors. Such transformations have direct practical value (see below). Here we introduce main points from [7] (relevant to this paper).LetX
≠
∅ be a compact Hausdorff space and K
(
X
) a set of nonempty compact subsets of X. Let I
=
{
1
,
…
,
N
} be a finite set of positive integers, I
∞ a set of infinite sequences of numbers from I, and f
i
:
X
→
X
,
i
∈
I, continuous mappings. Then F
=
(
X
;
f
1
,
…
,
f
N
) is callediterated function system (IFS).Due to decreasing monotone inclusion of corresponding compact subsets one can correctly define the mapping(18)
Π
:
I
∞
⟶
K
X
,
σ
=
σ
1
σ
2
⋯
⟼
⋂
k
=
1
∞
‍
f
σ
1
∘
f
σ
2
∘
⋯
∘
f
σ
k
X
.
If, for all σ
∈
I
∞, Π
(
σ
) is a singleton, then the IFS is calledpoint-fibred. In this case a mapping
(19)
π
:
I
∞
⟶
A
=
π
(
I
∞
)
⊂
X
,
{
π
(
σ
)
}
=
Π
(
σ
)
,
is called thecoding map of F, I
∞ thecode space of F, and σ
∈
I
∞ theaddress of the point π
(
σ
)
∈
A.For point-fibred IFS on a compact Hausdorff space there exists a unique setA
∈
K
(
X
) such that
(20)
A
=
⋃
i
∈
I
‍
f
i
(
A
)
,
and A
=
π
(
I
∞
) (see [7]). This set is called theattractor of the given IFS.IFS attractor often happens to be a fractal set or even self-similar one, which is usually of huge interest.Henceforth, we constrain ourselves to point-fibred IFS on some compact Hausdorff space only (however, this is rather typical, cf. Remark2.5 in [7]).A pointx
∈
A may have more than one address (even uncountably many). The following definition will be useful to make the choice of address unique. A subset Ω
⊂
I
∞ is called theaddress space of the IFS F if π
|
Ω
:
Ω
→
A is bijective. Then the inverse mapping
(21)
τ
:
A
⟶
Ω
,
x
⟼
π
Ω
-
1
(
x
)
,
is called thesection of
π.If there are two point-fibred IFSF
=
{
X
;
f
1
,
…
,
f
N
} and G
=
{
Y
;
g
1
,
…
,
g
N
} (with common I
∞) on compact Hausdorff spaces X and Y, A
F and A
G being their attractors, π
G the coding mapping of G, and τ
F the section of π
F, then we can define thefractal transformation (under this transformation the fractal dimension of a set could be changed) between attractors of F and G:
(22)
T
F
G
:
A
F
⟶
A
F
,
x
⟼
π
G
∘
τ
F
(
x
)
.The paper [7] gives a continuity criteria for T
F
G and also describes some applications of fractal transformations for conversion and filtering images and steganography (hidden data transmission, e.g., packing several images into one).The choice of the address spaceΩ
F of F defines a fractal transformation. In [7] two methods for construction of Ω
F are proposed, and they lead to sections τ
F with good properties.One of the methods is to usetop addresses: sequences from I
∞ may be put in lexicographic order, which lets us choose a unique (“top”) element from π
-
1
(
x
) for all x
∈
A (see [5, 8]). This method is computationally simple and can be easily implemented on computer. However, only a few certain sections can be obtained in this way.Let us consider the second method in more detail. LetF be a point-fibred IFS with injective maps f
i
,
i
∈
I. A collection of subsets M
=
{
M
i
⊂
A
,
i
∈
I
} is called themask of F if(1)
M
i
⊂
f
i
(
A
)
,
i
∈
I;
(2)
M
i
∩
M
j
=
∅
,
i
,
j
∈
I
,
i
≠
j;
(3)
∪
i
∈
I
M
i
=
A.For allx
∈
A, there exists a unique i
∈
A such that x
∈
M
i
⊂
f
i
(
A
). The mapping
(23)
T
:
A
⟶
A
,
x
⟼
f
1
-
1
(
x
)
,
x
∈
M
1
⋮
f
N
-
1
(
x
)
,
x
∈
M
N
,
is called themasked dynamical system for F.This system is used to construct a sectionτ
:
A
→
τ
(
A
)
⊂
I
∞ by following the orbit T
n
(
x
)
=
T
∘
⋯
∘
T
︸
n
times
(
x
) of point x; namely,
(24)
τ
x
=
σ
x
=
σ
1
x
σ
2
x
…
,
where
x
∈
T
k
-
1
-
1
M
σ
k
x
,
k
=
1,2
,
…
.
In this case π
(
σ
(
x
)
)
=
x (see [7]).Thus the maskM of dynamical system connected with IFS is a special case of endowment α, when α
i
=
χ
M
i, i
∈
I. We can also consider stochastic mask defined by endowment weight functions: if supp
⁡
α
i
⊂
f
i
(
A
), i
∈
I, then
(25)
T
~
~
(
x
)
=
f
1
-
1
(
x
)
with
probability
α
1
(
x
)
,
⋮
f
N
-
1
(
x
)
with
probability
α
N
(
x
)
.Let us describe the connection between this mask construction and2-transformation S. Consider the following IFS (see Figure 6):
(26)
X
=
0,1
;
f
1
x
=
1
-
a
x
,
f
2
x
=
1
-
a
x
+
a
.IFS equation (26) (a) and multivalued (without mask) dynamical system (coincides with S) connected with it (b).
(a)
(b)This is point-fibred IFS with injective functionsf
1
,
f
2, and its attractor is the interval A
=
X
=
[
0,1
]. Consider f
1
-
1
,
f
2
-
1 for construction of masked dynamical system T. As might be seen on Figure 6, this dynamical system is the object of this paper. Let M
=
{
M
1
,
M
2
} be a mask of this IFS. Then obviously, [
0
,
a
)
⊂
M
1 and (
1
-
a
,
1
]
⊂
M
2. Define M
1
∩
[
a
,
1
-
a
] and M
2
∩
[
a
,
1
-
a
] arbitrarily (M
1
∩
M
2
=
∅, M
1
,
M
2
∈
B). The example of a mask and the process of finding masked address of a point x
∈
A are illustrated on Figure 7.Figure 7
Example of masked dynamical system for IFS equation (26), M
1
=
[
0,0.5
), M
2
=
[
0.5,1
], and τ
(
x
)
=
222211
….As we have already mentioned, mask endowmentM of F in this case coincides with endowment α
=
{
α
1
(
x
)
=
χ
M
1
(
x
)
,
α
2
(
x
)
=
χ
M
2
(
x
)
} of S.Then the following question arises.Question 2.
Is there an invariant measure for this masked dynamical system?We give an example of such a measure in Section2.


## 2. The Case ofn
=
2

Here we consider the case ofn
=
2 in detail. The main ideas of this section can be used further for other values of n. The conditions (7)-(8) now can be written as
(27)
p
x
-
a
+
p
x
+
p
x
+
a
=
1
1
-
a
p
x
-
a
1
-
a
+
p
x
1
-
a
,
x
∈
[
a
,
1
-
a
)
;
(28)
p
x
~
-
a
+
p
x
~
=
1
1
-
a
p
x
~
-
a
1
-
a
,
x
~
∈
1
-
a
,
2
a
,
or in equivalent way,
(29)
p
x
+
p
x
+
a
+
p
x
+
2
a
=
1
1
-
a
p
x
1
-
a
+
p
x
+
a
1
-
a
,
x
∈
[
0,1
-
2
a
)
;
(30)
p
x
~
+
p
x
~
+
a
=
1
1
-
a
p
x
~
1
-
a
,
x
~
∈
1
-
2
a
,
a
.To make it simple, we consider special shift, according to the scheme on Figure4. In our case n
=
2, a
=
(
3
-
5
)
/
2; see Figure 8.Figure 8
Interval placement,a
=
(
3
-
5
)
/
2, n
=
2.Here we introduce a scheme to construct a densityp
(
x
) satisfying equations (29)-(30); see Figure 9. Consider the following marks on the x-axis: a, 1
-
a, 2
a, 2
-
3
a, x
k
=
a
(
1
-
a
)
k, k
≥
1, (x
1
=
1
-
2
a).(i)
Fix functionsp
0
*
,
p
1
*
∈
L
1, p
0
*
,
p
1
*
≥
0, arbitrarily, and define
(31)
p
(
x
)
=
p
1
(
x
)
=
1
1
-
a
p
0
*
(
x
1
-
a
)
-
p
1
*
(
x
+
a
)
,
x
∈
(
1
-
2
a
,
a
]
,
p
0
*
(
x
)
,
x
∈
(
a
,
1
-
a
]
,
p
1
*
(
x
)
,
x
∈
(
1
-
a
,
2
a
]
.
(ii)
Fix functionp
3
*
∈
L
1, p
3
*
≥
0, arbitrarily, and define
(32)
p
x
=
p
3
*
(
x
)
,
x
∈
(
2
-
3
a
,
1
]
,
p
2
(
x
)
=
1
1
-
a
p
1
x
1
-
a
+
p
3
*
x
+
a
1
-
a
-
p
0
*
(
x
+
a
)
-
p
3
*
(
x
+
2
a
)
,
x
∈
(
x
2
,
1
-
2
a
]
.
(iii)
Fix functionp
2
*
∈
L
1, p
2
*
≥
0, arbitrarily, and define
(33)
p
x
=
p
2
*
(
x
)
,
x
∈
(
2
a
,
2
-
3
a
]
,
p
3
(
x
)
=
1
1
-
a
p
2
x
1
-
a
+
p
2
*
x
+
a
1
-
a
-
p
0
*
(
x
+
a
)
-
p
3
*
(
x
+
2
a
)
,
x
∈
(
x
3
,
x
2
]
.
(iv)
Define for eachk
≥
4
(34)
p
x
=
p
k
x
=
1
1
-
a
p
k
-
1
x
1
-
a
+
p
1
*
x
+
1
1
-
a
-
p
0
*
(
x
+
a
)
-
p
2
*
(
x
+
2
a
)
,
where
x
∈
(
x
k
,
x
k
-
1
]
.
(v)
Fix the valuep
(
0
)
≥
0 arbitrarily.Figure 9
Scheme to construct a densityp
(
x
), with auxiliary intervals marked, a
=
(
3
-
5
)
/
2, n
=
2.By construction,p
(
x
) satisfies the conditions (29)-(30) (perhaps except at the most countable number of points on intervals boundaries). Notice that the function p
(
x
) is defined arbitrarily on (
a
,
1
] and is restored on [
0
,
a
] after that. We need the partition p
0
*
,
p
1
*
,
p
2
*
,
p
3
* of the function p
(
x
) to study its properties in more detail.Proposition 9.
Ifp
0
*
,
p
1
*
,
p
2
*
,
p
3
* are constants, then p
2 is a constant, and p
3
=
p
4
=
⋯ are constants.Proof.
We denoteA
=
1
/
(
1
-
a
); then
(35)
p
1
=
A
p
0
*
-
p
1
*
,
p
2
=
A
(
p
1
+
p
3
*
)
-
p
0
*
-
p
3
*
,
p
3
=
A
(
p
2
+
p
2
*
)
-
p
0
*
-
p
3
*
,
p
k
=
A
(
p
k
-
1
+
p
1
*
)
-
p
0
*
-
p
2
*
,
k
≥
4
.
Consider the following difference:
(36)
p
4
-
p
3
=
A
p
3
-
p
2
+
p
1
*
-
p
2
*
-
p
2
*
+
p
3
*
=
A
A
p
2
-
p
1
+
p
2
*
-
p
3
*
+
p
1
*
-
p
2
*
-
p
2
*
+
p
3
*
=
A
A
A
p
1
+
p
3
*
-
p
0
*
-
p
3
*
-
p
1
+
p
2
*
-
p
3
*
+
p
1
*
-
p
2
*
-
p
2
*
+
p
3
*
=
A
A
p
0
*
_
A
A
p
1
+
p
3
*
-
p
0
*
_
-
p
3
*
-
p
1
+
p
2
*
-
p
3
*
+
A
p
0
*
_
-
p
1
-
p
2
*
A
A
p
1
+
p
3
*
-
p
0
*
_
-
p
3
*
-
p
1
+
p
2
*
-
p
3
*
-
p
2
*
+
p
3
*
=
A
A
A
p
1
+
p
3
*
+
p
2
*
-
p
3
*
-
p
1
+
p
3
*
-
p
1
+
p
3
*
-
p
2
*
-
p
3
*
-
(
p
2
*
-
p
3
*
)
=
A
2
-
A
-
1
A
p
1
+
p
3
*
+
p
2
*
-
p
3
*
.
To simplify the calculations henceforth, we need the following equalities:(37)
1
-
a
2
=
a
,
a
2
=
3
a
-
1
,
1
-
2
a
1
-
a
=
a
,
A
2
-
A
-
1
=
1
1
-
a
2
1
-
(
1
-
a
)
-
1
-
a
2
=
0
.
Thus the last expression in equalities (36) equals zero.
Thenp
k
+
1
-
p
k
=
A
p
k
-
p
k
-
1
=
⋯
=
A
k
-
3
(
p
4
-
p
3
)
=
0, k
≥
4.However, the values of functionp
(
x
) we obtain can be negative. In the case of piecewise constant density we give the following criterion for p
(
x
) to be nonnegative.Proposition 10.
Letp
0
*
,
p
1
*
,
p
2
*
,
p
3
*
∈
R be constants, and function p
(
x
) is obtained according to the scheme above. Then p
(
x
)
≥
0 for all x
∈
[
0,1
] if and only if
(38)
p
0
*
,
p
1
*
,
p
2
*
,
p
3
*
≥
0
,
1
1
-
a
p
0
*
-
p
1
*
≥
0
,
1
1
-
a
(
p
3
*
+
p
0
*
-
p
1
*
)
-
p
3
*
≥
0
,
1
1
-
a
(
p
2
*
+
p
0
*
-
p
1
*
)
-
p
1
*
≥
0
.
These inequalities define unbounded convex set in R
4
∋
{
p
0
*
,
p
1
*
,
p
2
*
,
p
3
*
}.Proof.
In view of (37), it is sufficient to notice that
(39)
p
2
=
A
p
1
+
p
3
*
-
p
0
*
-
p
3
*
=
A
A
p
0
*
-
p
1
*
+
p
3
*
-
p
0
*
-
p
3
*
=
A
2
-
1
p
0
*
+
A
p
3
*
-
p
1
*
-
p
3
*
=
A
(
p
3
*
+
p
0
*
-
p
1
*
)
-
p
3
*
,
p
3
=
A
p
2
+
p
2
*
-
p
0
*
-
p
3
*
=
A
A
p
3
*
+
p
0
*
-
p
1
*
-
p
3
*
+
p
2
*
-
p
0
*
-
p
3
*
=
A
2
-
1
p
3
*
+
p
0
*
+
A
p
2
*
-
p
3
*
-
A
2
p
1
*
=
A
p
2
*
+
p
0
*
-
A
+
1
p
1
*
=
A
(
p
2
*
+
p
0
*
-
p
1
*
)
-
p
1
*
.Now consider obtaining a functionp
(
x
) with the property of continuity. This is discussed in Propositions 11–14.Proposition 11.
Given functionp
(
x
) obtained by the scheme above, then p
(
x
) is continuous on (
0,1
] if and only if (i)
parametersy
2
,
y
3
,
y
4
≥
0 satisfy the equation
(40)
y
2
+
y
4
=
1
1
-
a
y
3
;
(ii)
y
5
≥
0 is arbitrary;
(iii)
graphs of continuous functionsp
0
*
(
x
)
,
p
1
*
(
x
)
,
p
2
*
(
x
)
,
p
3
*
(
x
) connect points (
a
,
y
2
), (
1
-
a
,
y
3
), (
2
a
,
y
4
), (
2
-
3
a
,
y
5
), and (
1,0
); see Figure 11.Proof.
Letp
(
x
) be continuous; then we substitute x
=
x
~
=
1
-
2
a into (29)-(30) and obtain
(41)
p
1
-
2
a
+
p
1
-
a
+
p
1
=
1
1
-
a
p
1
-
2
a
1
-
a
+
p
1
,
p
1
-
2
a
+
p
1
-
a
=
1
1
-
a
p
1
-
2
a
1
-
a
,
wherefrom p
(
1
)
=
0. By substituting x
~
=
a into (30) and taking into account a
/
(
1
-
a
)
=
1
-
a, we have
(42)
p
(
a
)
+
p
(
2
a
)
=
1
1
-
a
p
(
1
-
a
)
,
which is equal to (40).
To prove the backward implication, letg
(
x
) be a piecewise function, made of functions p
0
*
,
p
1
*
,
p
2
*
,
p
3
* “glued together.”
By construction,p
1
(
x
)
=
(
1
/
1
-
a
)
g
(
x
/
(
1
-
a
)
)
-
g
(
x
+
a
) on (
1
-
2
a
,
a
]. Then p
1
(
x
) is continuous, because g
(
x
) is continuous, and
(43)
p
1
a
=
1
1
-
a
g
a
1
-
a
-
g
2
a
=
1
1
-
a
y
3
-
y
4
=
y
2
=
g
(
a
)
.
Now add functionp
1
(
x
) leftside into the set of functions which define g
(
x
).
By construction,p
2
(
x
)
=
(
1
/
1
-
a
)
(
g
(
x
/
(
1
-
a
)
)
+
g
(
(
x
+
a
)
/
(
1
-
a
)
)
)
-
g
(
x
+
a
)
-
g
(
x
+
2
a
) on (
x
2
,
1
-
2
a
], wherefrom p
2
(
x
) is continuous, and (considering g
(
1
)
=
0)
(44)
p
2
1
-
2
a
=
1
1
-
a
g
1
-
2
a
1
-
a
+
g
1
-
g
1
-
a
-
g
1
=
1
1
-
a
g
1
-
2
a
1
-
a
+
-
g
1
-
a
+
=
g
(
1
-
2
a
+
)
.
Fork
≥
2, by construction, p
k
+
1
(
x
)
=
(
1
/
1
-
a
)
(
g
(
x
/
(
1
-
a
)
) + g
(
(
x
+
a
)
/
(
1
-
a
)
)
)
-
g
(
x
+
a
)
-
g
(
x
+
2
a
) on (
x
k
+
1
,
x
k
], where g
(
x
) is made of functions p
k
,
…
,
p
2
,
p
1
,
p
0
,
p
1
*
,
p
2
*
,
p
3
*. Thus p
k
+
1
(
x
) is continuous, and
(45)
p
k
+
1
x
k
=
1
1
-
a
g
x
k
1
-
a
+
g
x
k
+
a
1
-
a
-
g
x
k
+
a
-
g
x
k
+
2
a
=
1
1
-
a
g
x
k
1
-
a
+
+
g
x
k
+
a
1
-
a
+
-
g
(
x
k
+
a
+
)
-
g
(
x
k
+
2
a
+
)
=
g
(
x
k
+
)
.Proposition 12.
Letp
(
x
) satisfy (29)-(30). If p
:
[
0,1
]
→
R is continuous, then p
(
0
)
=
p
(
1
)
=
0.Proof.
It suffices to showp
(
0
)
=
0. We substitute x
=
0 into (29) and obtain
(46)
p
0
+
p
a
+
p
2
a
=
1
1
-
a
p
0
+
p
a
1
-
a
,
wherefrom we have p
(
0
)
=
0 (considering (42)).However, the next question arises.Question 3.
Under which conditions does our construction yieldp
(
0
+
)
=
0?Notice that if by construction of densityp
(
x
) the equality p
(
0
+
)
=
0 is not fulfilled, then p
(
x
) is continuous on (
0,1
] but does not have finite limit at 0. Its graph is unbounded and (or) oscillates greatly in neighborhood of 0.Such situation is quite typical while constructingp
(
x
); see Figure 12. However, the following Proposition 13 gives an example of a density with good properties.Proposition 13.
Let functionp
(
x
) be obtained according to the scheme above. If functions p
0
*
,
p
1
*
,
p
2
*
,
p
3
* form a spline of degree 1 with p
3
*
(
1
)
=
0, then function p
(
x
) is also a spline of degree 1; furthermore (i)
p
:
[
0,1
]
→
R is continuous function;
(ii)
p
1
,
p
2 are linear functions;
(iii)
graphs of the functionsp
3
,
p
4
,
… make up one graph of linear function, which connects points (
0,0
) and (
x
2
,
p
2
(
x
2
)
); see Figure 13.Proof.
Again letA
=
1
/
(
1
-
a
), and denote the slope of spline on corresponding intervals by p
k
′
∈
R (for p
k
(
x
)). Then from the construction scheme of p
(
x
) itself, we obtain formulae equal to those from the proof of Proposition 9:
(47)
p
1
′
=
A
2
p
0
*
′
-
p
1
*
′
,
p
2
′
=
A
2
(
p
1
′
+
p
3
*
′
)
-
p
0
*
′
-
p
3
*
′
,
p
3
′
=
A
2
(
p
2
′
+
p
2
*
′
)
-
p
0
*
′
-
p
3
*
′
,
p
k
′
=
A
2
(
p
k
-
1
′
+
p
1
*
′
)
-
p
0
*
′
-
p
2
*
′
,
k
≥
4
.
SubstitutingA by A
2 in (36), we get
(48)
p
4
′
-
p
3
′
=
(
A
4
-
A
2
-
1
)
(
A
2
(
p
1
′
+
p
3
*
′
)
+
(
p
2
*
′
-
p
3
*
′
)
)
.
We need to show that the last multiplier equals zero. Let y
1
=
p
(
1
-
2
a
), y
2
=
p
(
a
), y
3
=
p
(
1
-
a
), y
4
=
p
(
2
a
), and y
5
=
p
(
2
-
3
a
). We use equalities (37) again:
(49)
p
2
*
′
-
p
3
*
′
+
A
2
p
1
′
+
p
3
*
′
=
y
5
-
y
4
2
-
5
a
-
y
5
1
-
3
a
+
A
2
y
2
-
y
1
3
a
-
1
+
y
5
1
-
3
a
=
1
2
-
5
a
1
-
3
a
×

A
2
1
-
3
a
y
5
-
y
4
-
2
-
5
a
y
5
+
A
2
2
-
5
a
y
1
-
y
2
+
y
5
=
1
2
-
5
a
1
-
3
a
×
y
5
2
a
-
1
+
A
2
2
-
5
a
-
y
4
1
-
3
a
+
A
2
2
-
5
a
y
1
-
y
2
y
5

=
1
2
-
5
a
1
-
3
a
×
-
A
y
2
-
y
1
1
-
3
a
+
A
2
2
-
5
a
y
1
-
y
2
=
1
2
-
5
a
1
-
3
a
×
y
1
-
y
2
1
-
3
a
A
+
A
2
2
-
5
a
=
1
2
-
5
a
1
-
3
a
×
(
y
1
-
y
2
)
(
1
-
3
a
)
A
+
(
A
+
1
)
(
2
-
5
a
)
.
Taking into account (37), we have
(50)
1
-
3
a
A
+
A
+
1
2
-
5
a
=
A
1
-
3
a
+
2
-
5
a
+
2
-
5
a
=
A
3
-
8
a
+
1
-
a
2
-
5
a
=
A
3
-
8
a
+
2
-
7
a
+
5
a
2
=
5
A
(
a
2
-
3
a
+
1
)
=
0
.
Thenp
k
+
1
′
-
p
k
′
=
A
p
k
′
-
p
k
-
1
′
=
⋯
=
A
k
-
3
(
p
4
′
-
p
3
′
)
=
0, k
≥
4.
Since the second statement of Proposition11 holds true (by the construction scheme of the spline), function p is continuous on (
0,1
]. Since it is linear on (
0
,
x
2
], then limit p
(
0
+
) exists, and, by Proposition 12, p
(
0
)
=
p
(
0
+
)
=
0.Figure14 shows an example of nontrivial density discussed in Proposition 13. Obviously such function is integrable. To accomplish the topic, we add nonnegativity criterion.Proposition 14.
Letp
(
x
) satisfy the conditions of Proposition 13. Denote y
3
=
p
(
1
-
a
), y
4
=
p
(
2
a
), y
5
=
p
(
2
-
3
a
). Then p
(
x
)
≥
0 for all x
∈
[
0,1
] if and only if
(51)
y
3
≥
y
4
≥
0
,
y
5
≥
11
a
-
4
2
-
5
a
y
4
-
29
a
-
11
8
a
-
3
y
3
11
a
-
4
2
-
5
a
≈
2.24
,
29
a
-
11
8
a
-
3
≈
1.38
y
5
≥
0
.
These inequalities define unbounded convex set in R
3
∋
{
y
3
,
y
4
,
y
5
}.Proof.
Lety
0
=
p
(
x
2
)
,
y
2
=
p
(
a
). In view of Proposition 13, this statement is equivalent to nonnegativeness of spline values p
(
x
) at the vertices y
k
≥
0, k
=
0
,
…
,
5.
Lety
1
=
p
(
1
-
2
a
), and substitute x
~
=
1
-
2
a into (30). Then using (
1
-
2
a
)
/
(
1
-
a
)
=
a, we have
(52)
y
1
+
y
3
=
1
1
-
a
y
2
.
We use expression (37) to simplify the quantities henceforth:
(53)
y
1
=
1
1
-
a
y
2
-
y
3
=
1
1
-
a
1
1
-
a
y
3
-
y
4
-
y
3
=
1
1
-
a
2
-
1
y
3
-
1
1
-
a
y
4
=
1
1
-
a
(
y
3
-
y
4
)
≥
0
,
wherefrom y
1
≥
0 if and only if y
3
≥
y
4.
Further,y
2
=
(
1
/
1
-
a
)
y
3
-
y
4
≥
y
3
-
y
4
≥
0 (1
/
(
1
-
a
)
≈
1.6); thus condition y
2
≥
0 holds true if y
3
≥
y
4.
To get the last restriction of the proposition, we consider an equality(54)
y
0
+
p
(
x
2
+
a
)
+
p
(
x
2
+
2
a
)
=
1
1
-
a
(
y
1
+
y
5
)
.
Since x
2
+
a
∈
[
a
,
1
-
a
], then
(55)
p
x
2
+
a
=
y
2
+
y
3
-
y
2
x
2
+
a
-
a
1
-
a
-
a
=
y
2
+
(
1
-
a
)
(
y
3
-
y
2
)
.
Since x
2
+
2
a
=
(
1
-
2
a
)
(
1
-
a
)
+
2
a
=
2
a
2
-
a
+
1
=
5
a
-
1
≈
0.91
>
2
-
3
a
≈
0.85, then x
2
+
2
a
∈
[
2
-
3
a
,
1
], and
(56)
p
(
x
2
+
2
a
)
=
y
5
-
y
5
5
a
-
1
-
2
+
3
a
1
-
2
+
3
a
=
2
-
5
a
3
a
-
1
y
5
.
Thus
(57)
y
0
=
1
1
-
a
y
1
+
y
5
-
y
2
-
1
-
a
y
3
-
y
2
-
2
-
5
a
3
a
-
1
y
5
=
1
1
-
a
1
1
-
a
y
2
-
y
3
-
y
3
1
-
a
-
a
y
2
+
1
1
-
a
-
2
-
5
a
3
a
-
1
y
5
=
1
1
-
a
2
-
a
y
2
-
y
3
(
1
1
-
a
+
1
-
a
)
+
2
-
5
a
3
a
-
1
1
-
a
y
5
=
1
1
-
a
+
1
-
a
y
2
-
y
3
+
2
-
5
a
(
3
a
-
1
)
(
1
-
a
)
y
5
.
Finally, y
0
≥
0 if and only if -
(
2
-
5
a
/
3
a
-
1
1
-
a
)
y
5
≤
(
1
/
(
1
-
a
)
+
1
-
a
)
(
y
2
-
y
3
), which leads to y
5
≥
11
a
-
4
/
2
-
5
a
(
y
3
-
y
2
) (2
-
5
a
,
3
a
-
1,1
-
a
>
0) and equals
(58)
y
5
≥
11
a
-
4
2
-
5
a
y
3
-
y
2
=
11
a
-
4
2
-
5
a
y
3
-
1
1
-
a
y
3
+
y
4
=
11
a
-
4
2
-
5
a
y
4
-
29
a
-
11
8
a
-
3
y
3
.Notice that functions in Figure12 differ a little from those in Figures 10 and 14: p
3
*
(
x
) is slightly changed in both cases. Such change leads to great oscillation and (or) unboundedness of p
(
x
). We formulate here the following questions.Example of densityp
(
x
) from Proposition 9 (p
1
=
1, p
1
*
=
2, p
3
*
=
3, p
2
*
=
2.5) (a) and corresponding function α
1
(
x
) (on [
a
,
1
-
a
]) (b).
(a)
(b)Figure 11
Illustration for Proposition11.Oscillating and unbounded functions.
(a)
(b)Figure 13
Illustration for Proposition13.Example of spline densityp
(
x
) from Proposition 13 (a). Here y
2
=
1.1, y
3
=
1.2, y
5
=
1, and y
4
=
(
1
/
(
1
-
a
)
)
y
3
-
y
2. Corresponding function α
1 (on [
a
,
1
-
a
]) (b).
(a)
(b)Question 4.
Explain such “bad” behavior of functionp
(
x
). Are there locally nonlinear densities (for them we need to check conditions (29)-(30), nonnegativeness and integrability of p
(
x
))?Question 5.
How can we provideα
1
(
x
)
∈
[
0,1
] in cases above (α
1
(
x
) is derived from (9))?In conclusion, consider the case whenα is an endowment by characteristic functions of sets of IFS (26) mask. Let M
=
{
M
1
,
M
2
} be the IFS mask: [
0
,
a
)
⊂
M
1 and (
1
-
a
,
1
]
⊂
M
2, M
1
′
=
M
1
∩
[
a
,
1
-
a
] and M
2
′
=
M
2
∩
[
a
,
1
-
a
] are arbitrary (M
1
∩
M
2
=
∅, M
1
,
M
2
∈
B).Letα
i
(
x
)
=
χ
M
i
(
x
), i
=
1,2. Condition (9) for n
=
2 turns into
(59)
α
1
x
=
1
p
x
p
x
-
a
+
p
x
-
1
1
-
a
p
x
-
a
1
-
a
,
∀
x
∈
a
,
1
-
a
.
On the set M
1, α
1
(
x
)
≡
1, and (59) implies
(60)
p
(
x
-
a
)
=
1
1
-
a
p
(
x
-
a
1
-
a
)
,
x
∈
[
a
,
1
-
a
)
∩
M
1
.
Similarly, on the set M
2, α
1
(
x
)
≡
0, and (59) yields
(61)
p
x
-
a
+
p
x
=
1
1
-
a
p
x
-
a
1
-
a
,
x
∈
a
,
1
-
a
∩
M
2
.One can see that condition (27) splits into two (as sketched in Figure 15):
(62)
on
⁡
[
a
,
1
-
a
)
∩
M
1
p
(
x
-
a
)
=
1
1
-
a
p
x
-
a
1
-
a
,
p
(
x
)
+
p
(
x
+
a
)
=
1
1
-
a
p
(
x
1
-
a
)
;
(63)
on
⁡
a
,
1
-
a
∩
M
2
p
(
x
-
a
)
+
p
(
x
)
=
1
1
-
a
p
(
x
-
a
1
-
a
)
,
p
(
x
+
a
)
=
1
1
-
a
p
(
x
1
-
a
)
.Figure 15
Scheme of conditions (62) (above) and (63) (below).Thus we can introduce the following scheme of constructionp
(
x
), which is slightly changed version of the one above.Consider the following marks on thex-axis: x
k
=
a
(
1
-
a
)
k, x
k
*
=
1
-
a
(
1
-
a
)
k, k
≥
1 see Figure 16.(i)
Fix functionsp
0
*
,
p
1
*
∈
L
1, p
0
*
,
p
1
*
≥
0, arbitrarily, and define
(64)
p
x
=
p
1
(
x
)
=
1
1
-
a
p
0
*
(
x
1
-
a
)
-
p
1
*
(
x
+
a
)
,
x
∈
(
1
-
2
a
,
a
]
,
p
0
*
(
x
)
,
x
∈
(
a
,
1
-
a
]
,
p
1
*
x
,
x
∈
(
1
-
a
,
2
a
]
.
(ii)
By induction onk
≥
2, define
(65)
p
x
=
p
k
x
=
1
1
-
a
p
k
-
1
(
x
1
-
a
)
,
x
∈
(
x
k
,
x
k
-
1
]
,
x
+
a
∈
M
1
′
,
1
1
-
a
p
k
-
1
(
x
1
-
a
)
-
p
0
*
(
x
+
a
)
,
x
∈
(
x
k
,
x
k
-
1
]
,
x
+
a
∈
M
2
′
.
(iii)
By induction onk
≥
2, define
(66)
p
x
=
p
k
*
x
=
1
1
-
a
p
k
-
1
*
(
x
-
a
1
-
a
)
,
x
∈
(
x
k
-
1
*
,
x
k
*
]
,
x
-
a
∈
M
2
′
,
1
1
-
a
p
k
-
1
*
(
x
-
a
1
-
a
)
-
p
0
*
x
-
a
,
x
∈
(
x
k
-
1
*
,
x
k
*
]
,
x
-
a
∈
M
1
′
.
(iv)
Fix valuesp
(
0
)
,
p
(
1
)
≥
0 arbitrarily.Figure 16
Construction scheme for densityp
(
x
), extra intervals marked, a
=
(
3
-
5
)
/
2, case of n
=
2.Thus functionp
(
x
) is completely defined by its values on (
1
-
2
a
,
1
-
a
], which are defined by functions p
0
* and p
1
*.Here next question arises.Question 6.
Under which conditions dop
(
x
)
∈
L
1 and p
(
x
)
≥
0? Is it possible to construct such function for any mask M?The examples of two masks (see Figure17) are the partial answer to it. In these examples masks M are connected with partition structure of interval [
0,1
] over iteration process of density construction. Namely, if M
1
=
[
0,4
a
-
1
), M
2
=
[
4
a
-
1,1
] (4
a
-
1
=
x
2
+
a), p
1
=
c
≥
0, and p
0
=
1
/
1
-
a
c, then one can show that p
1
*
=
(
1
/
1
-
a
)
c, p
(
x
)
≡
0 outside [
1
-
2
a
,
x
2
*
]. If M
1
=
[
0
,
a
)
∪
[
4
a
-
1,1
-
a
], M
2
=
[
a
,
4
a
-
1
)
∪
(
1
-
a
,
1
], and p
1
=
p
0
=
c, then p
1
*
=
(
1
-
a
)
c, p
2
*
=
c, and p
(
x
)
≡
(
1
/
1
-
a
)
c outside [
1
-
2
a
,
x
2
*
].Graphs of invariant measure densities with mask endowment (n
=
2) (see text): M
1
=
[
0,4
a
-
1
), M
2
=
[
4
a
-
1,1
], c
=
1.2 (a) and M
1
=
[
0
,
a
)
∪
[
4
a
-
1,1
-
a
], M
2
=
[
a
,
4
a
-
1
)
∪
(
1
-
a
,
1
], c
=
1.2 (b).
(a)
(b)We have not found an example of densityp
(
x
) for arbitrary mask (for instance, that in Figure 7): the function constructed had negative values, unbounded and (or) oscillated greatly.


## 3. The Case ofn
>
2

In conclusion of the paper, we introduce one of the possible construction schemes for densityp
(
x
) for all n
≥
2 and any a:
(67)
1
n
+
1
<
a
≤
1
n
.Consider the following variables (see Figure18):
(68)
x
k
=
(
1
-
n
a
)
1
-
a
k
-
1
,
k
∈
Z
.Figure 18
Construction scheme for densityp
(
x
) in the case of arbitrary n
≥
2.Lemma 15.
There exists a unique numberK
∈
Z, K
≤
1, such that x
K
<
a, x
K
-
1
≥
a.Proof.
Sincex
k
=
(
1
-
a
)
x
k
-
1, it is sufficient to consider the chain of inequalities:
(69)
(
1
-
a
)
a
≤
(
1
-
n
a
)
1
-
a
k
-
1
<
a
,
(
1
-
a
)
a
1
-
n
a
≤
1
-
a
k
-
1
<
a
1
-
n
a
,
log
⁡
1
-
a
a
1
-
n
a
+
1
≥
k
-
1
>
log
⁡
1
-
a
a
1
-
n
a
.
According to (67), we have a
/
(
1
-
n
a
)
>
1; hence log
⁡
1
-
a
(
a
/
1
-
n
a
)
<
0. Then (69) completes the proof.Now we introduce the following scheme to constructp
(
x
), with (7)-(8) satisfied.(i) Fixg
∈
L
1, g
≥
0, arbitrarily, and define
(70)
p
(
x
)
=
g
x
on
⁡
(
a
,
1
]
.(ii) Fork
=
K, define (see formula (8))
(71)
p
x
=
p
K
x
=
1
1
-
a
g
x
1
-
a
+
∑
i
=
1
n
-
2
‍
g
x
+
i
a
1
-
a
-
∑
i
=
1
n
-
1
‍
g
x
+
i
a
,
x
∈
(
x
K
,
a
]
.(iii) By induction onk
=
K
+
1
,
…
,
1, define (see (8))
(72)
p
x
=
p
k
x
=
1
1
-
a
p
k
-
1
x
1
-
a
+
∑
i
=
1
n
-
2
‍
g
x
+
i
a
1
-
a
-
∑
i
=
1
n
-
1
‍
g
(
x
+
i
a
)
,
x
∈
(
x
k
,
x
k
-
1
]
.(iv) By induction onk
=
0,1
,
…, define (see (7))
(73)
p
x
=
p
k
x
=
1
1
-
a
p
k
-
1
x
1
-
a
+
∑
i
=
1
n
-
1
‍
g
x
+
i
a
1
-
a
-
∑
i
=
1
n
‍
g
(
x
+
i
a
)
,
x
∈
(
x
k
,
x
k
-
1
]
.(v) Fix valuep
(
0
)
≥
0 arbitrarily.Here the following question appears.Question 7.
Under which conditions dop
(
x
)
∈
L
1 and p
(
x
)
≥
0? Under which conditions does α
1
(
x
)
∈
[
0,1
] (α
1
(
x
) is derived from (9))?Obvious “mirror” change of this scheme is shown in Figure19 (replacing x
k by x
k
*
=
1
-
x
k); compare with Corollary 8.Figure 19
“Mirror” construction scheme of densityp
(
x
) in the case of arbitrary n
≥
2.


## 4. Conclusion

Section1 contains motivation part. It overviews previously derived criteria of measure invariance and some related results, as well as connection between endowment and mask. In Section 2 we consider the case of a
=
(
3
-
5
)
/
2, and example of mask is given. Section 3 introduces construction scheme for densities with arbitrary a
∈
(
0
,
1
/
2
].



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*Source: 513061-2014-11-19.xml*