Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations
Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures...
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| Date: | 2013 |
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| author | Blackmore, D. Golenia, J. Prykarpatsky, A. K. Prykarpatsky, Ya. A. Блекмор, Д. Голеня, Й. Прикарпатський, А. К. Прикарпатський, Я. А. |
| author_facet | Blackmore, D. Golenia, J. Prykarpatsky, A. K. Prykarpatsky, Ya. A. Блекмор, Д. Голеня, Й. Прикарпатський, А. К. Прикарпатський, Я. А. |
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| datestamp_date | 2020-03-18T19:14:46Z |
| description | Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator.
New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed. |
| first_indexed | 2026-03-24T02:22:45Z |
| format | Article |
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UDC 517.9
D. Blackmore (New Jersey Inst. Technology, Newark, USA),
J. Golenia (Univ. Sci. and Technology, Krakow, Poland),
A. K. Prykarpatsky (Univ. Sci. and Technology, Krakow, Poland and Ivan Franko Ped. State Univ., Drohobych,
Ukraine),
Ya. A. Prykarpatsky (Inst. Math. Nath. Acad. Sci. Ukraine, Kyiv and Univ. Agriculture, Krakow, Poland)
INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS
AND ERGODIC PROPERTIES OF GENERALIZED BOOLE-TYPE
TRANSFORMATIONS
IНВАРIАНТНI МIРИ ДЛЯ ДИСКРЕТНИХ ДИНАМIЧНИХ СИСТЕМ
ТА ЕРГОДИЧНI ВЛАСТИВОСТI УЗАГАЛЬНЕНИХ ПЕРЕТВОРЕНЬ
БУЛЕВOГО ТИПУ
Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generati-
ng function approach based on special solutions for the Frobenius – Perron operator. New two-dimensional Boole-type
transformations are introduced, and their invariant measures and ergodicity properties are analyzed.
Вивчаються ергодичнi мiри для узагальнених перетворень булевoго типу iз використанням пiдходу твiрних функцiй
iнварiантних квазiмiр, що базується на спецiальних розв’язках для оператора Фробенiуса – Перрона. Запропоновано
новi двовимiрнi перетворення булевoго типу та дослiджено їхнi iнварiантнi мiри та ергодичнi властивостi.
1. Invariant measures: introductory setting. It is well known that discrete dynamical systems on
finite-dimensional manifolds play an important role [8, 9, 12, 21] in describing evolution properties
of many processes in the applied sciences. Of particular interest are discrete dynamical systems
on manifolds with invariant measures, often possessing additional properties such as ergodicity or
mixing, which allow to explain such phenomenon as chaotic behavior and instability of the physical
objects being studied. Therefore, methods of constructing invariant (with respect to a given discrete
dynamical system) measures, such as those we develop in the sequel, are of crucial importance.
Suppose that a topological phase space M is endowed with a structure of a measurable space, that
is a σ-algebraA(M) of subsets inM, on which there is a finite normalized measure µ : A(M)→ R+,
µ(M) = 1. As is well known [25], a measurable mapping ϕ : M →M of the measurable space (M,
A(M)) is called an ergodic discrete dynamical system if µ-almost everywhere (µ-a.e.) there exists
an x ∈M limit
lim
n→∞
1
n
n−1∑
k=0
f(ϕkx) (1.1)
for any bounded measurable function f ∈ B(M ;R).
We now assume that the limit (1.1) exists µ-a.e., that is one can define a bounded measurable
function fϕ ∈ B(M ;R), where
lim
n→∞
1
n
n−1∑
k=0
f(ϕkx) := fϕ(x) (1.2)
for all x ∈M the function (1.2) defines a finite measure µϕ : A(M)→ R+ on M such that
c© D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY, 2013
44 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 45∫
M
fϕ(x)dµ(x) :=
∫
M
f(x)dµϕ(x). (1.3)
Actually, the Lebesgue – Helley theorem on bounded convergence [22] implies that∫
M
fϕ(x)dµ(x) = lim
n→∞
∫
M
f(x)dµn,ϕ(x), (1.4)
where
µn,ϕ(A) :=
1
n
n−1∑
k=0
µ(ϕ−kA), (1.5)
is the Schur average, n ∈ Z+, A ⊂ A(M), and ϕ−kA := {x ∈ M : ϕkx ∈ A} for any k ∈ Z+.
The limit on the right-hand side of (1.4) obviously exists for any bounded measurable function
f ∈ B(M ;R). Consequently, the equality
µϕ(A) := lim
n→∞
µn,ϕ(A), (1.6)
for any A ⊂ A(M) defines on the σ-algebra A(M) an additive non-negative bounded mapping
µϕ : A(M) → R+. Besides, from the existence of a uniform approximation of arbitrary measurable
bounded function by means of finite-valued measurable (simple) functions, one immediately infers
the equality (1.3) for any f ∈ B(M ;R). The requirement for countable additivity of the mapping
µϕ : A(M)→ R+ follows from the equivalent expression [23]
lim
k→∞
sup
n∈Z+
µn,ϕ(Ak) = 0
for any monotonic sequence of sets Aj ⊃ Aj+1, j ∈ Z+, of A(M) with empty intersection.
The measure µϕ : A(M) → R+ defined by (1.6), has the following invariance property with
respect to the dynamical system ϕ : M →M :
µϕ(ϕ−1A) = µϕ(A) (1.7)
for any A ∈ A(M), which follows from simple identity
µn,ϕ(ϕ−1A) =
n+ 1
n
µn+1,ϕ(A)− 1
n
µ(A),
upon taking the limit as n→∞. It is easy to see that (1.7) is completely equivalent to the equality∫
M
f(ϕx)dµϕ(x) =
∫
M
f(x)dµϕ(x)
for any f ∈ B(M ;R). Moreover, if a σ-measurable set A ∈ A(M) is invariant with respect to the
mapping ϕ : M →M, that is ϕ−1(M) = M, then evidently µϕ(A) = µ(A).
Therefore, the existence of the ϕ-invariant measure µϕ : A(M)→ R+, coinciding with the mea-
sure µ : A(M) → R+ on the σ-algebra I(M) ⊂ A(M) of invariant (with respect to the dynamical
system ϕ : M → M ) sets, is a necessary condition of the convergence µ-a.e. on M of the mean
values (1.1) as n→∞ for any f ∈ B(M ;R). That the converse is also true follows from a theorem
of Birkhoff [22]: if the mapping ϕ : M → M conserves a finite measure µϕ : A(M) → R+, the
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
46 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
mean values (1.1) are convergent µϕ-a.e. on M, and the convergence set is invariant. Thus, if the
reduction of the measure µ : A(M) → R+ upon the invariant σ-algebra I(M) ⊂ A(M) is abso-
lutely continuous with respect to that of the measure µϕ : A(M)→ R+, the convergence holds µ-a.e.
on M.
2. An invariant measure generating construction. Assume we are given a discrete dynamical
system ϕ : M → M and a sequence of associated measures µn,ϕ : A(M) → R+, n ∈ Z+, defined
by (1.5). Then one can define measure generating functions (m.g.f.) µn,ϕ(λ;A), n ∈ Z+, where for
any A ∈ A(M), λ ∈ C,
µn,ϕ(λ;A) :=
n−1∑
k=0
λk µ(ϕ−kA). (2.1)
Define now the following measure generating function:
µϕ(λ;A) := lim
n→∞
n−1∑
k=0
λk µ(ϕ−kA), (2.2)
where A ∈ A(M), and |λ| < 1 to insure the finiteness of the expression (2.2). It is easy now to
prove the following result.
Lemma 2.1. The m.g.f. (2.2) satisfies the functional equation
µϕ(λ;A) = λµϕ(λ;ϕ−1A) + µ(A) (2.3)
for any A ∈ A(M) and |λ| < 1.
Proof. From (2.3) one finds by iteration directly that
µϕ(λ;A)−
n−1∑
k=0
λk µ(ϕ−kA) = λn+1µϕ(λ;ϕ−k−1A) (2.4)
for any n ∈ Z+, A ∈ A(M) and |λ| < 1. Taking the limit in (2.4) as n → ∞, one arrives at the
determining expression (2.2) that completes the proof.
Corollary 2.1. Assume we are given a mapping µ(s) := µ−s µ◦ϕ−1 on A(M), where |s| < 1.
Then the equality
µ(s)
ϕ (s;A) = µ(A) (2.5)
holds for all A ∈ A(M), |s| < 1.
Proof. This follows from a straightforward substitution of the mapping µ(s) : A(M) → R for
|s| < 1 into (2.3).
Example 2.1. The induced functional expansion.
Let M = [0, 1] ⊂ R and ϕ : M →M is the “baker” transformation, that is
ϕ(x) :=
2x, if x ∈ [0, 1/2),
2(1− x), if x ∈ [1/2, 1].
(2.6)
Take now a mapping f : M →M, given as
f(x) := 2x− x2 (2.7)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 47
for any x ∈ M and construct the convolution of (2.5) with the function (2.7) at the parametric
measure µ(A;x) :=
∫
A
dϑx(y), A ∈ A(M), x ∈ M, where ϑx : M → R, x ∈ M, is the standard
Heavyside function with the support supp ϑx = {y ∈M : y − x ≥ 0}. Then the decomposition
f(x) = (2− 4s)
∑
n∈Z+
snϕn(x) + (4s− 1)
∑
n∈Z+
snϕn(x)ϕn(x)
holds [26] for any x ∈M. In the cases s = 1/2 and s = 1/4, one readily obtains for any x ∈M the
decompositions ∑
n∈Z+
(1/2)nϕn(x)ϕn(x) = 2x− x2 =
∑
n∈Z+
(1/4)nϕn(x),
which are useful for some applied set-theoretical considerations. Note here also that a similar expan-
sion given by ∑
n∈Z+
(1/2)nϕn(x) := ξ(x)
for any x ∈ M, yields the well-known Weierstrass function ξ : [0, 1] → [0, 1], which is continuous
but nowhere differentiable [24] on M = [0, 1] ⊂ R.
3. Representation of invariant measures. Assume now that the limit (1.6) exists owing to (1.7)
being measure preserving on A(M). Then the following important Tauberian type [11] result holds.
Theorem 3.1. Let the measure generating function µϕ : C× A(M) → C, corresponding to a
discrete dynamical system ϕ : M → M, exist and satisfy the invariance condition (1.7). Then the
limit expression
lim
λ↑1 (Imλ=0)
(1− λ)µϕ(λ;A) = µϕ(A) (3.1)
holds for any A ∈ A(M). Moreover, the converse is also true.
Proof. Since all coefficients of the series (2.1) are bounded, that is are of O(1), then it follows
from a well-known Tauberian theorem of [11] Hardy that
lim
λ↑1 (Imλ=0)
(1− λ)µϕ(λ;A) = lim
n→∞
1
n
n−1∑
k=0
µ(ϕ−kA) := µϕ(A) (3.2)
for any A ∈ A(M), which completes the proof.
We can now use the above theorem to produce an invariant measure µϕ : A(M)→ R+ on M by
means of the measure generating function µϕ : C× A(M)→ C defined by (2.1) for a given discrete
dynamical system ϕ : M → M. Also, observe that the series (2.1) generates an analytic function
when |λ| < 1 such that for any λ ∈ (−1, 1) and A ∈ A(M),
Imµϕ(λ;A) = 0. (3.3)
Now, using classical analytic function theory [16, 17], one can readily verify the following result.
Theorem 3.2. Let a measure generating function µϕ : C×A(M) → C satisfy the condi-
tion (3.3). Then the following representation holds:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
48 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
µϕ(λ;A) =
2π∫
0
(1− λ2)dσϕ(s;A)
1− 2λ cos s+ λ2
(3.4)
for any A ∈ A(M), where σϕ(◦;A) : [0, 2π]→ R+ is a function of bounded variation
0 ≤ σϕ(s;A) ≤ µ(A) (3.5)
for any s ∈ [0, 2π] and A ∈ A(M).
This theorem appears to be exceptionally interesting for applications since it reduces the problem
of detecting the invariant measure µϕ : A(M)→ R+ defined by (1.6) to a calculation of the following
complex analytical limit:
µϕ(A) = lim
λ↑1 (Imλ=0)
2π∫
0
2(1− λ)2dσϕ(s;A)
1− 2λ cos s+ λ2
, (3.6)
where A ∈ A(M) and σϕ : [0, 2π] ×A(M)→ R+ — some Stieltjes measure on [0, 2π], generated by
a given a priori dynamical system ϕ : M →M and a measure µ : A(M)→ R+.
Example 3.1. The Gauss mapping.
Consider the case of the Gauss mapping ϕ : M → M, where M = [0, 1] and for any x ∈ (0, 1],
ϕ(x) := {1/x}, ϕ(0) = 0 (here “{·}” means taking the fractional part of a number x ∈ [0, 1]). One
can show by means of simple but somewhat cumbersome calculations that it is indeed ergodic [22]
and possesses the following invariant measure on M :
µϕ(A) =
1
ln 2
∫
A
dx
1 + x
,
which obviously yields the well-known Gauss measure µϕ : A(M) → R+ on M = (0, 1]. As a
result, the following limit for arbitrary f ∈ L1(0, 1) obtains:
lim
n→∞
n−1∑
k=0
f(ϕnx)
a.e.
=
1
ln 2
1∫
0
f(x)dx
1 + x
.
The analytical expression (3.6) obtained above for the invariant measure µϕ : A(M)→ R+, gen-
erated by a discrete dynamical system ϕ : M →M, should be quite useful for concrete calculations.
In particular, it follows directly from (3.4) that the Stieltjes measure σϕ(◦;A) : [0, 2π] → [0, µ(A)],
A ∈ A(M), generates for any s ∈ [0, 2π] a new positive definite measure on A ∈ A(M) as
σϕ(s)(A) = σϕ(s;A),
which can be regarded as smearing the measure µ : A(M) → R+ along the unit circle S1 in the
complex plane C.
An important still open problem, which is closely linked with the expression (3.6), is the following
inverse measure evaluation question: How can one retrieve the dynamical system ϕ : M →M which
generated the above smeared Stieltjes measure σϕ : [0, 2π] ×A(M)→ R+ via the expression (3.4)?
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 49
4. New generalizations of the Boole transformation and their ergodicity. In this section
we will study invariant measures and ergodicity properties of both the one-dimensional generalized
Boole transformation
y → ϕ(y) := αy + a−
N∑
j=1
βj
y − bj
∈ R, (4.1)
where a and bj ∈ R are real and α, βj ∈ R+ are positive parameters, 1 ≤ j ≤ N, and naturally
generalized two-dimensional Boole-type transformations
(x, y)→ (x− 1/x, y − 1/y) ∈ R2,
(x, y)→ (x− 1/y, y − 1/x) ∈ R2,
(4.2)
defined whenever xy 6= 0. They generalize the classical Boole transformation [10] y → ϕ(y) :=
:= y − 1/y ∈ R, defined for y 6= 0, which was proved to be ergodic [6] with respect to the invariant
standard infinite Lebesgue measure on R. In the case α = 1, a = 0, the analogous ergodicity
result was proved in [1 – 3] making use of the specially devised inner function approach. The related
spectral properties were in part studied in [3]. In spite of these results, the case α 6= 1 still persists
as a challenge. In fact, the only related result [4] concerns the following special case of (4.1):
y → ϕ(y) := αy−β/y ∈ R for 0 < α < 1 and arbitrary β ∈ R+, where the corresponding invariant
measure appeared to be finite absolutely continuous with respect to the Lebesgue measure on R and
equal to
dµ(x) :=
√
β(1− α)dx
π[x2(1− α) + β]
, (4.3)
where x ∈ R. The ergodicity for the invariant measure (4.3) now can be easily proved. It should
be recalled here that for a general nonsingular mapping ϕ : R → R, the problem of constructing
invariant ergodic measures is analyzed [4, 13] by studying the spectral properties of the adjoint
Frobenius – Perron operator T̂ϕ : L2(R;R)→ L2(R;R), where
T̂ϕρ(x) :=
∑
y∈{ϕ−1(x)}
ρ(y)J−1
ϕ (y) (4.4)
for any ρ ∈ L2(R;R+) and J−1
ϕ (y) :=
∣∣∣∣dϕ(y)
dy
∣∣∣∣, y ∈ R. Then if T̂ϕρ = ρ, ρ ∈ L2(R;R+), the
expression dµ(x) := ρ(x)dx, x ∈ R, will be an invariant (in general infinite) measure with respect
to the mapping ϕ : R→ R.
Another way of finding a general algorithm for obtaining such an invariant measure was devised
in [18, 19] using the generating measure function method.
Below we study some other special cases of the generalized Boole transformation (4.1), for
which we derive the corresponding invariant measures and prove the related ergodicity and spectral
properties.
4.1. Invariant measures and ergodic transformations. We will start with analyzing the follow-
ing Boole-type surjective transformation:
R 3 y → ϕ(y) := αy + a− β
y − b
∈ R (4.5)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
50 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
for any a, b ∈ R and 2β := γ2 ∈ R+. The transformation (4.5) for α = 1/2 and b = 2a ∈ R is
measure preserving with respect to a measure like (4.3). Namely, the following lemma holds.
Lemma 4.1. The Boole-type mapping (4.5) is measure preserving with respect to the measure
dµ(x) :=
|γ|dx
π[(x− 2a)2 + γ2]
, (4.6)
where x ∈ R and γ2 := 2β ∈ R+.
Proof. A proof follows easily from the fact that the function
ρ(x) :=
γ
π[(x− 2a)2 + γ2]
(4.7)
satisfies for all x ∈ R\{2a} the determining condition (4.4):
T̂ϕρ(x) :=
∑
I
ρ(y±)|y′±(x)|, (4.8)
where ϕ(y±(x)) := x for any x ∈ R. The relationship (4.8) is manifestly equivalent to the invariance
condition ∑
±
dµ(y±(x)) = dµ(x) := µ(dx) (4.9)
for any infinitesimal subset dx ⊂ R.
Lemma 4.1 is proved.
The question about the ergodicity of the mapping (4.5) is solved here easily by the following
theorem.
Theorem 4.1. The measure (4.7) is ergodic with respect to the transformation (4.5) at α = 1/2
and b = 2a ∈ R as such one is equivalent to the canonical ergodic mapping R/Z 3 s :→ ψ(s) := 2s
mod Z ∈ R/Z with respect to the standard Lebesgue measure on R/Z.
Proof. Define R/Z 3 s :→ ξ(s) = y ∈ R, where
ξ(s) := γ cotπs+ 2a. (4.10)
Then the transformation (4.5) for α = 1/2, b = 2a ∈ R and γ2 := 2β ∈ R+ yields under the
mapping (4.10)
ϕ(y) = ϕ(ξ(s)) =
γ
2
cotπs+ 2a− γ
2
tanπs =
γ(cos2 πs− sin2 πs)
2 sinπs cosπs
+ 2a =
= γ
cos 2πs
sin 2πs
+ 2a = γ cot 2πs+ 2a := ξ(2s) (4.11)
for any s ∈ R/Z. The result (4.10) means that the transformation (4.5) is conjugated [3, 13] with the
transformation
R/Z 3 s :→ ψ(s) = 2s mod Z ∈ R/Z; (4.12)
that is, the following diagram is commutative:
R/Z ψ→ R/Z
ξ ↓ ↓ ξ
R ϕ→ R,
(4.13)
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INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 51
that is ξ ◦ ψ = ϕ ◦ ξ, where ξ : R/Z→ R is the conjugate map defined by (4.10). It is easy now to
check that the measure (4.6) under the conjugation (4.13) transforms into the standard normalized
Lebesgue measure on R/Z :
dµ(x)|x=γ cotπs+2a =
dsγ2
∣∣d(cotπs)/ds
∣∣
(γ2 cot2 πs+ γ2)
=
sin2 πs · (sinπs)−2 ds
cos2 πs+ sin2 πs
= ds, (4.14)
where s ∈ R/Z. The infinitesimal measures ds on R/Z and the infinitesimal measure (4.6) on R are
normalized, so they are both probability measures. Now it suffices to make use of the fact that the
measure ds on R/Z on the interval [0, 1] ' R/Z is ergodic [4, 13] in order to obtain the desired
result.
4.2. Ergodic measures: the inner function approach. Assume that there exists a function ρω ∈
∈ H2(C+;C), holomorphic in parameter ω ∈ C+, satisfying the following identity:
T̂ϕρω = ρϕ̃(ω) (4.15)
for any ω ∈ C+ for some induced transformation C+ 3 ω → ϕ̃(ω) ∈ C+. If we now take ω :=
:= ω̄ ∈ C+ as a fixed point of the mapping ϕ̃ : C+ → C+, then it follows directly from (4.15) that
T̂ϕρω̄ = ρω̄, which means
dµ(x) := Im ρω̄(x)dx (4.16)
for x ∈ R is an invariant measure for the transformation ϕ : R → R. There is no general rule for
constructing such functions ρω ∈ H2(C+;C), analytic in ω ∈ C+, and the related induced mappings
ϕ̃ : C+ → C+. Nevertheless, for solving this problem one can adapt some natural ideas related to the
exact functional form of the determining Frobenius – Perron operator T̂ϕ : L2(R;R)→ L2(R;R). To
explain this, let us consider the following Boole-type transformation:
R 3 ϕ(y) := αy + a− β
y − b
∈ R, (4.17)
where a, b ∈R and β ∈ R+. It is easy to see that the Frobenius – Perron operator action on any
ρω ∈ H2(C+;C) can be represented as follows:
T̂ϕρω := ρω(y+)y′+ + ρω(y−)y′− =
=
(ω − y+)ρω(y+)(ω − y−)y′−
(ω − y+)(ω − y−)
+
ρω(y−)(ω − y+)(ω − y−)y′−
(ω − y+)(ω − y−)
=
=
k(ω − y−)y′+ + k(ω − y+)y′−
(ω − y+)(ω − y−)
=
−k[(ω − y+)(ω − y−)]′
(ω − y+)(ω − y−)
=
= −k d
dx
ln
[
(ω − y+)(ω − y−)
]
, (4.18)
where
ρω(x) =
k
ω − x
(4.19)
for all ω ∈ C+\{x}, x ∈ R, and some parameter k ∈ R. As a result of (4.19), one can take
ρω(y+)(ω − y+) = k = ρω(y−)(ω − y−), (4.20)
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52 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
for all x ∈ R and ω ∈ C+. Since the root functions y+ and y− : R → R satisfy, by definition, the
same equation
ω(y±(x)) = x, (4.21)
for all x ∈ R, the following identity for all ω ∈ C+ easily follows from (4.21) owing to the general
form of (4.17):
α(ω − y+)(ω − y−) = [ϕ(ω)− x](ω − b), (4.22)
where
y+(x) + y−(x) = b+
x− a
2
, y+(x)y−(x) =
bx− ab− β
2
. (4.23)
Whence, taking into account the expression (4.18), one computes that
T̂ϕρω = −k d
dx
ln([ϕ(ω)− x](ω − b)) =
k(ω − b)
[ϕ(ω)− x](ω − b)
=
k
ϕ(ω)− x
= ρϕ(ω), (4.24)
for all x ∈ R and ω ∈ C+. Therefore, the induced mapping ϕ̃ : C+ → C+ is exactly the transforma-
tion ϕ : C+ → C+, extended naturally from R to the complex plane C+.
Now let ω̃ ∈ C+ be a fixed point of the induced mapping ϕ : C+ → C+, that is ϕ(ω̄) = ω̄ ∈ C+.
Then from (4.24), one finds that T̂ϕρω̄ = ρω̄, or the corresponding invariant quasi-measure on R has
the form
dµ(x) := Im
kdx
ω̄ − x
(4.25)
for all x ∈ R and a suitable parameter k ∈ C. As Imρω̄ ∈ L2(R;R+) at any ω̄ ∈ C+\R and some
k ∈ C, the invariant quasi-measure (4.25) transforms into an actual invariant measure. These results
can be formulated as follows:
Theorem 4.2. The quasi-measure (4.25) is invariant with respect to the transformation (4.17)
for any α ∈ R+\{1}; for α = 1 at the condition a 6= 0, Im k 6= 0, it is reduced upon the set R/πZ,
being equivalent to the standard Gauss measure.
Proof. The desired infinitesimal quasi-measure dµ(x) exist if there is at least one fixed point of
the equation ϕ(ω) = ω for ω ∈ C+. If α 6= 1, this equation is equivalent to
(α− 1)ω2 − ω[(α− 1)b− a]− (ab+ β) = 0, (4.26)
which always has a solution ω̄ ∈ C+, for which ϕ(ω̄) = ω̄. When α = 1, the unique solution
ω̄ = (ab+ β)/a ∈ R exists only if a 6= 0 and Imk 6= 0, at which the quasi-measure (4.25) becomes
degenerate and reduces to the standard Gauss measure [1, 13] on R/πZ.
Theorem 4.2 is proved.
Theorem 4.2 states only that the quasi-measure (4.25) is invariant with respect to the transforma-
tion (4.17), so its ergodicity still needs to be proved separately using only the additional property that
the corresponding invariant measure is unique. Below we will proceed to study the general case of
the transformation (4.1), searching for a suitable invariant quasi-measure that is actually a measure
for some ω̄ ∈ C+\R, k ∈ C.
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INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 53
4.3. Invariant measures: the general case. Consider the equation
ϕ(y) = x, (4.27)
where x, y ∈ R and the mapping ϕ : C+ → C+ is given by expression (4.1) for a fixed integer
N ∈ Z+\{0, 1}. The equation (4.27) can be rewritten as
α
N+1∏
j=1
(y − yj) = [ϕ(y)− x]
N∏
j=1
(y − bj) (4.28)
for all x, y ∈ R and some functions yj : R → R, 1 ≤ j ≤ N + 1. Then the relationship (4.28) is
naturally extended on the complex plane C+ as
α
N+1∏
j=1
(ω − yj) = [ϕ(ω)− x]
N∏
j=1
(ω − bj) (4.29)
for any ω ∈ C+.
Consider now the relationship (4.15) in the manner of Section 3; namely
T̂ϕρω =
N∑
j=1
ϕ(ω)(yj)y
′
j =
N+1∑
j=1
ϕ(ω)(yj)(ω − yj
∏N+1
k 6=j (ω − yk)y′j∏N+1
k=1 (ω − yk)
=
=
N+1∑
j=1
ϕ(ω)(yj)(ω − yj
∏N+1
k 6=j (ω − yk)y′j∏N+1
k=1 (ω − yk)
=
N+1∑
j=1
k
∏N+1
k 6=j (ω − yk)y′j∏N+1
k=1 (ω − yk)
=
= −k
d
dx
∏N+1
k 6=j (ω − yk)∏N
k=1(ω − yk)
= −k d
dx
π
N+1∏
k=1
(ω − yk), (4.30)
where we have put, as before,
ρω(yj)(ω − yj) = k, (4.31)
for all j = 1, . . . , N + 1, ω ∈ C+, and some parameters k ∈ C. This clearly means that
ρω(y) =
k
ω − y
(4.32)
for any y ∈ R and ω ∈ C+.
Upon substituting the expression (4.29) into (4.30), one readily finds that
T̂ϕρω(x) =
k
ϕ(ω)− x
= ρϕ(ω)(x), (4.33)
for all x ∈ R and any ω ∈ C+. Thus, the invariant quasi-measure for the discrete dynamical
system (4.1) is given by the same expression (4.25) when ω̄ ∈ C+ is a fixed point of the mapping
ϕ : C+ → C+. This means that
αω̄ + a−
N∑
j=1
βj
ω̄ − bj
= ω̄, (4.34)
or, equivalently,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
54 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
αω̄
N∏
j=1
(ω̄ − bj) + a
N∏
j=1
(ω̄ − bj)−
N∑
j=1
βj
N∏
k 6=j
(ω̄ − bk) = ω̄
N∏
j=1
(ω̄ − bj), (4.35)
for some ω̄ ∈ C+. Assume now that α 6= 1; then it is easy to see that the algebraic equation (4.35)
possesses exactly N + 1 ∈ Z+ roots, which can be used to constructing the invariant quasi-measure
(4.25). When α = 1, the condition becomes
a
N∏
j=1
(ω̄ − bj) =
N∑
j=1
βj
N∏
k 6=j
(ω̄ − bk), (4.36)
which always possesses roots for arbitrary a ∈ R if N ≥ 2. This leads directly to the following
characterization for N ≥ 2:
Theorem 4.3. The expression (4.25) for some k ∈ C determines, in general, the infinitesimal
invariant quasi-measure for the generalized Boole transformation (4.1) for all N ≥ 2 with arbitrary
parameters a, bj ∈ R and α, βj ∈ R+, 1 ≤ j ≤ N + 1.
It is an important now to find in the set of invariant quasi-measures (4.25) that we obtained, those
that are positive and ergodic with respect to the transformation (4.1) for N ≥ 2. For positivity, the
determining equation (4.35) must possess at least one pair of complex conjugate roots. A thorough
analysis of the roots of equation (4.35) leads to the following result, which is analogous to that proved
in [4].
Theorem 4.4. The generalized Boole transformation (4.1) for anyN ≥ 1 is necessarily ergodic
with respect to the measure (4.25) for some ω̄ ∈ C+\R and k ∈ C iff α = 1 and a = 0. If α = 1 and
a 6= 0, the transformation (4.1) is not ergodic since it is totally dissipative, that is the wandering set
D(ϕ) :=
⋃
Wϕ = R, where Wϕ ⊂ R are such subsets such that ϕ−n(Wϕ), n ∈ Z, are disjoint.
Proof (sketch). It is easy to see that for N ≥ 2, α = 1 and a = 0 the determining algebraic
equation (4.35) always possesses exactly N − 1 real roots ω̄j ∈ R, j = 1, . . . , N − 1. Therefore, the
invariant quasi-measure expression (4.25) is degenerate for all of the ω̄j ∈ R, which leads directly to
the conclusion that the corresponding invariant measure dµ(x) = dx, x ∈ R, is the standard Lebesgue
measure on R. Its ergodicity with respect that transformation (4.1) then follows from the fact that the
corresponding dissipative set D(ϕ) = ∅ and the unique invariant set subalgebra I(ϕ) = {∅,R}.
Results similar to those above can also be obtained for the most generalized Boole-type transfor-
mation
R 3 y → ϕ(y) := αy + a+
∫
R
dν(s)
s− y
∈ R, (4.37)
where a ∈ R, α ∈ R+ and the measure ν on R has the compact support supp ν ⊂ R such that the
natural conditions [1, 14] ∫
R
dν(s)
1 + s2
= a,
∫
R
dν(s) <∞, (4.38)
hold. Concerning the extension of the transformation (4.37) on the upper part C+ of the complex
plane C so that that Imϕ(ω) ≥ 0 for all ω ∈ C+, the following representation:
ϕ(ω) = αω + a+
∫
R
1 + sω
s− ω
dσ(s), (4.39)
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INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 55
holds [1, 5], where the measure dσ on R is closely related to the measure dν.
The general properties of the mapping (4.39) were in part studied in [4] in the framework of the
theory of inner functions. The invariant measures corresponding to (4.37) and their ergodic properties
can be also treated effectively by making use of the analytical and spectral properties of the associated
Frobenius – Perron transfer operator (4.4).
5. Two-dimensional generalizations of the Boole transformation. Consider the two-dimensional
Boole-type transformations ϕ2, ψσ(2) : R2\{0, 0} → R2
ϕ2(x, y) := (x− 1/x, y − 1/y) (5.1)
and
ψσ(2) (x, y) := (x− 1/y, y − 1/x). (5.2)
It is easy to see that the infinitesimal (product) measure
dµ(x, y) := dxdy, (5.3)
is invariant with respect to the first mapping (5.1) since it is the product of two measures, each
of which is invariant with respect to the corresponding classical Boole transformation. Therefore,
the generalized Boole-type transformation (5.1) is ergodic too. In the case of the generalized two-
dimensional transformation (5.2), the invariance property of the measure (5.3) is a direct consequence
of the following result.
Lemma 5.1. The mapping (5.2) satisfies the infinitesimal invariance property
µ(ψ−1
σ(2)([u, u+ du]× [v, v + dv]) = dudv = µ([u, u+ du]× [v, v + dv])
with respect to the product measure defined in (5.3) for all infinitesimal rectangles in the image
of ψσ(2).
Proof. Representing the map (5.2) in the form
ψσ(2) (x, y) := (u, v) = (x− 1/y, y − 1/x),
and “inverting”, we obtain the solutions
x± = x±(u, v) := (1/2)
[
u±
√
u2 + (4u/v)
]
:= (1/2) [u±X(u, v)] ,
y± = y±(u, v) := v + (1/x±(u, v)) = (1/2)
[
v ±
√
v2 + (4v/u)
]
:= (1/2) [v ± Y (u, v)] ,
(5.4)
where mappingsX : R2 → R and Y : R2 → R have the obvious definitionsX(u, v) =
√
u2 + (4u/v)
and Y (u, v) =
√
v2 + (4v/u), respectively. In order to make effective use of these formulas, it is
convenient to define the mappings ψ̂+, ψ̂− : R2 → R2 as
ψ̂+(u, v) := (x+(u, v), y+(u, v)) , ψ̂−(u, v) := (x−(u, v), y−(u, v)) (5.5)
for (u, v) ∈ R2. It follows directly from (5.4) that for all points (u, v) ∈ R2 in the image of
ψσ(2) : R2 → R2, except for those of a subset of (product) measure zero, the preimage ψ−1
σ(2) (u, v}) ⊂
⊂ R2 is comprised of two points in the domain of the map. For such points, if we choose an
infinitesimal rectangle dR := [u, u+ du]× [v, v + dv] (sufficiently small by definition), the union
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 1
56 D. BLACKMORE, J. GOLENIA, A. K. PRYKARPATSKY, YA. A. PRYKARPATSKY
ψ−1
σ(2) (dR) = ψ̂+ (dR) t ψ̂− (dR)
is disjoint, which implies that
µ
(
ψ−1
σ(2) (dR)
)
= µ
(
ψ̂+ (dR)
)
+ µ
(
ψ̂− (dR)
)
,
which is owing to (5.5) infinitesimally equivalent to the following equation involving the product
measure:
µ
(
ψ−1
σ(2) (dR)
)
= dx+dy+ + dx−dy−.
It is useful to observe from (5.4) that we have the following algebraic relationships:
x+ + x− = u, y+ + y− = v, x+/y+ = u/v,
x+x− = u/v, y+y− = v/u, x−/y− = u/v.
Next, we perform an infinitesimal measure computation to verify the invariance making extensive use
of the fact that for any real-valued Lebesgue measurable function ϕ : R2 → R, d(ϕdϕ) = dϕdϕ = 0:
µ
(
ψ−1
σ(2) (dR)
)
= dx+dy+ + dx−dy− = d (x+dy+ + x−dy−) =
= d [(u/vy+) dy+ + (u/vy−) dy−] = (1/2) d (u/v) d
[
y2
+ + y2
−
]
= (1/2)d (u/v) d
(
v2 − 2v/u
)
=
= d (u/v) [vdv − d(v/u)] =
(
du/v − udv/v2
)
vdv − d(u/v)d(v/u) =
= dudv − (u/v)dvdv + (v2/u2)d(u/v)d(u/v) = dudv.
Lemma 5.1 is proved.
Consequently, it follows by a simple modification of the proof of the main theorem in [6] that
dµ(x, y), (x, y) ∈ R2, is the unique absolutely continuous invariant measure for the Boole-type
transformation (5.2). This, in particular, implies that the map (5.2) is ergodic with respect to the
infinitesimal measure dµ(x, y), (x, y) ∈ R2, and so we have the following result.
Proposition 5.1. The generalized two-dimensional Boole-type transformations (5.1) and (5.2)
are ergodic with respect to the standard infinitesimal measure dµ(x, y) = dxdy for (x, y) ∈ R2. In
particular, the following equalities:∫
R2
f(ϕ2(x, y))dxdy =
∫
R2
f(x, y)dxdy =
∫
R2
f(ψσ(2) (x, y))dxdy
hold for any integrable function f ∈ L1(R2;R).
The above result strongly suggests the validity of the following conjecture.
Conjecture 5.1. Let σ ∈ Σn be any element (permutation) of the symmetric group Σn, n ∈ Z+.
Then the following generalized Boole-type transformation ψσ : Rn\{0, 0, . . . , 0} → Rn, where
ψσ (x1, x2, . . . , xn) := (x1 − 1/xσ(1), x2 − 1/xσ(2), x3 − 1/xσ(3), . . . , xn − 1/xσ(n)),
is ergodic with respect to the standard infinitesimal measure dµ(x1, x2, . . . , xn) := dx1dx2 . . . dxn,
(x1, x2, . . . , xn) ∈ Rn.
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INVARIANT MEASURES FOR DISCRETE DYNAMICAL SYSTEMS AND ERGODIC PROPERTIES . . . 57
6. Acknowledgements. D. Blackmore acknowledges the National Science Foundation (Grant
CMMI-1029809), A. Prykarpatsky and Ya. Prykarpatsky acknowledge the Scientific and Technolog-
ical Research Council of Turkey (TUBITAK/NASU-111T558 Project) for a partial support of their
research.
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Received 27.09.12
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|
| id | umjimathkievua-article-2403 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:45Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/49/250a123c972a0d698cd3f8fbcb870249.pdf |
| spelling | umjimathkievua-article-24032020-03-18T19:14:46Z Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations Інварiантнi мiри для дискретних динамiчних систем та ергодичнi властивостi узагальнених перетворень булевoго типу Blackmore, D. Golenia, J. Prykarpatsky, A. K. Prykarpatsky, Ya. A. Блекмор, Д. Голеня, Й. Прикарпатський, А. К. Прикарпатський, Я. А. Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed. Вивчаються ергодичнi мiри для узагальнених перетворень булевoго типу iз використанням пiдходу твiрних функцiй iнварiантних квазiмiр, що базується на спецiальних розв’язках для оператора Фробенiуса – Перрона. Запропоновано новi двовимiрнi перетворення булевoго типу та дослiджено їхнi iнварiантнi мiри та ергодичнi властивостi. Institute of Mathematics, NAS of Ukraine 2013-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2403 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 1 (2013); 44-57 Український математичний журнал; Том 65 № 1 (2013); 44-57 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2403/1573 https://umj.imath.kiev.ua/index.php/umj/article/view/2403/1574 Copyright (c) 2013 Blackmore D.; Golenia J.; Prykarpatsky A. K.; Prykarpatsky Ya. A. |
| spellingShingle | Blackmore, D. Golenia, J. Prykarpatsky, A. K. Prykarpatsky, Ya. A. Блекмор, Д. Голеня, Й. Прикарпатський, А. К. Прикарпатський, Я. А. Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title | Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title_alt | Інварiантнi мiри для дискретних динамiчних систем та ергодичнi властивостi узагальнених перетворень булевoго типу |
| title_full | Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title_fullStr | Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title_full_unstemmed | Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title_short | Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations |
| title_sort | invariant measures for discrete dynamical systems and ergodic properties of generalized boole-type transformations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2403 |
| work_keys_str_mv | AT blackmored invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT goleniaj invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT prykarpatskyak invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT prykarpatskyyaa invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT blekmord invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT golenâj invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT prikarpatsʹkijak invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT prikarpatsʹkijâa invariantmeasuresfordiscretedynamicalsystemsandergodicpropertiesofgeneralizedbooletypetransformations AT blackmored ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT goleniaj ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT prykarpatskyak ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT prykarpatskyyaa ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT blekmord ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT golenâj ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT prikarpatsʹkijak ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu AT prikarpatsʹkijâa ínvariantnimiridlâdiskretnihdinamičnihsistemtaergodičnivlastivostiuzagalʹnenihperetvorenʹbulevogotipu |