Group of continuous transformations of real interval preserving tails of G₂-representation of numbers
In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs g₀ < 1 and g₁ = g₀ − 1. Transformations (bijections of the set to itself) of interval [0, g₀] preserving tails of this representation of numbers are studied. We prove constructively th...
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irk-123456789-1885052023-03-04T01:27:15Z Group of continuous transformations of real interval preserving tails of G₂-representation of numbers Pratsiovytyi, M.V. Lysenko, I.M. Maslova, Yu.P. In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs g₀ < 1 and g₁ = g₀ − 1. Transformations (bijections of the set to itself) of interval [0, g₀] preserving tails of this representation of numbers are studied. We prove constructively that the set of all continuous transformations from this class with respect to composition of functions forms an infinite non-abelian group such that increasing transformations form its proper subgroup. This group is a proper subgroup of the group of transformations preserving frequencies of digits of representations of numbers. 2020 Article Group of continuous transformations of real interval preserving tails of G₂-representation of numbers / M.V. Pratsiovytyi, I.M. Lysenko, Yu.P. Maslova // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 99–108. — Бібліогр.: 10 назв. — англ. 1726-3255 DOI:10.12958/adm1498 2010 MSC: 11H71, 26A46, 93B17 http://dspace.nbuv.gov.ua/handle/123456789/188505 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs g₀ < 1 and g₁ = g₀ − 1. Transformations (bijections of the set to itself) of interval [0, g₀] preserving tails of this representation of numbers are studied. We prove constructively that the set of all continuous transformations from this class with respect to composition of functions forms an infinite non-abelian group such that increasing transformations form its proper subgroup. This group is a proper subgroup of the group of transformations preserving frequencies of digits of representations of numbers. |
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Pratsiovytyi, M.V. Lysenko, I.M. Maslova, Yu.P. |
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Pratsiovytyi, M.V. Lysenko, I.M. Maslova, Yu.P. Group of continuous transformations of real interval preserving tails of G₂-representation of numbers Algebra and Discrete Mathematics |
| author_facet |
Pratsiovytyi, M.V. Lysenko, I.M. Maslova, Yu.P. |
| author_sort |
Pratsiovytyi, M.V. |
| title |
Group of continuous transformations of real interval preserving tails of G₂-representation of numbers |
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Group of continuous transformations of real interval preserving tails of G₂-representation of numbers |
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Group of continuous transformations of real interval preserving tails of G₂-representation of numbers |
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Group of continuous transformations of real interval preserving tails of G₂-representation of numbers |
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Group of continuous transformations of real interval preserving tails of G₂-representation of numbers |
| title_sort |
group of continuous transformations of real interval preserving tails of g₂-representation of numbers |
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Інститут прикладної математики і механіки НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188505 |
| citation_txt |
Group of continuous transformations of real interval preserving tails of G₂-representation of numbers / M.V. Pratsiovytyi, I.M. Lysenko, Yu.P. Maslova // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 99–108. — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT pratsiovytyimv groupofcontinuoustransformationsofrealintervalpreservingtailsofg2representationofnumbers AT lysenkoim groupofcontinuoustransformationsofrealintervalpreservingtailsofg2representationofnumbers AT maslovayup groupofcontinuoustransformationsofrealintervalpreservingtailsofg2representationofnumbers |
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2025-07-16T10:35:47Z |
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| fulltext |
“adm-n1” — 2020/5/14 — 19:35 — page 99 — #107
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 1, pp. 99–108
DOI:10.12958/adm1498
Group of continuous transformations of real
interval preserving tails of G2-representation of
numbers
M. V. Pratsiovytyi, I. M. Lysenko, and Yu. P. Maslova
Communicated by A. P. Petravchuk
Dedicated to the 70th anniversary
of Leonid Andriyovych Kurdachenko
Abstract. In the paper, we consider a two-symbol system
of encoding for real numbers with two bases having different signs
g0 < 1 and g1 = g0 − 1. Transformations (bijections of the set to
itself) of interval [0, g0] preserving tails of this representation of
numbers are studied. We prove constructively that the set of all
continuous transformations from this class with respect to compo-
sition of functions forms an infinite non-abelian group such that
increasing transformations form its proper subgroup. This group
is a proper subgroup of the group of transformations preserving
frequencies of digits of representations of numbers.
Introduction
Many two-symbol systems of encoding (representation) of fractional
part of real numbers are known. They use the alphabet A = {0, 1} and
are based on expansions of numbers in series, infinite products, continued
fractions, etc. These systems identify a number as a sequence of zeros and
2010 MSC: 11H71, 26A46, 93B17.
Key words and phrases: two-symbol system of encoding for real numbers
with two bases having different signs (G2-representation), tail of representation of
number, continuous transformation of interval, left and right shift operators, continuous
transformation preserving tails of representations.
https://doi.org/10.12958/adm1498
“adm-n1” — 2020/5/14 — 19:35 — page 100 — #108
100 Group of transformations preserving tails
ones, i.e., as element of space L = A×A×. . . of sequences of zeros and ones.
Every such a system has some advantages and restrictednesses as well
as certain conveniences for solving some problems of number-theoretical,
topological-metric and probabilistic kind. A system with two positive
bases (q0 ∈ (0, 1), q1 = 1− q0) is among them. It generalizes classic binary
system, has a self-similar geometry and zero redundancy. This is a system
of Q2-representation of numbers based on representation of a number by
the series
x = α1q1−α1
+
∞∑
k=2
(αkq1−αk
k−1∏
j=1
qαj
) ≡ ∆Q2
α1α2...αk...
.
For q0 = 0,5, it is a classic binary system.
In expression
∏k
j=1 qαj
= q
N0(x,k)
1 q
N1(x,k)
1 , where N1(x, k) = α1 + α2 +
. . .+αk and N0(x, k) = k−N1(x, k), one can see that this is a system with
two bases. This system of encoding of numbers has various applications in
metric and probabilistic theory of numbers, function theory and measure
theory, fractal analysis and fractal geometry. For this system, the left shift
operator is a discontinuous function, linear and increasing on cylinders of
rank 1: ∆Q2
0 = [0, q0] and ∆Q2
1 = [q0, 1].
In the papers [7], [8], an analogue of Q2-representation of numbers,
namely, system of encoding of numbers in interval [0, g0] with bases g0 ∈
(0, 1) and g1 = g0 − 1 having different signs is introduced. It is based on
expansion of a number in the series
x = α1g1−α1
+
∞∑
k=2
(αkg1−αk
k−1∏
j=1
gαj
) ≡ ∆G2
α1α2...αk...
. (1)
This system also has a zero redundancy (any number has at most two rep-
resentations, and set of numbers having two representations is countable).
Generally speaking, series (1) has positive as well as negative terms.
So G2-representation is not topologically equivalent to Q2-representation
and it is not a simple reencoding of Q2-representation. At the same time
metric theories of these representations are similar. A vivid peculiarity of
G2-representation is the fact that left shift operator of G2-representation
is a continuous function on a whole interval [0, g0]. This is a fundamental
difference between this system and previously studied systems.
Real numbers in interval [0, g0] having two G2-representations are
called G2-binary numbers, and numbers having a unique representation
“adm-n1” — 2020/5/14 — 19:35 — page 101 — #109
M. V. Pratsiovytyi, I . M. Lysenko, Yu. P. Maslova 101
are called G2-unary numbers. It is known [10] that the set B of all G2-
binary numbers is countable and consists of numbers with the following
G2-representation: ∆G2
c1...cm01(0) = ∆G2
c1...cm11(0).
Algorithm for comparison of numbers in terms of their G2-representa-
tions is the following.
Theorem 1 ([10]). Numbers ∆G2
c1c2...cm1d1d2...
= x1 6= x2 = ∆G2
c1c2...cm0d′
1
d′
2
...
are in the relation
x1 > x2 if σm ≡ c1 + c2 + . . .+ cm = 2k,
x1 6 x2 if σm ≡ c1 + c2 + . . .+ cm = 2k − 1.
In this paper, we consider continuous transformations of interval [0, g0],
i.e., bijections of interval to itself, preserving “tails of G2-represen-
tation of numbers”, namely, we study group properties of the family
of these transformations. Groups of transformations preserving tails of
different representations of numbers were studied in the papers [2], [3], [4].
Specific properties of G2-representation generate a peculiar corresponding
transformation group having a non-trivial subgroup of increasing functions
unlike other systems.
1. Tails and tail sets
Definition 1. We say that G2-representations of numbers x = ∆G2
α1...αn...
and y = ∆G2
β1β2...βn...
have the same tail if there exist positive integers k
and m such that
αk+j = βm+j (2)
for any j ∈ N . We denote it symbolically by x ∼ y.
Definition 2. Suppose that k and m are the smallest numbers satisfying
condition (2). Then G2-representation and corresponding number
z ≡ [x ∧ y] = ∆G2
αk+1αk+2...
= ∆G2
βm+1βm+2...
is called a common tail of representations of numbers x and y.
It is evident that binary relation ∼ (“has the same tail”) is an equiva-
lence relation, i.e., it is reflexive, symmetric and transitive. Thus it provides
a partition of the set Z of all G2-representations of numbers in interval
[0, g0] into the equivalence classes, and they form together a quotient set
W = Z� ∼. Any element of the set W is called a tail set, this set is
uniquely determined by an arbitrary its element.
“adm-n1” — 2020/5/14 — 19:35 — page 102 — #110
102 Group of transformations preserving tails
Theorem 2. Any tail set is a countable everywhere dense in interval
[0, g0] set. The set W of all tail sets is a continuum set.
Proof. Suppose Kx is a tail set containing the number x, K1 is a set of
all numbers y such that [y ∧ x] = x, and Kn is a set of all numbers y such
that [y ∧∆G2
αn+1(x)αn+2(x)...
] = ∆G2
αn+1(x)αn+2(x)...
.
It is evident that every set Kn (n ∈ N) is countable and Kx =
⋃
n
Kn.
Thus Kx is a countable set because it is a countable union of countable
sets.
An arbitrary cylinder ∆G2
c1...cn
contains points of tail set Kx because
points ∆G2
c1...cnα1(x)α2(x)...
, ∆G2
c1...cnα2(x)α3(x)...
, and ∆G2
c1...cnαk(x)αk+1(x)...
be-
long to this cylinder. This proves that the set Kx is everywhere dense in
interval [0, g0].
The set W is a continuum set. Indeed, suppose that it is a countable
set. Then we see that interval [0, g0] is a countable set because it is
a countable union of countable sets. This contradicts the fact that interval
is a continuum set.
Note that all G2-binary numbers belong to the same tail set.
2. Left and right shift operators of G2-representation of
numbers
Theorem 3 ([8]). Left shift operator ω of G2-representation of numbers
in interval [0, g0] defined by equality
ω(∆G2
α1α2...αn...
) = ∆G2
α2α3...αn...
in space of G2-representations, is analytically expressed in the form
ω(x) =
1
gα1(x)
x−
δα1(x)
gα1(x)
, is a continuous well-defined function on [0, g0],
is linear on every cylinder of rank 1, is increasing on ∆0 and decreasing
on ∆1.
Numbers
0 = ∆G2
(0) and ∆G2
(1) = g0 + g0g1 + g0g
2
1 + ... =
g0
2 + g0
are invariant points of left shift operator ω.
Remark 1. The last theorem shows an essential difference between G2-
representation and other known two-symbol representations, in particular,
Q∗
2-representation, Q̃-representation [9] and A2-continued fractions [3].
“adm-n1” — 2020/5/14 — 19:35 — page 103 — #111
M. V. Pratsiovytyi, I . M. Lysenko, Yu. P. Maslova 103
Let n be a positive integer that is greater than 1. Put
ωn(x) = ω(ωn−1(x)) = ∆G2
αn+1(x)αn+2(x)...
.
Since
x = δα1(x) +
n∑
k=2
δαk(x)
k−1∏
j=1
gαj(x)
+
n∏
j=1
gαj(x)
ωn(x),
we have
ωn(x) =
x
Pn(x)
−
1
Pn(x)
δα1(x) +
n∑
k=2
δαk(x)
k−1∏
j=1
gαj(x)
.
Theorem 4 ([10]). Function ωn is well defined by equality
ωn(x) = ωn(∆G2
α1(x)α2(x)...αn(x)...
) ≡ ∆G2
αn+1(x)αn+2(x)...
,
is analytically expressed in the form ωn(x) =
1
Pn
x −
Bn
Pn
, where Pn =
∏n
j=1 gαj(x),
Bn = δα1(x) +
n∑
k=2
(
δαk(x)
∏k−1
j=1 gαj(x)
)
, is continuous on interval [0, g0]
and linear on every cylinder of rank n.
Remark 2. In terms of dynamical systems, relation “has the same tail”
can be defined in the following form:
x = ∆G2
α1α2...αn
∼ y = ∆G2
β1β2...βn
⇔ Ox ∩Oy 6= ∅,
where Ou = {u, ω1(u), ω2(u), . . .} is an orbit of point u under mapping ω,
and [x ∧ y] = Ox ∩Oy.
Definition 3. Function τi defined on [0, g0] by equality
τi(x) = τi
(
∆G2
α1(x)α2(x)...αn(x)...
)
= ∆G2
iα1(x)α2(x)...αn(x)...
,
where i ∈ {0, 1}, is called a right shift operator of G2-representation of
numbers with parameter i (in the sequel, we just say “right shift operator”).
Since τi
(
∆G2
c1c2...cm01(0)
)
= τi
(
∆G2
c1c2...cm11(0)
)
, we see that function τi
is well defined. It is evident that the set of values of function τi is cylinder
∆G2
i . In particular,
τ0(0) = τ
(
∆G2
(0)
)
= 0, τ1(0) = τ1
(
∆G2
(0)
)
= ∆G2
1(0) = g0.
“adm-n1” — 2020/5/14 — 19:35 — page 104 — #112
104 Group of transformations preserving tails
Lemma 1. Function τi is continuous at any point of interval [0, g0] and
analytically expressed in the form τi(x) = δi + gix.
Indeed, since
τi
(
∆G2
α1α2...αn...
)
= ∆G2
iα1α2...αn...
= δi + gi∆
G2
α1α2...αn...
,
we have τi(x) = δi + gix, i.e., τ0(x) = g0x, τ1(x) = g0 + g1x. So, it is
evident that function τi is continuous on cylinders of rank 1. From equality
τi
(
∆G2
c1c2...cm01(0)
)
= τi
(
∆G2
c1c2...cm11(0)
)
it follows that the function is
continuous at G2-binary points that are endpoints of cylinders.
Corollary 1. Function τ0 is increasing and function τ1 is decreasing.
Moreover, τ0(g0) = τ1(g0).
The following equalities are evident: ω(δi(x)) = x and τα1(x)(ω(x)) =
x.
Equation τi(x) = ω(x) has two solutions: x = ∆G2
(ji), where j ∈ {0, 1}.
Equation τi(x) = ωm(x) has 2m solutions: x = ∆G2
(j1,j2,...,jmi), where jk ∈
{0, 1}, k = 1,m.
Let (i1, i2, . . . , in) be a tuple of zeros and ones. Function τi1i2...in defined
by equality τi1i2...in(x) = ∆G2
i1i2...inα1(x)α2(x)...
is called a right shift operator
with parameters (i1, i2, . . . , in). By induction, from equality τi1i2...in(x) =
τi1(τi2...in(x)) it follows that operator τi1i2...in(x) is well defined.
Operator τi1i2...in is analytically expressed in the form
τi1i2...in(x) = δi1 +
n∑
k=1
δik
k−1∏
j=1
gij
+
n∏
j=1
gij
x,
is a linear function, is increasing if Pn =
∏n
j=1 gij > 0 (this is equivalent
to i1 + i2 + . . . + in is even number) and decreasing if Pn < 0 (this is
equivalent to i1 + i2 + . . .+ in is odd number).
For example, consider n = 2 and corresponding functions τ00, τ01, τ10,
τ11. Functions τ00 = g20x and τ11 = g20x + g20 are linear increasing, but
functions τ01 = g0g1x+ g20 and τ10 = g0g1x+ g0 are linear decreasing.
3. Continuous functions and transformations of interval
[0, g0] preserving tails of G2-representation of numbers
We say that function y = f(x) preserves tails of G2-representation of
numbers in interval [0, g0] (or is a tail function) if any number x ∈ [0, g0]
and its image y = f(x) have the same tail.
“adm-n1” — 2020/5/14 — 19:35 — page 105 — #113
M. V. Pratsiovytyi, I . M. Lysenko, Yu. P. Maslova 105
Left and right shift operators ωn, τi1i2...in for any positive integer n
and for any tuple (i1, i2, . . . , in) of zeros and ones are simple examples
of continuous functions preserving tails of G2-representation of numbers.
Various “joinings” of these functions are the same. For example, function
f(x) =
ω(x) if 0 6 x 6 x1,
τ1(x) if x1 6 x 6 x2,
ω(x) if x2 6 x 6 g0,
where x1 and x2 are solutions of equation ω(x) = τ1(x), i.e., x1 = ∆G2
(01),
x2 = ∆G2
(1), preserves tails of G2-representation.
But not every continuous function defined on interval [0, g0] is its
transformation, i.e., bijection of the interval to itself. It is clear that above
mentioned functions are not transformations.
It is clear that continuous transformations of interval [0, g0] can be
only strictly monotonic (increasing and decreasing) functions such that
their domain and set of values coincide with this interval.
Lemma 2. Decreasing function
f1(x) =
τ1(x) if x 6 x1 = ∆G2
(1) =
g0
2− g0
,
ω(x) if x > x1 = ∆G2
(1) =
g0
2− g0
,
is a continuous tail transformation of interval [0, g0].
Proof. Number x1 is a solution of equation τ1(x) = ω(x) being equivalent
to system of equations 1 = α2(x) = α4(x) = . . . , α1(x) = α3(x) =
α5(x) = . . .. So this equation has two solutions: x = ∆G2
(α11)
, α1 ∈ {0, 1}.
First solution x0 = ∆G2
(01) belongs to interval of decrease of function τ1 and
to interval of increase of function ω, and second solution x1 belongs to
interval of decrease of function τ1 and to interval of decrease of function ω.
Thus f1 is a continuous and strictly decreasing function. Moreover, f1(0) =
g0 and f1(g0) = 0. Hence f1 is a continuous transformation of interval
[0, g0].
Example 1. Function
f2(x) =
{
τ1(x) if 0 6 x 6 ∆G2
(101),
ω2(x) if ∆G2
(101) 6 x 6 g0,
is a continuous decreasing tail transformation.
“adm-n1” — 2020/5/14 — 19:35 — page 106 — #114
106 Group of transformations preserving tails
Indeed, τ1 is a continuous decreasing tail function and ∆G2
(101) is a so-
lution of equation τ1(x) = ω2(x) belonging to the last interval of decrease
of function ω2(x).
Example 2. Decreasing function
f3(x) =
τ10 . . . 0
︸ ︷︷ ︸
k
(x) if 0 6 x 6 xk ≡ ∆G2
i10 . . . 0
︸ ︷︷ ︸
k
,
ω(x) if xk 6 x 6 g0, i ∈ A,
is a continuous tail transformation of interval [0, g0].
Indeed, ω and τ10...0 are continuous tail functions, number xk is a so-
lution of equation τ10 . . . 0
︸ ︷︷ ︸
k
(x) = ω(x) belonging to intervals of decrease
of both functions. Hence f3 is a continuous decreasing tail function.
Theorem 5. The set C of all continuous bijections of interval [0, g0]
preserving tails of G2-representation of numbers with respect to composition
(superposition) ◦ forms an infinite non-abelian group such that increasing
functions form its non-trivial subgroup.
Proof. It is known that the set of all bijections of interval forms a group
such that an identity transformation is its neutral element and an inverse
transformation is its symmetric element. It is evident that composition
of tail transformations is a tail transformation. The same is true for
inverse transformation. Thus, by the subgroup test, (C, ◦) is a group.
From example 3 (where k is an arbitrary positive integer) it follows that
this group is infinite.
To prove that group (C, ◦) is non-abelian it is enough to provide two
transformations in the set C that are not commute. To this end we consider
function f1 and f2 from examples 1 and 2 and number x0 that is less than
x1. Then we have
f2(f1(∆
G2
01(0))) = f2(τ1(∆
G2
01(0))) = f2(∆
G2
101(0))
= τ1(∆
G2
101(0)) = ∆G2
1101(0)
because of ∆G2
101(0) < ∆G2
(101);
f1(f2(∆
G2
01(0))) = f1(τ1(∆
G2
01(0))) = f1(∆
G2
101(0))
= ω(∆G2
101(0)) = ∆G2
01(0),
“adm-n1” — 2020/5/14 — 19:35 — page 107 — #115
M. V. Pratsiovytyi, I . M. Lysenko, Yu. P. Maslova 107
because of ∆G2
101(0) > ∆G2
(1). Hence f2(f1(x0)) 6= f1(f2(x0)).
To prove that there exists a non-trivial subgroup of increasing functions
it is enough to give an example of non-trivial increasing bijection f ∈ C.
A such function is the following:
f5(x) =
ω(x) if 0 6 x 6 ∆G2
(011),
τ11(x) if ∆G2
(011) 6 x 6 ∆G2
(1),
x if ∆G2
(1) 6 x 6 g0,
because functions ω, τ11 and f(x) = x are increasing on given intervals,
∆G2
(011) is a solution of equation ω(x) = τ11(x), and ∆G2
(1) is a solution of
equation τ11(x) = x.
Remark 3. The group (C, ◦) is a proper subgroup of the group of transfor-
mations of interval [0, g0] preserving frequencies of digits of representation.
References
[1] M. Iosifescu, C. Kraaikamp, Metric properties of Denjoy’s canonical continued
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Contact information
Mykola
V. Pratsiovytyi,
Iryna M. Lysenko,
Yuliya P. Maslova
Institute of Mathematics,
National Academy of Sciences of Ukraine,
Tereschenkivska str. 3,
Institute of Physics and Mathematics,
National Pedagogical Mykhailo Drahomanov
University, 9 Pyrohova St., Kyiv, 01601,
Ukraine
E-Mail(s): prats4444@gmail.com,
iryna.pratsiovyta@gmail.com,
julia0609mas@gmail.com
Received by the editors: 21.11.2019
and in final form 11.01.2020.
M. V. Pratsiovytyi, I. M. Lysenko, Yu. P. Maslova
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