Bernoulli Algebra on Common Fractions and Generalized Oscillations

Bernoulli algebra on set of proper common fractions with fixed denominator has been introduced and investigated. This algebra is one of most important components of a discrete dynamical system called logistic bipendulum.


Introduction
A dynamical system was introduced and investigated in (Kozlov, Buslaev, & Tatashev, 2015).This system is called a bipendulum.The bipendulum consists of two cells V 0 and V 1 .A channel connects the cells.There are two particles P 0 and P 1 .Each particle is in one of the cell at every instant.Two numbers a (0) and a (1) are given.These numbers belong to the segment [0, 1].The binary representation of the number a (i)   a (i) = 0.a (i)  1 a (i) 2 . . ., i = 0, 1, is called the plan of the particle P i , i = 0, 1.If there are no delays, then the particle P i is located in the cell a (i)  T at the instant T, i = 0, 1; T = 1, 2, . . .The plan implementation can be delayed.If one of the particles is located in the cell V 0 and, according to its plan, this particle has to come to the cell V 1 at next instant, and the other particle is in the cell V 1 and plans to come to the cell V 0 , then only one particle moves in accordance with a given rule.The choice is realized in accordance with priorities (a dynamical system) or equiprobably (a Markov process).The average (in a certain sense) number of delays is interesting for investigations.Suppose that the plans of particles are periodical fractions a (i) = 0.a (i)  1 . . .a (i)  l (a l+1 , . . ., a l+m ), i.e. proper common fractions.Shift onto a position to the left is equivalent to applying of the Bernoulli operation to a proper fraction.If we apply the Bernoulli shift to a fraction, we multiply the fraction by 2 and exclude the integer part.An algebra on proper fractions k/N, k = 0, 1, . . ., N − 1, with respect to the Bernoulli shift has been introduced in (Kozlov, Buslaev, & Tatashev, 2015), and is called a Bernoulli algebra.The set of the algebra elements can be divided into disjoint subalgebras.It is found in (Kozlov, Buslaev, & Tatashev, 2015) that, if the plans are determined by elements of the same subalgebra, then the system comes to the state of synergy, i.e., there are no delays in the present instant and in the future.If plans of particles are given by elements of different orbits, then the tape velocities depend on orbits containing these elements.Bernoulli algebras are investigated in this paper.
The aim of this paper is to represent set of proper common fractions as Bernoulli algebras.These algebras are sets of logistic plans of bipendulums which was introduced earlier and are discrete dynamical systems with two positions and two pendulums.We discuss connection with Markov chains, graph theory, binary positional representational of numbers, and theory of functions.We have proved theorems about Bernoulli algebras.

Algebra of a Markov Process
In this section, we introduce a classification of elements of an algebra.This classification is similar to a known classification of Markov chain states.
Suppose algebra G is a set of elements together with a unary operation ω.We shall give definitions that allow to classify elements of the algebra.This classification is similar to the classification of Markov chain states (Gantmacher, 2004), (Borovkov, 1986).Suppose x ∈ G.The sequence T (x) = ω(x), ω(ω(x)), . . . is called the trajectory of an element x.We write x → y if y ∈ T (x).Elements x and y, x y are called communicating with each other if y ∈ T (x) and x ∈ T (y), i.e., x → y and y → x.The element x is called inessential if x T (y) for ∀ y ∈ T (x).The other elements are essential.The set of all essential elements can be divided into disjoint sets such that any two elements of the same set communicate with each other, and any two elements of different sets do not communicate with each other.These sets are called classes of communicating essential elements or orbits.An element x is called absorbing if ω(x) = x.If a class of communicating essential elements contains d ≥ 1 elements, then this class is called a periodical class of communicating essential elements or an orbit with period d.

Bernoulli Algebra on Common Fractions
In Section 3, we give definition of Bernoulli algebra.We introduce the concept of connected components of algebras.This concept is similar to concept of connected components in graph theory.
Consider the set of proper common fractions with denominator N with a unary operation, called the Bernoulli shift, (Schuster, 1984), where [2x] is the integral part of 2x.
Suppose W N is the algebra on the set G N with the operation FB, Figure 1.The number i/N (0 ≤ i ≤ N − 1) generates a subalgebra.Denote by W N (i) the subalgebra generated by the element i/N, i = 0, 1, . . ., N − 1.If i/N and j/N communicate with each other, then W N ( j) = W N (i), i, j = 1, 2, . . ., N − 1.If j/N W N (i) and i/N W N ( j), then W N (i) and W N ( j) are disjoint subalgebras.Identifying subalgebras W N (i) and W N ( j) with each other for W N (i) = W N ( j), we divide the algebra W N into k N disjoint subalgebras Each of these subalgebras contains a class of communicating essential elements and a set of inessential elements.This set of inessential elements can be empty.These subalgebras are called connected components of the algebra.We shall introduce the concept of the algebra W N .graph.The graph of the algebra W N is directed graph (Harary, 1969) which contains N vertices v 0 , v 2 , . . ., v N−1 .The graph contains the arc (v i , v j ), where v i is the tail of the arc and v j is the head of the arc if and only if FB(i/N) = j/N.A connected component of the algebra W N graph corresponds to a connected component of this algebra.

Binary Representation and Trajectory of the Element in the Bernoulli Algebra
In Section 4, we consider binary representation and trajectories of elements in Bernoulli algebra.We give geometric interpretation of Bernoulli algebras in the form of graphs.
Consider a common fraction k/N, N ≥ 1, 0 ≤ k < N. The following theorem allows to find the binary representation of this fraction where a i is equal to 0 or 1, i = 1, 2, . . .
Proof.The fractional part of the number 2 i−1 k N is equal to that value if we apply i − 1 times the operation FB(•), i.e., this value equals to b and, therefore, b  We have formulated an approach to find the binary representation of a common fraction.This approach is similar to the approach of converting the decimal representation of a common fraction to the p-ary representation, (Broido & Ilyina, 2006).It is easy to reconstruct the trajectory of the element k/N of the algebra W N if we know the binary representation of the number k/N, Figure 2.

Variation of the Binary Representation
In Section 5, we consider functional representation of plans.We give definition of variation which is one of most important characteristics of function.We also give definition of variation for binary representation of periodical fractions.This definition is equivalent to the general definition.
Suppose the binary representation of the number k/N is The variation of this representation is defined as where the addition, in indexes, is meant modulo m.The limit is called the variation of the binary representation of a real number a ∈ [0, 1) It is easy to give an example of a real number for that the variation of its binary representation is not defined.Suppose Then the variation of the binary representation of the number a is not defined.Some number theory problems, related to binary representation of numbers, are considered in (Uteshev, Cherkasov, Shaposhnikov, 2001).

Markov Chain Interpretation of Algebra W N
In Section 6, we give Markov chain interpretation of Bernoulli algebras.
A Markov chain, (Gantmacher, 2004), (Borovkov, 1986), corresponds to the algebra W N .The chain contains N states S 0 , S 1 , . . ., S N−1 .The state S i corresponds to the element i/N of the algebra W N , i = 0, 1, . . ., N − 1.The behavior of this chain is deterministic and depends only on the initial state.If at the instant t − 1 the system is in the state S i , and then, at the instant t, the chain will be at the state S j , i, j = 0, 1, . . ., N − 1, with probability 1.Therefore, if and the chain is in the state S k , k = 0, 1, . . ., N − 1, at the initial instant t = 0, then, at the instant t, the change will be in the state corresponding to the value b t , t = 1, 2, . . .Denote by p i j the probability of the chain comes from the state S i to the state S j for a step, i, j = 1, . . ., N .Suppose P is the transition matrix of the chain On the one hand, the matrix P is a stochastic matrix, and each element of this matrix is equal to 0 or 1.On the other hand, each row of the matrix P contains exactly one 1.If N is odd, then the matrix P is a permutation matrix.Since it follows that the state S 0 is absorbing, i.e., p 00 = 1.If the state S k is inessential, and the chain comes from this state to the absorbing state S 0 for l steps, then the binary representation of the number k/N contains just l positive digits Values of the variation functions are equal to The transition matrix of the Markov chain of the algebra W 8 have the form 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 The state S 0 is absorbing.The other states are inessential, Figure 3.In Section 8, we prove a theorem which allow to find the form of algebra W 2N if the form of algebra W N is known.
Let us compare algebras W N and W 2N with each other.
Theorem 2. Algebras W N and W 2N contain the same number of connected components and, for any d, the same number of orbits with period d.The algebra W 2N contains a subalgebra isomorphic to the algebra W N .This subalgebra consists of elements 2i/2N, i = 0, 1, . . ., N − 1.The algebra W 2N also contains N inessential elements x = 2i + 1/2N, i = 0, 1, . . ., N − 1, such that FB(x) = y for an element y of the subalgebra isomorphic to algebra W N .
Proof.The subalgebra W 2N , with elements 2i/2N, i = 0, 1, . . ., N − 1, is isomorphic to the algebra W N as the values of the elements 2i/2N of the algebra W 2N and the element i/N of the algebra W N , i = 0, 1, . . ., N − 1, are the same.The elements 2i + 1/2N, i = 0, 1, . . ., N − 1, are inessential as, applying the Bernoulli shift, we obtain an element which belongs to the subalgebra isomorphic to the algebra W N .Theorem 2 has been proved.9. Structure of the Algebra W N for an Odd N In Section 9, we prove theorems about Bernoulli algebras in the case of odd values of N.

Preliminary Definitions and Results
Suppose the canonical representation of the number N, N ≥ 2, has the form where 2 < p 1 < • • • < p l are prime numbers, s 1 , . . ., s l are natural numbers.We write N = p s , where p = (p 1 , . . ., p l ), s = (s 1 , . . ., s l ).Denote by E(N) the Euler's function (Vinogradov, 1972), Euler's theorem on numbers, (Vinogradov, 1972).The value of E(N) is equal to number of positive integers less than N that are coprime to N.
This function can be also represented as ) .
Euler's theorem on divisibility, (Vinogradov, 1972).Suppose k and N are coprime natural numbers.Then N is a divisor of k E(N) − 1.
Fermat's little theorem is a special case of Euler's theorem on divisibility.

It is clear that any divisor of e(N) is a divisor of E(N).
The generalized Euler's divisibility theorem, (Vinogradov, 1972).Let k and N be coprime positive integers.Then N is a divisor of k E(N) − 1.
According to this theorem, if k and N are coprime then there exists a positive number l such that N is a divisor of k l − 1.
The smallest positive number l such that N is a divisor of k l − 1 is called the order of k modulo N, (Vinogradov, 1972).Denote by m(N, k) the order of k modulo N.
The generalized Euler's order theorem, (Vinogradov, 1972).Let k and N be coprime positive integers.Then m(N, k) is a divisor of e(N).
Denote by m(N) the order of 2 modulo N, i.e., m(N) = m(N, 2).The number m(N) is the smallest number m such that N, is a divisor of 2 m − 1.The elements of the algebra W N can be divided into two classes.They are the class of noncancelable fractions and the class cancelable fractions Consider the set of all elements k/N of the algebra W N such that k/N is a noncancelable fraction.The number of these elements is equal to the number of positive integers less than N and coprime to N, i.e., the function e(N) is equal to Euler's function.Suppose (2) In accordance with section 2 an orbit of period d is a set of elements of the algebra ) .
Denote by O(N, k) the number of orbits with period k (k ≥ 2).

Simple Case
Suppose N = p s , where p ≥ 3 is prime, s is a natural number; Theorem 3. The following formula is true The proof is based on Lemma 1.
Lemma 1. Suppose r ≤ s, k/p r is an element of the algebra W p r such that it is noncancelable fraction.Then this element belongs to an orbit of period m(p r ).The set of cancelable fractions of the algebra W p r is a subalgebra isomorphic to the algebra W p r−1 .
Proof.Suppose the number a satisfies the equation The number a is natural if and only if the number N is the divisor of 2 m − 1.We take into account that k and N are comprime, and k < N. Applying to a noncancelable element the Bernoulli shift operation m(N) times, we obtain the same element.If the Bernoulli shift is applied to this element less than m(N) times, then the same element cannot be obtained.
It is obviously that the subalgebra of cancelable fractions of the algebra W p r and the algebra W p r−1 are isomorphic to each other.Lemma 1 has been proved.
The graphs of algebras W 13 , W 23 , W 27 are shown in Figures 4 -6.

Common Case
Let N be as (1).
Lemma 2. Any element k/N of the algebra W N belongs to an orbit, and, if k/N ∈ W copr N , i.e., the element k/N is noncancelable fraction, then the period of this orbit equals m(N).
Proof.Apply the Bernoulli shift operation to the element k/N i times.We obtain the same element if and only if the equation contains natural number a.If i = e(N) in (3), then the number a is natural.Take into account, that N is coprime to 2. Hence there exists an orbit of period not greater than e(N), such that this orbit contains the element k/N.If the fraction k/N is a nocancelable fraction, then i = m(N, 2) = m(N) is the smallest value of i such that the number a is a natural number.From this Lemma 2 follows.
Lemma 3. Let N be an odd natural number.Two elements a ∈ W copr N , b ∈ W canc N of the algebra W N cannot belong the same orbit, i.e., if one of these elements is a noncancelable fraction, then the other element is a cancelable fraction.Proof.If an element is cancelable fraction, and we apply the Bernoulli shift operation to this element any times, we can obtain no noncancelable fraction.
Denote by δ d,N the number of orbits with period d in the subalgebra W copr N .In accordance with Lemmas 2 and 3, If d m(p r ), then δ d,p r = 0.
Suppose R = R(N) is the set of vectors r = (r 1 , . . ., r l ) with integer nonnegative numbers, 0 ≤ r 1 ≤ s 1 , . . ., 0 ≤ r l ≤ s l , and at least one of numbers r 1 , . . ., r l is positive.Theorem 9. Let d be a divisor of the number N. Then the number of orbits with period m, the number of the connected components and the number of elements W N are not less than in the algebra W d .

Calculation of Numbers and Periods of Orbits of the Algebra W 225
In Section 12, we give an example.We consider algebra W 225 .We calculate number of orbits of each period.

Figure 1 .
Figure 1.Algebra W 5 , 5 = 1[1] + 1[4] Theorem 1 has been proved.Suppose numbers l (l ≥ 1) and m (m ≥ 1) are such that b l = b l+m , and there are no numbers i and j that i, j < l + m and b i = b j .Then a i = a i+m for any i ≥ l.Hence the binary representation of the common fraction k/N contains an aperiodic part with length l − 1 and a repeating part with length m.We write k N = 0.a 1 . . .a l−1 (a l . . .a l+m−1 ).
a 1 . . .a l = 0, a 1 . . .a l (0).If the initial state S k belongs to a periodic class of communicating states, and this class contains d ≥ 1 states, then the repeating part of the number k/N binary representation contains d digits k N = 0.(a 1 . . .a d ).If the initial state S k is inessential and the chain comes from this state after l steps to a class of communicating states with period d, then the binary representation of the number k/N is k N = 0.a 1 . . .a l (a l+1 . . .a l+d ),i.e., the aperiodic part of the binary representation contains l digits, and the length of the repeating part equals d.7.Binary Representation of Common Fractions and Division of the Algebra W N into ClassesIn Section 7, we consider connection between Bernoulli algebras, binary representation, and Markov chains.Consider examples.Suppose N = 8, Figure3.There is an absorbing element 0/8 = 0.(0

Theorem 4 .
Suppose O(N, k)  is the number of orbits with period k(k ≥ 2) in algebra W N .Then O(N, k) = ∑ r∈R δ(k, p r ),and jd of the algebra W d belong to different orbits, then the elements algebra W N also belong to the different orbits.If the element i/d of the algebra W d is inessential, then the element algebra W N belong to the same components if and only if the elements i/N and j/N of the algebra W N also belong to the same connected component.Proof.The value of the element N d i N of the algebra W N and the value of element i/d of the algebra W d are the same for any i = 1, 2, . . ., d − 1.From this, Theorem 8 follows.