The dynamical system generated by the floor function ⌊λx⌋
aa r X i v : . [ m a t h . D S ] M a r THE DYNAMICAL SYSTEM GENERATED BY THE FLOORFUNCTION ⌊ λx ⌋ ROZIKOV U.A., SATTAROV I.A., USMONOV J.B.
Abstract.
We investigate the dynamical system generated by the function ⌊ λx ⌋ defined on R and with a parameter λ ∈ R . For each given m ∈ N we show thatthere exists a region of values of λ , where the function has exactly m fixed points(which are non-negative integers), also there is another region for λ , where there areexactly m + 1 fixed points (which are non-positive integers). Moreover the full set Z of integer numbers is the set of fixed points iff λ = 1. We show that depending on λ and on the initial point x the limit of the forward orbit of the dynamical systemmay be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbitor (iii) ±∞ . Introduction and Preliminaries
Let X ⊂ R and f be a map from X to itself. The set X need not be closed orbounded interval, although this is usually assumed in the literature. The point of viewof dynamical systems is to study iterations of f : if f n denotes the n -fold composition(iteration) of f with itself, then for a given point x one investigates the sequence x, f ( x ) , f ( x ) , f ( x ), and so on. This sequence is called one-dimensional discrete timedynamical system or the forward orbit of x , or just the orbit of x for short.The theory of one-dimensional non-linear dynamical systems underwent considerableprogress, as the result of the efforts of theorists from several fields -in particular fromphysics- to get a better understanding, by making use of the notion of the ”Hopf’sbifurcation” of the appearance of cycles and of the transition to aperiodic or ”chaotic”behavior in physical, biological or ecological systems. These new developments seemto be potentially very useful for the study of periodic and aperiodic phenomena ineconomics. Parts of this theory have been indeed already used in economic or gametheory [1]- [4].In the theory of the dynamical system the main problem is to know the set of limitpoints of { f n ( x ) } n ≥ for each initial point x , It is particularly interesting when the orbitrepeats. In this case x is a periodic point. If f has a periodic point of period m , then m is called a period for f . Given a continuous map of an interval one may ask whatperiods it can have, this question was answered in Sharkovsky’s well-known theorem.One of the implications of the theorem is that if a discrete dynamical system on the Mathematics Subject Classification.
Key words and phrases.
Dynamical systems; floor function; fixed point; trajectory. real line has a periodic point of period 3, then it must have periodic points of everyother period. But this theorem works only for continuous functions. The dynamicalsystems generated by discontinuous functions are rather difficult to study, and eachsuch system requires a specific method.In this paper we shall study a one-parametric family of discontinuous functions,which is defined by the floor function as f λ ( x ) = ⌊ λx ⌋ , λ ∈ R .For convenience of the reader let us give necessary definitions and properties of thefloor function. The floor function of x ∈ R is defined by ⌊ x ⌋ = max { m ∈ Z : m ≤ x } . The following are properties of the floor function which we shall use in this paper: ⌊ x ⌋ = m if and only if m ≤ x < m + 1 , ⌊ x ⌋ = m if and only if x − < m ≤ x,x < m if and only if ⌊ x ⌋ < m,m ≤ x if and only if m ≤ ⌊ x ⌋ . The above are not necessarily true if m is not an integer.The floor function has been applied in the study of mod operator, quadratic reci-procity, rounding, number of digits, Riemann function etc. Moreover this function isuseful to give formulas for prime numbers, here are some of them (see [6]): there is anumber θ = 1 . ... (Mills’ constant) and a number ω = 1 . ... with the propertythat (cid:4) θ (cid:5) , (cid:4) θ (cid:5) , (cid:4) θ (cid:5) , . . . ⌊ ω ⌋ , (cid:4) ω (cid:5) , j ω k , . . . are all prime.These various applications of the floor function gave a motivation to study dynamicalsystems generated by such functions. The parameter λ makes rich the behavior of ourdynamical system generated by ⌊ λx ⌋ : in subsection 2.1 for each given m ∈ N we showthat there exists a region of values of λ , where the function has exactly m fixed points(which are non-negative integers), also there is another region for λ , where there areexactly m + 1 fixed points (which are non-positive integers). Moreover the full set Z ofinteger numbers is the set of fixed points iff λ = 1. In the rest subsections of the Section2 we show that depending on λ and on the initial point x the limit of the forward orbitof the dynamical system may be a fixed point or a two-periodic orbit or ±∞ .2. The dynamical system
In this paper we consider the dynamical system associated with the function f : R → R defined by f ( x ) ≡ f λ ( x ) = ⌊ λx ⌋ , (2.1)where λ ∈ R is a parameter. HE DYNAMICAL SYSTEM OF FLOOR FUNCTION 3
Fixed points.
A point x ∈ R is called a fixed point of f if f ( x ) = x . The set ofall fixed points is denoted by Fix( f ). The following lemma gives all fixed points of thisfunction. Lemma 1.
For the set of fixed points the following hold If λ ≤ then Fix ( f ) = { } ; If m − m < λ ≤ mm +1 for some m ∈ N then Fix ( f ) = { , − , − , ..., − m } ; If λ = 1 then Fix ( f ) = Z ; If m +1 m ≤ λ < mm − for some m ∈ N then Fix ( f ) = { , , , ..., m − } .Proof.
1) Let λ ≤
0. In the case λ = 0 we have f ( x ) = 0, i.e., only x = 0 is fixed point.Moreover, x = 0 is a fixed point independently on value of λ . For λ < x < f ( x ) = ⌊ λx ⌋ ≥
0, consequently f ( x ) = x ;b) If x > f ( x ) = ⌊ λx ⌋ <
0, consequently f ( x ) = x .Thus if λ ≤ ⌊ λx ⌋ = x has a unique solution x = 0.2) Let 0 < λ <
1. Since solutions of ⌊ λx ⌋ = x are integer numbers we consider thefollowing partition of the set Z = Z − ∪ { } ∪ Z + . If x ∈ Z + then by λ ∈ (0 ,
1) wehave 0 < λx < x and ⌊ λx ⌋ < x . For each λ ∈ (0 ,
1) there exists m ∈ N such that m − m < λ ≤ mm +1 holds, because(0 ,
1) = ∞ [ m =1 (cid:18) m − m , mm + 1 (cid:21) . (2.2)Assume m − m < λ ≤ mm +1 then ∀ x ∈ Z − we have x − xm + 1 ≤ λx < x − xm . (2.3)From (2.3) for any x ∈ {− , − , ..., − m } we obtain x < x − xm + 1 ≤ λx < x − xm ≤ x + 1 . Thus each x ∈ {− , − , ..., − m } satisfies ⌊ λx ⌋ = x .Let now x < − m then there exists l ∈ N such that x = − ( m + l ). By (2.3) we get x + m + lm + 1 ≤ λx < x + m + lm and x + 1 ≤ λx < x + 1 + lm . Thus ⌊ λx ⌋ ≥ x + 1 > x. This completes the proof of part 2).3) Straightforward.
ROZIKOV U.A., SATTAROV I.A., USMONOV J.B. λ >
1. For x ∈ Z − by λ > λx < x , hence ⌊ λx ⌋ < x . Since(1 , + ∞ ) = ∞ [ m =1 (cid:20) m + 1 m , mm − (cid:19) (2.4)there is m ∈ N such that m +1 m ≤ λ < mm − and for x ∈ Z + we have x + xm ≤ λx < x + xm − . (2.5)Consequently for x ∈ { , , ..., m − } we have x < x + xm ≤ λx < x + xm − ≤ x + 1 . Thus each x ∈ { , , ..., m − } is a fixed point.Now assume x > m −
1, then there exists p ∈ N such that x = m − p . Thus by(2.5) we obtain x + m + p − m ≤ λx < x + m − pm − . Since p − m ≥ x + 1 ≤ λx < x + 1 + pm − , consequently ⌊ λx ⌋ ≥ x + 1 > x. Thus if m +1 m ≤ λ < mm − for some m ∈ N then each x ∈ { , , , ..., m − } is a fixedpoint, these are all possible fixed points. (cid:3) The limit points.
For a given function f : R → R the ω -limit set of x ∈ R ,denoted by ω ( x, f ) or ω ( x ), is the set of cluster points of the forward orbit { f n ( x ) } n ∈ N of the iterated function f . Hence, y ∈ ω ( x ) if and only if there is a strictly increasingsequence of natural numbers { n k } k ∈ N such that f n k ( x ) → y as k → ∞ .In this section for function (2.1) we shall describe the set ω ( x ) for each given x ∈ R .2.2.1. The case λ ≤ . The case λ = 0 is trivial ω (0) = { } . Consider the case λ < Theorem 1. If λ < then the dynamical system generated by f has the followingproperties: If − < λ < then ∀ x ∈ R we have lim n →∞ f n ( x ) = 0 , i.e., ω ( x ) = { } . If λ = − then each non-zero integer has period two, i.e. f ( x ) = x for any x ∈ Z \ { } . Moreover f ( x ) = f ( x ) , for each x ∈ R , i.e., ω ( x ) = ( { x, f ( x ) } , if x ∈ Z { f ( x ) , f ( x ) } , if x ∈ R \ Z . HE DYNAMICAL SYSTEM OF FLOOR FUNCTION 5 If λ < − then ∀ x ∈ ( λ , we have f ( x ) = 0 , and ω ( x ) = ( { } , if x ∈ ( λ , {−∞ , + ∞} , if x ∈ R \ ( λ , . Proof.
1. Since for each x ∈ R the sequence { f n ( x ) } n ≥ is subset of Z , by the condition − < λ < x < f ( x ) > | f ( x ) | ≥ f ( x ) > | f ( x ) | ≥ f ( x ) > ... (2.6)and for x > | f ( x ) | ≥ f ( x ) > | f ( x ) | ≥ f ( x ) > | f ( x ) | ≥ ... (2.7)From (2.6) and (2.7) it follows that ∀ x ∈ R the sequence {| f n ( x ) |} n ≥ of non-negativeinteger numbers is non-increasing and bounded from below by 0. Hence there is limitlim n →∞ | f n ( x ) | = α ( x ) ∈ { } ∪ N . We shall show that α ( x ) = 0 for any x ∈ R . Supposethere is x ∈ R such that α ( x ) ≥
1. Then the sequence { f n ( x ) } n ≥ (without absolutevalue) has the set of limit points {− α ( x ) , α ( x ) } . Moreover by the properties a) andb) mentioned in the proof of Lemma 1 if x > x < k →∞ f k ( x ) = α ( x ) and lim k →∞ f k +1 ( x ) = − α ( x ) . (2.8)Since ⌊ λx ⌋ ≤ λx for any x ∈ R , we have ⌊ λf n ( x ) ⌋ ≤ λf n ( x ), i.e. f n +1 ( x ) ≤ λf n ( x ).Since λ <
0, using again the properties a) and b), for x > f ( x ) < f ( x ) ≤ λf ( x ) = | λ || f ( x ) | ,f ( x ) ≤ | λ || f ( x ) | < | λ | f ( x ) ,f ( x ) < | λ | f ( x ) < | λ | f ( x ) ,. . . . . .f k ( x ) ≤ | λ | k − f ( x ) ≤ | λ | k | f ( x ) | . (2.9)Write the last inequality for x = x , i.e f k ( x ) ≤ | λ | k | f ( x ) | . Since λ ∈ ( − , k → ∞ and using(2.8) we get lim k →∞ f k ( x ) = α ( x ) ≤
0, this contradicts to our assumption α ( x ) ≥ α ( x ) = 0 for any x ∈ R . Consequently lim n →∞ f n ( x ) = 0, since lim n →∞ | f n ( x ) | =0. 2. For λ = − f ( x ) = ⌊− x ⌋ . Then ∀ x ∈ Z we get f ( x ) = [ ⌊−⌊− x ⌋⌋ = x .Moreover it is easy to check that ⌊− x ⌋ = ⌊−⌊−⌊− x ⌋⌋⌋ for all x ∈ R . Hence f ( x ) = f ( x ).3. Assume λ < − λ < x ≤
0. In this case we have 1 > λx ≥
0. Consequently ⌊ λx ⌋ = 0. ROZIKOV U.A., SATTAROV I.A., USMONOV J.B.
Let now λ < − x ∈ R \ ( λ , x ≤ λ we get f ( x ) < | f ( x ) | ≤ f ( x ) < | f ( x ) | ≤ f ( x ) < ... (2.10)and for x > | f ( x ) | ≤ f ( x ) < | f ( x ) | ≤ f ( x ) < | f ( x ) | ≤ ... (2.11)Since {| f n ( x ) |} ⊂ N , from (2.10) and (2.11) it follows that lim n →∞ f n ( x ) = ∞ . More-over, using properties a) and b) one can see thatlim k →∞ f k ( x ) = (cid:26) + ∞ , if x > −∞ , if x ≤ λ lim k →∞ f k +1 ( x ) = (cid:26) −∞ , if x > ∞ , if x ≤ λ . (cid:3) The case < λ < . Note that for each λ ∈ (0 ,
1) there exists m ∈ N such that m − m < λ ≤ mm +1 . Theorem 2.
Let m − m < λ ≤ mm +1 for some m ∈ N . Then lim n →∞ f n ( x ) = , for all x ∈ [0 , + ∞ ) ,k, for all x ∈ [ kλ , k +1 λ ) , − m, for all x ∈ ( −∞ , − mλ )where k ∈ {− , − , . . . , − m } . Proof.
For any x ∈ [0 , + ∞ ) we have f ( x ) = ⌊ λx ⌋ ≤ λx , iterating this inequality we get0 ≤ f n ( x ) ≤ λ n x. Consequently0 ≤ lim n →∞ f n ( x ) ≤ lim n →∞ λ n x = 0 , i.e. lim n →∞ f n ( x ) = 0 . Consider now x and k ∈ {− , − , . . . , − m } such that kλ ≤ x < k +1 λ . Then by 0 < λ < k ≤ λx < k + 1. Consequently, ⌊ λx ⌋ = k . Since each k ∈ {− , − , . . . , − m } is a fixed point, we obtain lim n →∞ f n ( x ) = k. Consider now the case x < − mλ . Then f ( x ) = ⌊ λx ⌋ ∈ Z with f ( x ) < − m . Moreoverfor λ ∈ (0 ,
1) we have f ( x ) > x (see the proof of part 2 of Lemma 1). Iterating thelast inequality we obtain f n +1 ( x ) > f n ( x ), i.e. f n ( x ) is an increasing sequence, whichis bounded from above by − m . Since − m is the unique fixed point in ( −∞ , − m ], wehave lim n →∞ f n ( x ) = − m. (cid:3) HE DYNAMICAL SYSTEM OF FLOOR FUNCTION 7
The case λ ≥ . For λ = 1 we have f ( x ) = ⌊ x ⌋ and Fix( f ) = Z . It is easy to seethat lim n →∞ f n ( x ) = ⌊ x ⌋ , ∀ x ∈ R . Let now λ >
1. Because of (2.4) it is sufficient to study the dynamics of f at λ suchthat m +1 m ≤ λ < mm − , for some m ∈ N . Here for m = 1 we consider 2 ≤ λ < + ∞ . Theorem 3. If m +1 m ≤ λ < mm − for some m ∈ N then lim n →∞ f n ( x ) = k, for all x ∈ [ kλ , k +1 λ ) , −∞ , for all x ∈ ( −∞ , , + ∞ , for all x ∈ [ mλ , + ∞ ) , where k ∈ { , , . . . , m − } . Proof.
Take x ∈ [ kλ , k +1 λ ) for some k ∈ { , . . . , m − } . Then ⌊ λx ⌋ = k and since each k ∈ { , . . . , m − } is a fixed point, we obtainlim n →∞ f n ( x ) = k. In the case x < λx < x > λx ≥ ⌊ λx ⌋ = f ( x ) . Since f ( x ) is a non-decreasing function we get from the last inequality that f n ( x ) >f n +1 ( x ), i.e., the sequence f n ( x ) is decreasing. By Lemma 1 for λ > f in ( −∞ , n →∞ f n ( x ) = −∞ . Assume now mλ ≤ x < + ∞ . Since each f n ( x ), n ≥ x ≥ m we have ⌊ λx ⌋ > x (see the part 4) of proof of Lemma 1). From thisinequality it follows that f n ( x ) is an increasing sequence and by Lemma 1 there is nofixed point in [ m, + ∞ ), hence lim n →∞ f n ( x ) = + ∞ . (cid:3) Acknowledgements
U.Rozikov thanks Aix-Marseille University Institute for Advanced Study IM´eRA(Marseille, France) for support by a residency scheme. His work also partially sup-ported by the Grant No.0251/GF3 of Education and Science Ministry of Republic ofKazakhstan.
ROZIKOV U.A., SATTAROV I.A., USMONOV J.B.
References [1] J.Benhabib, R.H. Day,
Rational choice and erratic behaviour , Review of Economic Studies, ,(1981) 459-472.[2] R.H. Day, The emergence of chaos from classical economic growth , Quarterly Journal of Econom-ics, , (1983) 201-213.[3] R.L. Devaney, An introduction to chaotic dynamical system , Westview Press, 2003.[4] R.U. Jensen, R. Urban,
Chaotic price behaviour in a nonlinear cobweb model , Yale University.1982.[5] A.B. Katok, B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems , CambridgeUniv. Press, Cambridge, 1995.[6] P. Ribenboim,
The New Book of Prime Number Records , New York: Springer, 1996.
U. A. Rozikov, Institute of mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent,Uzbekistan.
E-mail address : [email protected] I.A. Sattarov and J.B. Usmonov, Namangan state university, Namangan, Uzbekistan.
E-mail address ::