Periodic structures for nonlinear piecewise contracting maps
PPERIODIC STRUCTURES FOR NONLINEAR PIECEWISE CONTRACTINGMAPS
FUMIHIKO NAKAMURA
Kitami Institute of Technology, Kitami, 090-8507, Japan
Abstract.
In this paper, we first show that any nonlinear monotonic increasing contracting mapswith one discontinuous point on a unit interval which has an unique periodic point with period n conjugates to a piecewise linear contracting map which has periodic point with same period.Second, we consider one parameter family of monotonic increasing contracting maps, and show thatthe family has the periodic structure called Arnold tongue for the parameter which is associatedwith the Farey series. This implies that there exist a parameter set with a positive Lebesgue measuresuch that the map has a periodic point with an arbitrary period. Moreover, the parameter set withperiod ( m + n ) exists between the parameter set with period m and n . Introduction
The piecewise linear contracting map on a unit interval defined by S α,β ( x ) = αx + β (mod 1) , with 0 < α, β < α + β >
1, and it is known that the map has a unique periodic point for almost every parameter α and β and a period with an arbitrary numbers can be found by choosing appropriate parameters[10]. The parameter region divided by a set of the parameters for which the map has periodic pointwith each period is known as Arnold tongue or Farey structure which shows a layered structurebased on a Farey series. More precisely, there is a periodic region with period n + n between theregions with period n and n if | l n − l n | = 1 for irreducible fractions l /n and l /n in itsparameter space.The Arnold tongue is observed in many researches, for instance [4, 5, 6] showed the structurefor the standard circle maps which describes a cardiac oscillation model. Such structure for thepiecewise linear model has already been studies in, for example [2, 7, 8, 10], especially Keener’sresults cover the nonlinear models and show for almost all parameters the system has a rationalrotation number so that it possesses a periodic point. Moreover, an irrational rotation number isachieved on the Cantor set in the parameter space.In this paper, we focus on the nonlinear model with a discontinuous point and show that anynonlinear system with certain conditions conjugates the linear contracting map (1). Furthermore,we prove that the family of parametrized nonlinear systems has the layered structure based on Fareyseries. In [10], although similar layered structures were numerically observed for non-linear models E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases. nonlinear contracting map, periodic point, Farey structure. a r X i v : . [ m a t h . D S ] J un F. NAKAMURA (e.g. T ( x ) = αx + β (mod 1), T ( x ) = α √ x + β (mod 1), etc), the rigorous proof for the mapscould not be accomplished because of the difficulty for the non-linearity.The organization of this paper is as follows. In section 2, we prepare some notations and previousresults for the linear model. In section 3, we show that almost any nonlinear monotonic increasingmaps with one discontinuous point conjugates to the linear contracting maps (1) for some parameterby constructing a concrete homeomorphism. This implies that any two nonlinear maps with aperiodic point with same period can be conjugated each other under certain conditions. Note thatthe nonlinear map does not require the contracting property. In section 4, we consider one parameterfamily of nonlinear monotonic increasing contracting maps, and give the proof of an existence ofFarey structure on the parameter space. This implies that we can find the map with an arbitraryperiod by choosing an appropriate parameter for a family of nonlinear maps, and justified a part ofFarey structure observed in [10]. 2. Preliminary
In this section, we introduce the facts for linear contracting maps (1) with a periodic point (See[10, 11] for the details). Let
P r ( n ) := { l < n | GCD ( n, l ) = 1 } for each n ∈ N . In this paper, if wewrite ( n, l ), then n and l always satisfy n ∈ N ≥ and l ∈ P r ( n ). We define two functions B Un,l ( α )and B Ln,l ( α ) and sets { D n,l } ( n,l ) as follows; B Un,l ( α ) = (1 − α ) (cid:32) − α n n − (cid:88) m =1 k m α m + 1 (cid:33) , (2) B Ln,l ( α ) = (1 − α ) (cid:32) − α n n − (cid:88) m =1 k m α m + 1 − α n − − α n − α n (cid:33) , (3)and D n,l = { ( α, β ) ∈ (0 , | B Ln,l ( α ) ≤ β < B Un,l ( α ) } , (4)where k m := (cid:20) ( m + 1) ln (cid:21) − (cid:20) mln (cid:21) for m ∈ Z , (5)where [ x ] is the integer part of x . In [10], we showed that if the parameter α and β are chosenin the set D n,l , then the map S α,β has periodic point with period n . It is clear that the set D n,l has a positive Lebesgue measure. Moreover, the set D n + n (cid:48) ,l + l (cid:48) exists between D n,l and D n (cid:48) ,l (cid:48) if | n (cid:48) l − nl (cid:48) | = 1. In this way, we can obtain explicit formula for the parameter region which themap S α,β has a periodic point for the linear maps, and show that these regions display a layeredstructure associated with Farey series. Moreover, when ( α, β ) ∈ D n,l , the n periodic points for themap S α,β is given by Per n ( S α,β ) = (cid:26) β − α − A i ( α ) (cid:12)(cid:12)(cid:12)(cid:12) i = 0 , · · · , n − (cid:27) , (6)where, for i = 0 , , · · · , n − A i ( α ) = 11 − α n (cid:32) i − (cid:88) m =0 k m α i − m − + n − (cid:88) m = i k m α n + i − m − (cid:33) . (7)We call the sequence { k i } i ∈ Z defined by (5) a rational characteristic sequence with respectto ( n, l ). The following propositions give the properties of rational characteristic sequences, andplays an important role to prove Theorem 4.2. ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 3
Proposition 2.1. ( [10] , Proposition 2.2) Let { k m } m ∈ Z be a rational characteristic sequence withrespect to ( n, l ) . We then have the following properties. (i) k m ± n = k m for m ∈ Z , (ii) k n − − m = k m for m ∈ Z , m / ∈ n Z , n Z − , (iii) k m − ˆ l = k m for m ∈ Z , m / ∈ n Z , n Z − ,where ˆ l := min { t ∈ N | tl = 1 (mod n ) } . Note that k = 0 and k n − = 1 always hold obviously. Proposition 2.2. ( [10] , Proposition 2.3) Let { k m } m ∈ Z be a rational characteristic sequence withrespect to ( n, l ) and { k (cid:48) m } m ∈ Z be another rational characteristic sequence with respect to ( n (cid:48) , l (cid:48) ) . If ln < l (cid:48) n (cid:48) and nl (cid:48) − n (cid:48) l = 1 , then the sequence { ˆ k m } m ∈ Z defined by ˆ k m := (cid:40) k m for m = 0 , · · · , n − k (cid:48) m − n for m = n, · · · , n + n (cid:48) − and ˆ k ¯ m := ˆ k m if ¯ m = m + t ( n + n (cid:48) ) with m = 0 , · · · , n + n (cid:48) − t ∈ Z \{ } is the rational characteristic sequence with respect to ( n + n (cid:48) , l + l (cid:48) ) . Remark 2.3.
From (ii) in Proposition 2.1, ( k k · · · k n − ) = ( k n − · · · k k ) holds when { k i } is arational characteristic sequence. From Proposition 2.2,(ˆ k · · · ˆ k n + n (cid:48) − ) = ( k · · · k n − k (cid:48) · · · k (cid:48) n (cid:48) − )= ( k · · · k n − k (cid:48) · · · k (cid:48) n (cid:48) − )since { ˆ k i } , { k i } and { k (cid:48) i } are rational characteristic sequences with respect to ( n + n (cid:48) , l + l (cid:48) ), ( n, l )and ( n (cid:48) , l (cid:48) ), using (ii) in Proposition 2.1 again, we have(ˆ k · · · ˆ k n + n (cid:48) − ) = ( k (cid:48) n (cid:48) − · · · k (cid:48) k n − · · · k )= ( k (cid:48) · · · k (cid:48) n (cid:48) − k · · · k n − )These calculations are used many times in the proof of Theorem 4.2.Next, considering the pre-images of 0 or the discontinuous point is useful to analyse the contract-ing maps. Indeed, we can write the pre-image of 0 for the map S α,β explicitly as follows. We alsoconsider the pre-image for the nonlinear map in the section 3. Proposition 2.4. ( [11] , Proposition 4) Assume that ( α, β ) ∈ D n,l , then S − iα,β (0) = i (cid:88) m =1 k n − i + m − − βα m ∈ [0 , , ( i = 1 , · · · , n − , (8) where { k m } is a rational characteristic sequence with respect to ( n, l ) .Moreover, for i = n , S − nα,β (0) is not in [0 , . In the end of this section, we introduce the results in [8, 9] which tell us that if parameters ( α, β )and ( α (cid:48) , β (cid:48) ) are chosen from same set D n,l , then the maps S α,β and S α (cid:48) ,β (cid:48) are conjugate each other.More precisely, the following proposition holds. Proposition 2.5 ([8], Theorem 7.1) . The followings hold: (i) If ( α, β ) , ( α, β ) ∈ int ( D n,l ) , S α,β and S α (cid:48) ,β (cid:48) are conjugate. (ii) If β = B Un,l ( α ) and β (cid:48) = B Un,l ( α (cid:48) ) , then S α,β and S α (cid:48) ,β (cid:48) are conjugate. (iii) If β = B Ln,l ( α ) and β (cid:48) = B Ln,l ( α (cid:48) ) , then S α,β and S α (cid:48) ,β (cid:48) are conjugate. F. NAKAMURA Conjugacy with nonlinear piecewise monotonic increasing maps
In this section, we show that nonlinear piecewise monotonic increasing maps satisfying someassumptions conjugate the linear contracting ones.Let f : [0 , → [0 ,
1] be a continuous map except with x = c (0 < c < f (0) > f (1) (called non-overlapping condition ),(A2) f ( c − ) = 1 and f ( c +) = 0,(A3) if x < y , then f ( x ) < f ( y ) for x, y ∈ [0 , c ) or x, y ∈ [ c, n ≥ f − i (0) is in [0 ,
1) for i =0 , , · · · , n − f − n (0) = ∅ . Remark 3.1.
In the above setting, we note that the pre-image for any point is unique if it exists.Then, we often use f − i ( x ) as the point iterating x by the inverse map f − i . Remark 3.2.
Clearly if x ∈ ( f (1) , f (0)), then f − ( x ) = ∅ . Moreover, since f − (0) = c , aboveassumption (A4) can be written by a pre-image of discontinuous point c such as(A4)’ there exists an integer n ≥ f − i ( c ) ∈ [0 , f (1)) ∪ [ f (0) ,
1) for i = 0 , , · · · , n − f − ( n − ( c ) ∈ ( f (1) , f (0)) .This form is used in the proof of Theorem 4.2.The main result in this section is the next theorem which conclude any nonlinear system satisfying(A1)-(A4) conjugates some linear systems. Theorem 3.3.
Let f : [0 , → [0 , be a continuous map except with x = c (0 < c < , satisfying(A1)-(A4). Then f conjugates S α,β with ( α, β ) ∈ int ( D n,l ) . To prove the theorem, we first prepare the following lemmas.
Lemma 3.4.
Let l be a number of elements of the set { i | f − i (0) ∈ [ c, } . Let { x i } n − li =1 and { y i } li =1 be n points of pre-images of zero, { f − i (0) } n − i =0 , such that, x < x < · · · < x n − l < y < y < · · · < y l . If n > l , then the following orbit relations hold; f − ( x i ) = y i for i = 1 , · · · , l, (9) f − ( x i ) = ∅ for i = l + 1 , (10) f − ( x i ) = x i − l for i = l + 2 , · · · , n − l, (11) f − ( y i ) = x n − l + i for i = 1 , · · · , l. (12) If n < l , then the following orbit relations hold; f − ( x i ) = y i for i = 1 , · · · , l, (13) f − ( y i ) = y i + l for i = 1 , · · · , l − n − , (14) f − ( y i ) = ∅ for i = 2 l − n − , (15) f − ( y i ) = x n − l + i for i = 2 l − n, · · · , l. (16) Proof.
In the case n > l , since x = 0, f − ( x ) = y = c and f − ( y l ) = x n − l , the relations(9) and (12) are immediately hold because of the monotonicity of the map. Then, one of x i , i = l + 1 , l + 2 , · · · , n − l is mapped to ∅ , and the others are mapped to x k , k = 2 , · · · , n − l by f − . By the monotonicity, the relations (10) and (11) must hold.For the case n < l , we can show similarly by substituting the role of x i and y i . (cid:3) ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 5
Figure 1.
Example of our target map with ( n, l ) = (9 , n, l ) = (9 ,
2) in Figure 1.
Lemma 3.5.
Let l be a number of elements of the set { i | f − i (0) ∈ [ c, } . Then l and n arerelatively prime numbers.Proof. When n is a prime number, n and l are always relatively prime. Then we assume that n = kn (cid:48) and l = kl (cid:48) ( k > , n (cid:48) ≥ , l (cid:48) ≥
1) where n (cid:48) and l (cid:48) are relatively prime. Clearly f (0) = 0 ∈ [0 , c )and f − (0) = c ∈ [ c, n > l , let x < · · · < x k < x < · · · < x k < · · · < x n (cid:48) − l (cid:48) < · · · < x n (cid:48) − l (cid:48) k < y < · · · < y k < · · · < y l (cid:48) < · · · < y l (cid:48) k be points in { f − i (0) } n − i =0 such that x ji ∈ [0 , c ) for i = 1 , · · · , k and j = 1 , · · · , n (cid:48) − l (cid:48) , and y ji ∈ [ c,
1] for i = 1 , · · · , k and j = 1 , · · · , l (cid:48) .Consider pre-images of all x ji and y ji by f . First, by the orbit relation (9) and (12), we have f − ( x ji ) = y ji for i = 1 , · · · , k and j = 1 , · · · , l (cid:48) , (17) f − ( y ji ) = x n (cid:48) − l (cid:48) + ji for i = 1 , · · · , k and j = 1 , · · · , l (cid:48) . (18)and for reminding x l (cid:48) +12 , · · · , x l (cid:48) +1 k and x ji for i = 1 , · · · , k and j = l (cid:48) + 2 , · · · , n (cid:48) − l (cid:48) , we have f − ( x ji ) = x j − l (cid:48) i by (11) . (19)Moreover, for x l (cid:48) +11 , we have x l (cid:48) +11 = f − ( n − (0) by (10) , (20)that is, there is no pre-image for x l (cid:48) +11 by f . However, since the index i of x ji or y ji is invariant bythese rules (17)-(18) of iteration, the pre-image of x (= 0) traces on only x · and y · , and reachesto x l (cid:48) +11 . This contradicts to the assumption that all x ji and y ji are elements of pre-images of 0, { f − i (0) | i = 0 , , · · · , n − } .For the case n < l , it can be shown similarly by substituting the role of x ji and y ji . (cid:3) F. NAKAMURA
Lemma 3.6.
Let l be a number given in previous Lemma 3.5. Let ρ be a permutation which arranges { S − iα,β (0) } n − i =0 in increasing order, that is, S − ρ (0) α,β (0) < S − ρ (1) α,β (0) < · · · < S − ρ ( n − α,β (0) . where ( α, β ) ∈ int ( D n,l ) . Then, ρ permutes { f − i (0) } n − i =0 in increasing order, that is, f − ρ (0) (0) < f − ρ (1) (0) < · · · < f − ρ ( n − (0) . Proof.
From Lemma 3.4, if two maps f and g satisfying the assumption (A1) - (A4) with samenumber n and l , then the orders of points { f − i (0) } n − i =0 and { g − i (0) } n − i =0 coincide. Especialy, thelinear map S α,β with ( α, β ) ∈ D n,l also satisfies the assumption (A1) - (A4) from the facts inprevious section. Therefore the orders of points { f − i (0) } n − i =0 and { S − iα,β (0) } n − i =0 coincide for anymaps f . (cid:3) Lemma 3.7.
Let { I if } ni =1 be partitions of [0 , determined by { f − i (0) } n − i =1 with point 0 and 1 suchthat each I if is closed interval and ∪ i I if = [0 , . Then, f has a periodic point with period n , andthese n points belongs to an interior of each interval I if one each.Proof. By Lemma 3.6 in [2], it has already known that if the set { f − i (0) | i = 0 , , · · · } is finite anda number of the set is n , then f has a periodic point with period n . Thus we shall show that eachinterval I if , i = 1 , · · · , n , possesses only one of points of a periodic point.Assume that there are two points p and p of the periodic point in some interval I if . Since themap has contracting property, all periodic points are stable and unstable fixed points or periodicpoints do not exist. Then there exists k such that f − k (0) must be between p and p . This iscontradiction since the partition { I if } i is made by the points { f − i (0) } n − i =1 . (cid:3) Proof of Theorem 3.3.
For convenience, we write S α,β by S . We shall construct the homeomor-phism H such that f ◦ H = H ◦ S . First, let { I if } ni =1 and { I iS } ni =1 be partitions of [0 ,
1] determinedby { f − i (0) } n − i =1 and { S − i (0) } n − i =1 respectively. Each subintervals I if and I iS have a periodic point inits interior by Lemma 3.7, we denote the periodic points by p if and p iS for i = 1 , · · · , n . Then set h ( p iS ) = p if , i = 1 , · · · , n. (21)Next, by Lemma 3.6, since the orders of S − i (0) and f − i (0) are corresponding, it is enable to set h ( S − i (0)) = f − i (0) , i = 0 , · · · , n − . (22)Considering a orbits of 0 by f , { f i (0) } ∞ i =0 , the sequence f k (0) , f n + k (0) , f n + k (0) , · · · goes to theperiodic point p kf from left side of the periodic point in I kf by the map f n . On the other hand,considering a orbits of 1 by f , { f i (1) } ∞ i =0 , the sequence f k (1) , f n + k (1) , f n + k (1) , · · · goes to theperiodic point p kf from right side of the periodic point in I kf by the map f n . Then we correspondeach orbits, that is, h ( S i (0)) = f i (0) , h ( S i (1)) = f i (1) , i = 1 , , · · · . Finally, we define the function h between these points { S i (0) } ∞ i = − ( n − and { p iS } ni =1 . Consider afurther partition for each I iS , i = 1 , · · · , n , by the points S i + mn (0) and S i + mn (1) for m = 0 , , · · · .We first define h − ( n − : [ S − ( n − (0) , S (0)] → [ f − ( n − (0) , f (0)] ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 7 as an arbitrary homeomorphism. Next we define h − ( n − : [ S − ( n − (0) , S (0)] → [ f − ( n − (0) , f (0)]by h − ( n − := f ◦ h − ( n − ◦ S − . Since f , S − and h − ( n − are all bijective, continuous and monotonicincreasing on each domain, h − ( n − becomes homeomorphism. and the following diagram holds.[ S − ( n − (0) , S (0)] S −−−−→ [ S − ( n − (0) , S (0)] h − ( n − (cid:121) h − ( n − (cid:121) [ f − ( n − (0) , f (0)] f −−−−→ [ f − ( n − (0) , f (0)]Inductively, we define h − ( n − m ) : [ S − ( n − m ) (0) , S m (0)] → [ f − ( n − m ) (0) , f m (0)] as h − ( n − m ) := f ◦ h − ( n − m ) ◦ S for m = 3 , , , · · · satisfying the following diagram.[ S − ( n − m +1) (0) , S m − (0)] S −−−−→ [ S − ( n − m ) (0) , S m (0)] h − ( n − m +1) (cid:121) h − ( n − m ) (cid:121) [ f − ( n − m +1) (0) , f m − (0)] f −−−−→ [ f − ( n − m ) (0) , f m (0)]Similarly, we construct the homeomorophism by using the image of 1. We define g − ( n − : [ S − ( n − (1) , S (1)] → [ f − ( n − (1) , f (1)]as an arbitrary homeomorphism. Next we define g − ( n − : [ S − ( n − (1) , S (1)] → [ f − ( n − (1) , f (1)]by g − ( n − := f ◦ g − ( n − ◦ S − . Since f , S − and g − ( n − are all bijective, continuous and monotonicincreasing on each domain, g − ( n − becomes homeomorphism, and the following diagram holds.[ S − ( n − (1) , S (1)] S −−−−→ [ S − ( n − (1) , S (1)] g − ( n − (cid:121) g − ( n − (cid:121) [ f − ( n − (1) , f (1)] f −−−−→ [ f − ( n − (1) , f (1)]Inductively, we define g − ( n − m ) : [ S − ( n − m ) (1) , S m (1)] → [ f − ( n − m ) (1) , f m (1)] as g − ( n − m ) := f ◦ g − ( n − m +1) ◦ S − for m = 3 , , , · · · satisfying the following diagram.[ S − ( n − m +1) (1) , S m − (1)] S −−−−→ [ S − ( n − m ) (1) , S m (1)] g − ( n − m +1) (cid:121) g − ( n − m ) (cid:121) [ f − ( n − m +1) (1) , f m − (1)] f −−−−→ [ f − ( n − m ) (1) , f m (1)]Finally, defining the map h by H = (cid:40) h i on [ S i − (0) , S n + i − (0)] g i on [ S i − (1) , S n + i − (1)] , i = − ( n − , − ( n − , · · · , we obtained the homeomorophism H satisfying f ◦ H = H ◦ S . (cid:3) Periodic structure for family of nonlinear contracting maps
In this section, we show that the family of nonlinear contracting maps constructed as followspossesses Farey structure for the parameter space.
F. NAKAMURA
Let g : R → R be a continuous monotonic increasing with g (0) = 0 and g ( c ∗ ) + 1 = c ∗ . Assumethat g has contracting property, that is, there exists κ < | g ( x ) − g ( y ) | ≤ κ | x − y | for x, y ∈ (0 , c ∗ ) , Define h ( x ) := g ( x ) + 1. For c ∈ (0 , c ∗ ), the map T c ( x ) = (cid:40) h ( x ) (if x < c ) g ( x ) (if x ≥ c ) (23)becomes a transformation on [ g ( c ) , h ( c )] which is a continuous monotonic increasing except with x = c . In figure 2, we draw the illustration of constructing the family of maps { T c } c ∈ (0 ,c ∗ ) . Figure 2.
The illustration of constructing the family of maps { T c } c ∈ (0 ,c ∗ ) .Next lemma implies that the transformation T c satisfies non-overlapping condition (A1). Lemma 4.1.
The inequality h ◦ g ( c ) > g ◦ h ( c ) holds for any c ∈ (0 , c ∗ ) .Proof. Since h ( x ) = g ( x ) + 1, the map h clearly has contracting property. Thus, we have | g ( c ) − gh ( c ) | ≤ κ | c − h ( c ) | , | hg ( c ) − h ( c ) | ≤ κ | g ( c ) − c | , which leads | g ( c ) − gh ( c ) | + | hg ( c ) − h ( c ) | ≤ κ ( | c − h ( c ) | + | g ( c ) − c | )= κ ( h ( c ) − g ( c )) . Since [ g ( c ) , gh ( c )] , [ hg ( c ) , h ( c )] ⊂ [ g ( c ) , h ( c )], we have h ◦ g ( c ) > g ◦ h ( c ). (cid:3) Note that, in this section, we omit the character for a composition, ◦ , that is, gh ( c ) implies g ◦ h ( c ). Under these setting, the following theorem holds. Theorem 4.2.
For the family of transformations { T c } c ∈ (0 ,c ∗ ) defined by (23) , there exist c Ln,l , c
Rn,l ∈ (0 , c ∗ ) for any n ∈ N ≥ and l ∈ P r ( n ) such that (i) if c ∈ ( c Ln,l , c
Rn,l ) , then T c conjugates S α,β with ( α, β ) ∈ int ( D n,l ) , (ii) if n (cid:48) l − nl (cid:48) = 1 , then c Rn,l < c Ln + n (cid:48) ,l + l (cid:48) < c Rn + n (cid:48) ,l + l (cid:48) < c Ln (cid:48) ,l (cid:48) . ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 9
Proof.
Since T c is clearly satisfied the assumptions (A1), (A2) and (A3) in Theorem 3.3, we haveto show (A4) for the item (i). We use the special type of inductions based on the Farey series. (STEP 1) For the case ( c L , , c R , ).Setting C , := (0 , c ∗ ). Since gh ( C , ) = ( g (1) , g ( c ∗ )) ⊂ C , and hg ( C , ) = (1 , hg ( c ∗ )) ⊂ C , and the compositions gh and hg are contraction mappings , there exist unique points c (cid:48) ∈ gh ( C , ) ⊂ C , and c (cid:48)(cid:48) ∈ hg ( C , ) ⊂ C , such that c (cid:48) = gh ( c (cid:48) ) and c (cid:48)(cid:48) = hg ( c (cid:48)(cid:48) ) from the Banach’s fixed pointtheorem. Moreover, we have that, for any c ∈ C , ,( ∗ ) , : (cid:40) if c < c (cid:48) , then c < gh ( c )if c > c (cid:48) , then c > gh ( c ) and (cid:40) if c < c (cid:48)(cid:48) , then c < hg ( c )if c > c (cid:48)(cid:48) , then c > hg ( c ) . Assume that c (cid:48) ≥ c (cid:48)(cid:48) , then there exists ˆ c ∈ ( c (cid:48)(cid:48) , c (cid:48) ) such that ˆ c > hg (ˆ c ) and ˆ c < gh (ˆ c ), that is, hg (ˆ c ) < gh (ˆ c ) holds, which is contradict to non-overlapping condition (Remark 4.1). We then have c (cid:48) < c (cid:48)(cid:48) . By taking c (cid:48) and c (cid:48)(cid:48) as c L , and c R , respectively, we find that, for any c ∈ ( c L , , c R , ), gh ( c ) < c < hg ( c ) holds which means that c is not in [ g ( c ) , gh ( c )] ∪ [ hg ( c ) , h ( c )]. Therefore, T c conjugate S α,β with ( α, β ) ∈ int ( D , ) by Theorem 3.3. (STEP 2) For the case ( c L , , c R , ).The idea is similar to (STEP 1). In the case, setting C , = (0 , c L , ), we have g − ( C , ) =(0 , h ( c L , )), gh ( C , ) = ( g (1) , c L , ) and hg ( C , ) = (1 , hg ( c L , )) so that gh ( C , ) ⊂ g − ( C , ) and hg ( C , ) ⊂ g − ( C , ) hold. From the Banach’s fixed point theorem, there exists c (cid:48) ∈ ggh ( C , ) ⊂ C , and c (cid:48)(cid:48) ∈ ghg ( C , ) ⊂ C , such that c (cid:48) = ggh ( c (cid:48) ) and c (cid:48)(cid:48) = ghg ( c (cid:48)(cid:48) ). Moreover, we have that,for any c ∈ C , ,( ∗ ) , : (cid:40) if c < c (cid:48) , then c < ggh ( c )if c > c (cid:48) , then c > ggh ( c ) and (cid:40) if c < c (cid:48)(cid:48) , then c < ghg ( c )if c > c (cid:48)(cid:48) , then c > ghg ( c ) . Assume that c (cid:48) ≥ c (cid:48)(cid:48) , then there exists ˆ c ∈ ( c (cid:48)(cid:48) , c (cid:48) ) such that ˆ c > ghg (ˆ c ) and ˆ c < ggh (ˆ c ),that is, hg (ˆ c ) < gh (ˆ c ) holds, which is contradict to non-overlapping condition (Remark 4.1). Wethen have c (cid:48) < c (cid:48)(cid:48) . Take c (cid:48) and c (cid:48)(cid:48) as c L , and c R , respectively. Since ( c L , , c R , ) ⊂ C , , we have c ∈ [ g ( c ) , gh ( c )] ∪ [ hg ( c ) , h ( c )] by ( ∗ ) , for any c ∈ ( c L , , c R , ). Moreover, we find ggh ( c ) < c < ghg ( c ),by ( ∗ ) , , that is, gh ( c ) < g − ( c ) < hg ( c ) holds which means that g − ( c ) is not in [ g ( c ) , gh ( c )].Therefore, T c conjugate S α,β with ( α, β ) ∈ int ( D , ) by Theorem 3.3. (STEP 3) For the case ( c Ln,n − , c Rn,n − ).Assume that we have already found a interval ( c Ln − ,n − , c Rn − ,n − ) so that c Ln − ,n − = g n − gh ( c Ln − ,n − ).For the case ( c Ln,n − , c Rn,n − ), setting C n,n − = (0 , c Ln − ,n − ), we have g − ( n − ( C n,n − ) = (0 , h ( c Ln − ,n − )), gh ( C n,n − ) = ( g (1) , gh ( c Ln − ,n − ) and hg ( C n,n − ) = (1 , hg ( c Ln − ,n − )) so that gh ( C n,n − ) ⊂ g − ( n − ( C n,n − )and hg ( C ) ⊂ g − ( n − ( C ) hold. From the Banach’s fixed point theorem, there exists c (cid:48) ∈ g n − gh ( C n,n − ) ⊂ C n,n − and c (cid:48)(cid:48) ∈ g n − hg ( C n,n − ) ⊂ C n,n − such that c (cid:48) = g n − gh ( c (cid:48) ) and c (cid:48)(cid:48) = g n − hg ( c (cid:48)(cid:48) ). More-over, we have that, for any c ∈ C n,n − ,( ∗ ) n,n − : if c < c (cid:48) , then c < g n − gh ( c )if c > c (cid:48) , then c > g n − gh ( c )if c < c (cid:48)(cid:48) , then c < g n − hg ( c )if c > c (cid:48)(cid:48) , then c > g n − hg ( c ) . Assume that c (cid:48) ≥ c (cid:48)(cid:48) , then there exists ˆ c ∈ ( c (cid:48)(cid:48) , c (cid:48) ) such that ˆ c > g n − hg (ˆ c ) and ˆ c < g n − gh (ˆ c ),that is, hg (ˆ c ) < gh (ˆ c ) holds, which is contradict to non-overlapping condition (Remark 4.1). Wethen have c (cid:48) < c (cid:48)(cid:48) . Take c (cid:48) and c (cid:48)(cid:48) as c Ln,n − and c Rn,n − respectively. We know that( c Ln,n − , c Rn,n − ) ⊂ C n − ⊂ C n − ,n − ⊂ · · · ⊂ C , ⊂ C , . Then, for any c ∈ ( c Ln,n − , c Rn,n − ), by ( ∗ ) m,m − , we have g − ( m − ( c ) ∈ [ g ( c ) , gh ( c )] for m = 2 , , · · · , n − , and by ( ∗ ) n,n − , g n − gh ( c ) < c < g n − hg ( c ), that is, gh ( c ) < g − ( n − ( c ) < hg ( c ) holds which meansthat g − ( n − ( c ) is not in [ g ( c ) , gh ( c )] ∪ [ hg ( c ) , h ( c )]. Therefore, T c conjugate S α,β with ( α, β ) ∈ int ( D , ) by Theorem 3.3. (STEP 4) For the case ( c L , , c R , ).Under setting C = ( c R , , c ∗ ), we can show the existences of c L , and c R , by the same way as(STEP 2) and substituting the roles of g and h . (STEP 5) For the case ( c Ln, , c Rn, ).Under setting C = ( c Rn − , , c ∗ ), we can show the existences of c Ln, and c Rn, by the same way as(STEP 3) and substituting the roles of g and h . (STEP 6) For the case ( c Ln,l , c
Rn,l ).We show that for any n ∈ N ≥ and l ∈ P r ( n ), there exist c Ln,l , c
Rn,l such that(i) ( n,l ) : c Ln,l = v v · · · v n − gh ( c Ln,l ) and c Rn,l = v v · · · v n − hg ( c Rn,l ),(ii) ( n,l ) : (cid:40) if c < c Ln,l , then c < v v · · · v n − gh ( c )if c > c Ln,l , then c > v v · · · v n − gh ( c ) , (cid:40) if c < c Rn,l , then c < v v · · · v n − hg ( c )if c > c Rn,l , then c > v v · · · v n − hg ( c ) , (iii) ( n,l ) : c Ln,l < c
Rn,l ,(iv) ( n,l ) : for c ∈ ( c Ln,l , c
Rn,l ), gh ( c ) < ( v v · · · v n − ) − ( c ) < hg ( c ),where { v i } n − i =1 is defined by v i := (cid:40) h if k i = 0 g if k i = 1 , (24)with { k i } is a rational characteristic sequence corresponding to ( n, l ).To show the above statement, we assume that the above holds for ( n, l ) and ( n (cid:48) , l (cid:48) ) with n (cid:48) l − nl (cid:48) =1. Then we will show the above statement for ( n + n (cid:48) , l + l (cid:48) ). Let { v i } n − i =1 and { v (cid:48) i } n (cid:48) − i =1 be given by(24) with respect to ( n, l ) and ( n (cid:48) , l (cid:48) ) respectively. Set C = ( c Rn,l , c Ln (cid:48) ,l (cid:48) ). From the assumption, weknow c Rn,l = v v · · · v n − hg ( c Rn,l ) , (25) c Ln (cid:48) ,l (cid:48) = v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c Ln (cid:48) ,l (cid:48) ) . (26) ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 11
First we show gh ( C ) ⊂ ( v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − ghv v · · · v n − ) − ( C ) (27) hg ( C ) ⊂ ( v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − ghv v · · · v n − ) − ( C ) (28)Since c Rn,l < c Ln (cid:48) ,l (cid:48) , by (ii) ( n (cid:48) ,l (cid:48) ) , we have c Rn,l < v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c Rn,l ), that is,( v (cid:48) · · · v (cid:48) n (cid:48) − ) − ( c Rn,l ) < gh ( c Rn,l )( v (cid:48) · · · v (cid:48) n (cid:48) − ) − g − h − ( v · · · v n − ) − ( c Rn,l ) < gh ( c Rn,l ) by (25)( v · · · v n − hgv (cid:48) · · · v (cid:48) n (cid:48) − ) − ( c Rn,l ) < gh ( c Rn,l )( v (cid:48) · · · v (cid:48) n (cid:48) − ghv · · · v n − ) − ( c Rn,l ) < gh ( c Rn,l ) (29)where we use the calculations pointed out in Remark 2.3.Since c Ln (cid:48) ,l (cid:48) > c Rn,l , by (ii) ( n,l ) , we have c Ln (cid:48) ,l (cid:48) > v v · · · v n − hg ( c Ln (cid:48) ,l (cid:48) ), that is,( v · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > hg ( c Ln (cid:48) ,l (cid:48) )( v · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > hgv (cid:48) · · · v (cid:48) n (cid:48) − gh ( c Ln (cid:48) ,l (cid:48) ) by (26)( v (cid:48) · · · v (cid:48) n (cid:48) − ) − g − h − ( v · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > gh ( c Ln (cid:48) ,l (cid:48) )( v · · · v n − ghv · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > gh ( c Ln (cid:48) ,l (cid:48) )( v (cid:48) · · · v (cid:48) n (cid:48) − ghv · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > gh ( c Ln (cid:48) ,l (cid:48) ) . (30)where we use the calculations pointed out in Remark 2.3. (29) and (30) imply (27). Similarly, since c Rn,l < c Ln (cid:48) ,l (cid:48) , by (ii) ( n (cid:48) ,l (cid:48) ) , we have c Rn,l < v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c Rn,l ), and by (25) and the calculation inRemark 2.3, ( v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − ghv v · · · v n − ) − ( c Rn,l ) < hg ( c Rn,l ) . (31)Since c Ln (cid:48) ,l (cid:48) > c Rn,l , by (ii) ( n,l ) , we have c Ln (cid:48) ,l (cid:48) > v v · · · v n − hg ( c Ln (cid:48) ,l (cid:48) ), and by (25) and the calculationin Remark 2.3, ( v (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − ghv v · · · v n − ) − ( c Ln (cid:48) ,l (cid:48) ) > hg ( c Ln (cid:48) ,l (cid:48) ) by (26) . (32)(31) and (32) imply (28).Thus, from the Banach’s fixed point theorem, there exists c (cid:48) ∈ v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( C ) ⊂ C and c (cid:48)(cid:48) ∈ v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − hg ( C ) ⊂ C such that c (cid:48) = v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c (cid:48) )and c (cid:48)(cid:48) = v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − hg ( c (cid:48)(cid:48) ). Moreover, we have that (cid:40) if c < c (cid:48) , then c < v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c )if c > c (cid:48) , then c > v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c ) , (cid:40) if c < c (cid:48)(cid:48) , then c < v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − hg ( c )if c > c (cid:48)(cid:48) , then c > v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − hg ( c ) . Assume that c (cid:48) ≥ c (cid:48)(cid:48) , then there exists ˆ c ∈ ( c (cid:48)(cid:48) , c (cid:48) ) such that ˆ c > v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − hg (ˆ c )and ˆ c < v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh (ˆ c ), that is, hg (ˆ c ) < gh (ˆ c ) holds, which is contradict to non-overlapping condition (Remark 4.1). We then have c (cid:48) < c (cid:48)(cid:48) . By taking c (cid:48) and c (cid:48)(cid:48) as c Ln + n (cid:48) ,l + l (cid:48) and c Rn + n (cid:48) ,l + l (cid:48) respectively, we find that, for any c ∈ ( c Ln + n (cid:48) ,l + l (cid:48) , c Rn + n (cid:48) ,l + l (cid:48) ), v v · · · v n − hgv (cid:48) v (cid:48) · · · v (cid:48) n (cid:48) − gh ( c ) Remark 4.3. If c = c Ln,l (or c Rn,l ), we can prove similarly that T c conjugates S α,β with β = B Ln,l ( α )(or B Un,l ( α )). Example 4.4. The family of maps { T c } defined by (23) for g ( x ) = α ( x + sin x ). Since g (cid:48) ( x ) = α ( + cos x ) and 0 < g (cid:48) ( x ) < 1, the map T c always satisfies the assumptions for Theorem 4.2. Themaximum value c ∗ depends on α and the relation α = c ∗ − c ∗ +sin c ∗ holds. In figure 3, we display theperiodic structure for the system. The number in each region implies the period. Figure 3. The periodic structure for the family of maps { T c } defined by (23) to-gether with g ( x ) = α ( x + sin x ).In the end of this section, we give another type of result with Theorem 4.2 in order to apply totransformations such as T ( x ) = αx + β (mod 1) or T ( x ) = α √ x + β (mod 1), which are introducedin [10].Let f : [0 , → [0 , 1] be a monotonically increasing continuous function such that f (0) = 0, f (1) < | f ( x ) − f ( y ) | ≤ κ | x − y | for some κ < x, y ∈ [0 , , 1] by T c ( x ) = (cid:40) f ( x ) − f ( c ) + 1 (0 ≤ x < c ) f ( x ) − f ( c ) ( c ≤ x < . (33) Theorem 4.5. For the family of transformations { T c } c ∈ (0 , defined by (33) , there exist c Ln,l , c Rn,l ∈ (0 , for any n ∈ N ≥ and l ∈ P r ( n ) such that (i) if c ∈ ( c Ln,l , c Rn,l ) , then T c conjugates S α,β with ( α, β ) ∈ int ( D n,l ) . (ii) if n (cid:48) l − nl (cid:48) = 1 , then c Rn,l < c Ln + n (cid:48) ,l + l (cid:48) < c Rn + n (cid:48) ,l + l (cid:48) < c Ln (cid:48) ,l (cid:48) .Proof. For the case ( n, l ) = (2 , C , = (0 , h c ( x ) = f ( x ) − f ( c ) − g c ( x ) = f ( x ) − f ( c ), let H ( c ) = g c h c ( c ) and G ( c ) = h c g c ( c ). In the proof of Theorem 4.2,substituting the role of hg and gh by G and H , we have G ( C , ) ⊂ C , and H ( C , ) ⊂ C , . Thenwe have c (cid:48) and c (cid:48)(cid:48) such that c (cid:48) = H ( c (cid:48) ) and c (cid:48)(cid:48) = G ( c (cid:48)(cid:48) ) by the Banach fixed point theorem. Bythe same way, taking c (cid:48) and c (cid:48)(cid:48) as c L , and c R , , we can show that for any c ∈ ( c L , , c R , ), inequality H ( c ) < c < G ( c ) holds which means that c is not in [ g c ( c ) , g c h c ( c )] ∪ [ h c g c ( c ) , h ( c )]. Therefore, T c conjugates S α,β with ( α, β ) ∈ int ( D , ) by Theorem 3.3. ERIODIC STRUCTURE FOR PIECEWISE CONTRACTING MAP 13 For any case ( n, l ), setting H ( c ) = v c v c · · · v cn − g c h c ( c ) and G ( c ) = v c v c · · · v cn − h c g c ( c ), where v ci = h c (if k i = 0) or g c (if k i = 1), we can give a similar proof with Theorem 4.2. (cid:3) Example 4.6. Consider the family of maps { T c } defined by (33) for f ( x ) = αx . Then, putting β = 1 − αc , the map becomes T c ( x ) = αx + β (mod 1), which is a one of numerical example in[10]. Since f (cid:48) ( x ) = 2 αx , the map T c satisfies the assumptions of Theorem 4.5 if α < / 2. In figure4, we display the periodic structure for the system. The number in each region implies the period.Although we can see the Farey structure for α ≥ / < α < / 2. Because Theorem 3.3 does not require the contractingproperty, and we can apply the Banach fixed point theorem in the proof of Theorem 4.2 if thetotal derivative of v · · · v n ghv (cid:48) · · · v (cid:48) n (cid:48) is less than one, we expect Theorem 4.2 can be hold underweakened condition. Figure 4.