Ultradiscretization of solvable one-dimensional chaotic maps

Ultradiscretization of solvable one-dimensional chaotic maps K enji K ajiw ara 1 , A tsushi N obe 2 and T er uhisa T sud a 1 1 Faculty of Mathematics, Kyushu Uni versity , 6-10- 1 Hakozaki, Fukuoka 812-8 581, Japan 2 Graduate School of Engineerin g S cience, Osaka University , 1-3 Machikaneyama- cho T oyonaka, Osaka 560-853 1, Japan Abstract W e consider the ultradiscretization of a solvable one-dimensional chaotic map which arises fr om the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously . A geometric interpretation of the dynamics of the tent map is gi v en in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding t o the m -th multipli cation formula of the elliptic functions is also discussed . 1 Introd uction In this article, we consider the following map z n + 1 = f ( z n ) = 4 z n (1 − z n )(1 − k 2 z n ) (1 − k 2 z 2 n ) 2 , (1.1) which admits the general solution z n = sn 2 (2 n u 0 ; k ) , (1.2) describing the orb it in [0 , 1]. Here sn( u ; k ) is Jacobi’ s sn function , 0 < k < 1 is the modulus, and u 0 is an ar bitrary constant. In fact, (1.1) can be reduced to the duplication formu la of sn functio n: sn(2 u ; k ) = 2sn( u ; k ) cn( u ; k ) dn( u ; k ) 1 − k 2 sn 4 ( u ; k ) , (1.3) cn 2 ( u ; k ) = 1 − sn 2 ( u ; k ) , dn 2 ( u ; k ) = 1 − k 2 sn 2 ( u ; k ) , (1.4) where cn( u ; k ) an d d n( u ; k ) are Jacobi’ s cn and dn function s, respectively . The map (1. 1) is a gen eralization o f th e logistic map (or Ulam-von Neumann map): z n + 1 = 4 z n (1 − z n ) , z n = sin 2 (2 n u 0 ) . (1.5) The map (1.1) was first conside red by Schr ¨ oder [28] in 1871, an d it has been studied by many author s [1 0, 1 2, 38, 39]. It is now classified as one of the (flexible) Latt ` e s maps [21]. In this article, we call (1.1) the Schr ¨ oder map . It is well-known that th e Schr ¨ oder map is conjugate to the tent map for X n ∈ [0 , 1 ] X n + 1 = T 2 ( X n ) = 1 − 2      X n − 1 2      =        2 X n 0 ≤ X n ≤ 1 2 , 2(1 − X n ) 1 2 ≤ X n ≤ 1 . (1.6) Namely , we have th e relation s ◦ f ◦ s − 1 = T 2 , s ( z ) = 1 K ( k ) sn − 1 ( √ z ; k ) , (1.7) where K ( k ) is the comp lete elliptic integral o f the first kind K ( k ) = Z 1 0 d x p (1 − x 2 )(1 − k 2 x 2 ) . (1.8) The p urpose o f th is ar ticle is to establish a new relationship between the Schr ¨ oder m ap an d th e ten t m ap th rough a certain limiting pro cedure called the ultradiscr etization [35]. The method of ultradiscretizatio n has achie ved a great success in the th eory o f integrable systems. From the integrable di ff eren ce equations, various interesting piecewise 1 linear dynam ical system s have been construc ted systematically , such as the s oliton cellular auto mata [4, 6, 17, 22, 29, 30, 31, 34, 37, 4 1] and pie cewis e lin ear version of th e Qu ispel-Roberts-Th ompson (QR T) maps [23, 2 6, 32, 3 6]. The resulting piecewise linear discrete dynamical s ystems can be expr essible in terms of the max and ± operatio ns, which we call the ultradiscrete systems. The ke y of the method is that one can o btain not only the equations but a lso their solutions simultaneou sly . It also allo ws us to unde rstand the underlying mathematical structures of the ultradiscrete systems [2, 3, 5, 8, 11, 13, 14, 15, 16, 25, 33]. In this article, we ap ply the ultra discretization to the Schr ¨ od er map (1.1) and its elliptic solution (1.2). As a r esult they are redu ced to the ten t map and its solution . W e also clarif y the tro pical geom etric nature of th e tent map; we show t hat the tent map can be regard ed as the du plication map on the Jacobian of a certain tropical curve. 2 Ultradiscr etization of the Schr ¨ oder map The key o f the ultradiscr etization is the following formula: lim ǫ → + 0 ǫ log  e A ǫ + e B ǫ + · · ·  = max( A , B , · · · ) , (2.1) where the ter ms in log mu st be po siti ve, and the do minant term survives under the limit. W e note that the o rbit of the map ( 1.1) is always r estricted in [0 , 1] if th e in itial value is in this in terval. Since this is somewhat too restrictiv e fo r ultradiscretization , we apply the fractional linear transform ation z n 7− → x n = z n 1 − z n , (2.2) which maps [0 , 1] → [ 0 , ∞ ). Th en the Schr ¨ oder map (1.1) and its solution (1.2) are re written as x n + 1 = φ ( x n ) = 4 x n (1 + x n )  1 + k ′ 2 x n   1 − k ′ 2 x 2 n  2 , k ′ 2 = 1 − k 2 , (2.3) x n = z n 1 − z n = sn 2 (2 n u 0 ; k ) 1 − sn 2 (2 n u 0 ; k ) = sn 2 (2 n u 0 ; k ) cn 2 (2 n u 0 ; k ) , (2.4) respectively . W e note th at the map (2 .3) can be o btained f rom (1 .1) b y re placing as z n − → − x n , k − → k ′ = √ 1 − k 2 . On the le vel of solution, this correspon ds to Jacob i’ s imaginar y transformation − i sn ( iu ; k ′ ) = sn( u ; k ) cn( u ; k ) . (2.5) Figure 1 shows th e m ap fu nctions of ( 1.1) and (2. 3). No te that f ( z ) and φ ( x ) have pole s at z = ± 1 / k and x = ± 1 / k ′ , respectively . -0.5 0.5 1 1.5 z -0.5 0.5 1 f H z L 1 2 3 4 5 x 2 4 6 8 10 Φ H x L Figure 1: Map fun ctions of (1.1) (left: k = 0 . 7) and (2.3)(right: k ′ = 0 . 8) Now we put x n = exp  X n ǫ  , k ′ = exp  − L 2 ǫ  , ( 0 < k ′ < 1 , L > 0) . (2.6) 2 Then (2.3) is rewritten as X n + 1 = F ǫ ( X n ) = ǫ log        4 e X n ǫ (1 + e X n ǫ )(1 + e X n − L ǫ ) (1 − e 2 X n − L ǫ ) 2        . (2.7) T akin g the limit ǫ → + 0 by using the formu la (2.1 ), we obtain X n + 1 = F ( X n ) = X n + m ax(0 , X n ) + ma x(0 , X n − L ) − 2 max (0 , 2 X n − L ) =                    X n X n < 0 , 2 X n 0 ≤ X n < L 2 , − 2 X n + 2 L L 2 ≤ X n < L , − X n + L L ≤ X n . (2.8) L 2 L X n L X n + 1 -1 -0.5 0.5 1 1.5 2 2.5 x -1 -0.5 0.5 1 1.5 2 2.5 F Ε H x L Figure 2: Lef t: map function of the u ltradiscrete Sch r ¨ oder map (2.8). Right: limit tr ansition o f the map fu nction F ǫ ( X ) for L = 1 . 5. Dashed line: ǫ = 0 . 3, dot-d ashed line: ǫ = 0 . 1, solid line: ǫ = 0 . 01. Remark 2.1 Although the te rms in log in the formula (2.1) must be positive in gene ral, th e ne gative terms ca n also exis t as lon g as the y ar e n ot dominan t in the limit. F o r example, we ha ve lim ǫ → + 0 ǫ log  e A ǫ − e B ǫ  2 = lim ǫ → + 0 ǫ log  e 2 A ǫ − 2 e A + B ǫ + e 2 B ǫ  = 2 max( A , B ) . (2.9) W e call the map (2 .8) the ultr adiscr ete Schr ¨ oder map . Figur e 2 sh ows the map function of (2.8) and limit transition of the function F ǫ ( X ). Th e dynamics of the map (2.8) is described as follows: if the initial value X 0 is in [0 , L ], the map is th e tent m ap and X n ∈ [0 , L ] f or all n . If X 0 ∈ ( −∞ , 0], then X n = X 0 for all n ≥ 1. Finally if X 0 ∈ [ L , ∞ ), then X 1 = − X 0 + L < 0 and X n = X 1 for all n ≥ 1. Theref ore the ultradiscrete Schr ¨ o der ma p (2.8) is essentially the tent map on [0 , L ] X n + 1 = L 1 − 2      X n L − 1 2      ! , X n ∈ [0 , L ] , (2.10) and otherwise the dynamics is trivial. Now let us c onsider the limit of the solution by using the ultradiscretization of the elliptic theta function s [32 ](see also [ 14, 2 4, 25]). Jaco bi’ s elliptic fun ctions are expressed in term s o f th e elliptic theta f unctions ϑ i ( ν ) ( i = 0 , 1 , 2 , 3) as sn( u ; k ) = ϑ 3 (0) ϑ 1 ( ν ) ϑ 2 (0) ϑ 0 ( ν ) , c n( u ; k ) = ϑ 0 (0) ϑ 2 ( ν ) ϑ 2 (0) ϑ 0 ( ν ) , (2.11) u = π ( ϑ 3 (0)) 2 ν , k 2 = ϑ 2 (0) ϑ 3 (0) ! 4 , (2.12) 3 where ϑ 0 ( ν ) = X n ∈ Z ( − 1) n q n 2 z 2 n , (2.13) ϑ 1 ( ν ) = i X n ∈ Z ( − 1) n q ( n − 1 / 2) 2 z 2 n − 1 , (2.14) ϑ 2 ( ν ) = X n ∈ Z q ( n − 1 / 2) 2 z 2 n − 1 , (2.15) ϑ 3 ( ν ) = X n ∈ Z q n 2 z 2 n , (2.16) and z = exp[ i πν ]. W e p arametrize the nome q as q = exp " − ǫ π 2 θ # , θ > 0 . (2.17) Applying Jacobi’ s imaginary transformatio n (or Poisson’ s summation for mula) the elliptic theta f unctions are rewritten as ϑ 0 ( ν ) = r θ ǫ π X n ∈ Z exp        − θ ǫ ( ν − n + 1 2 !) 2        , (2.18) ϑ 1 ( ν ) = r θ ǫ π X n ∈ Z ( − 1) n exp        − θ ǫ ( ν − n + 1 2 !) 2        , (2.19) ϑ 2 ( ν ) = r θ ǫ π X n ∈ Z ( − 1) n exp  − θ ǫ ( ν − n ) 2  , (2.20) ϑ 3 ( ν ) = r θ ǫ π X n ∈ Z exp  − θ ǫ ( ν − n ) 2  . (2.21) Asymptotic behaviour of these functions for ǫ → + 0 is given by ϑ 0 (0) ∼ 2 r θ ǫ π exp  − θ 4 ǫ  , (2.22) ϑ 2 (0) ∼ r θ ǫ π  1 − 2 exp  − θ ǫ  , (2.23) ϑ 3 (0) ∼ r θ ǫ π  1 + 2 exp  − θ ǫ  , (2.24) ( ϑ 0 ( ν )) 2 ∼ θ ǫ π exp        − 2 θ ǫ ( (( ν )) − 1 2 ) 2        , (2.25) ( ϑ 1 ( ν )) 2 ∼ θ ǫ π exp        − 2 θ ǫ ( (( ν )) − 1 2 ) 2        , (2.26) ( ϑ 2 ( ν )) 2 ∼ θ ǫ π  exp  − θ ǫ { (( ν )) } 2  − exp  − θ ǫ { (( ν )) − 1 } 2  2 , (2.27) where (( ν )) is the decimal part of ν , namely , (( ν )) = ν − Floor( ν ) , 0 ≤ (( ν )) < 1 . (2.28) Then we have k ′ 2 = exp  − L ǫ  = 1 − k 2 = 1 − ϑ 2 (0) ϑ 3 (0) ! 4 ∼ 16 e xp h − θ ǫ i  1 + 4 exp h − 2 θ ǫ i  1 + 2 exp h − θ ǫ i 4 , x n = exp  X n ǫ  = sn 2 ( u ; k ) cn 2 ( u ; k ) = ϑ 3 (0) ϑ 1 ( ν ) ϑ 0 (0) ϑ 2 ( ν ) ! 2 ∼  1 + 2 exp h − θ ǫ i 2 exp h 2 θ (( ν )) ǫ i 4  1 − exp h θ ǫ [2(( ν )) − 1] i 2 , 4 which yield in the limit ǫ → + 0 L = θ , (2.29) X n = θ 1 − 2      (( ν )) − 1 2      ! , ν = 2 n ν 0 , (2.30) respectively , wh ere ν 0 is an arbitrary constan t. W e note that in ta king th e limit of x n , we have put the arbitrar y co nstant u 0 as u 0 = θ ǫ ν 0 (2.31) so that ν = 2 n u 0 π ( ϑ 3 (0)) 2 = 2 n ν 0 θ ǫ π ( ϑ 3 (0)) 2 − → 2 n ν 0 ( ǫ → + 0) . (2.32) One ca n verify th at (2 .29) and (2 .30) actually satisfy the u ltradiscrete Schr ¨ oder map (2.8) or (2.10) by d irect calcu la- tion. Th erefor e we ha ve shown that throug h the ultradiscretization the Schr ¨ od er map (2 .3) and its solu tion (2.4) y ield the map (2.8) (or (2.10)) and its solution (2.30) simultaneou sly . Remark 2.2 (1) The fundamenta l period s of sn 2 ( u ; k ) cn 2 ( u ; k ) ar e 2 K ( k ) and 2 i K ( k ′ ) . In the ultr adiscr etization of the elliptic theta functions, we have parametrized the nome q as (2.17), which implies that the ratio o f half-period τ is given by τ = i ǫ π θ and K ( k ) = π 2 ( ϑ 3 (0)) 2 ∼ θ 2 ǫ , K ( k ′ ) = − π i 2 ( ϑ 3 (0)) 2 τ = π 2 ǫ 2 θ ( ϑ 3 (0)) 2 ∼ π 2 , (2.33) as ǫ → + 0 . Since we h ave u = θ ǫ ν , the fund amental periods with respect to ν tend to 1 and i ǫ π θ as ǫ → + 0 . This implies that the ultradiscr etization of the ellip tic function s is r e alized by co llapsing the imaginary period and keeping the r eal period fin ite. (2) The S chr ¨ o der map (1.1) is reduced to the logistic map (1.5) for k = 0 . Th is co rr esponds to the ultradiscr ete Schr ¨ oder ma p (2.8) with L = 0 , X n + 1 = − | X n | , (2.34) whose dyna mics is trivial, and the so lution (2.30) b ecomes X n = 0 . Ther efo r e ultradiscr etization of the logistic map do es not yield an interes ting ma p [9]. In fa ct, we see that this case is not consistent with the u ltradiscr ete limit, since the asymptotic behaviour of K ( k ) and K ( k ′ ) as k → 0 is g iven by K ( k ) ∼ π 2 , K ( k ′ ) ∼ log 4 k . (2.35) One can ap ply the sam e proced ure to the f ollowing map which o riginates from th e triplication for mula of sn 2 [12, 21, 39] z n + 1 = g ( z n ) = z n n k 4 z 4 n − 6 k 2 z 2 n + 4 ( k 2 + 1) z n − 3 o 2 n 3 k 4 z 4 n − 4 k 2 ( k 2 + 1 ) z 3 n + 6 k 2 z 2 n − 1 o 2 , z n = sn 2 (3 n u 0 ; k ) , (2.36) which is rewritten as x n + 1 = γ ( x n ) = x n n k ′ 4 x 4 n − 6 k ′ 2 x 2 n − 4 ( k ′ 2 + 1 ) x n − 3 o 2 n 3 k ′ 4 x 4 n + 4 k ′ 2 ( k ′ 2 + 1 ) x 3 n + 6 k ′ 2 x 2 n − 1 o 2 , x n = sn 2 (3 n u 0 ; k ) cn 2 (3 n u 0 ; k ) , (2.37) by the transfor mation (2.2). The m ap f unctions g ( z ) a nd γ ( x ) are illustrated in figu re 3. Th en ultrad iscretization of (2.37) yields the map X n + 1 = G ( X n ) = X n + 2 max(0 , X n , 4 X n − 2 L ) − 2 max(0 , 3 X n − L , 4 X n − 2 L ) =                          X n X n < 0 , 3 X n 0 ≤ X n < L 3 , − 3 X n + 2 L L 3 ≤ X n < 2 L 3 , 3 X n − 2 L 2 L 3 ≤ X n < L , X n L ≤ X n , (2.38) 5 -0.2 0.5 1 1.5 z -0.5 0.5 1 1.5 2 g H z L 2 4 6 x -1 -0.5 0.5 1 1.5 2 2.5 3 Γ H x L Figure 3: Map fun ctions of (2.36) (left: k = 0 . 7 ) and (2.37)(right: k ′ = 0 . 8) and its solution X n = L 1 − 2      (( ν )) − 1 2      ! , ν = 3 n ν 0 . (2.39) Figure 4 shows the map function G ( X n ) and the limit transition of the map function of X n + 1 = G ǫ ( X n ) = ǫ log           e X n ǫ n e 4 X n − 2 L ǫ − 6 e 2 X n − L ǫ − 4 ( e − L ǫ + 1 ) e X n ǫ − 3 o 2 n 3 e 4 X n − 2 L ǫ + 4 ( e − 2 L ǫ + e − L ǫ )) e 3 X n ǫ + 6 e 2 X n − L ǫ − 1 o 2           . (2.40) W e note that one can directly u ltradiscretize th e map (2.36) to obtain (2 .38), h owe ver , the solution x n = sn 2 (3 n u 0 ; k ) L 3 2 L 3 L X n L X n + 1 -1 -0.5 0.5 1 1.5 2 2.5 x -1 -0.5 0.5 1 1.5 2 2.5 G Ε H x L Figure 4: Left: map function of the map (2.38). Right: limit transition of the map function G ǫ ( X ) for L = 1 . 5 . Dashed line: ǫ = 0 . 3, dot-dashe d line: ǫ = 0 . 1, solid line: ǫ = 0 . 01. degenerates to th e trivial solu tion X n = 0. Thu s it is important to consider (2 .37) in o rder to obtain the lim it w hich is consistent with the solution. It is possible to apply ultradiscretization to the m aps ar ising f rom th e m -th multiplicatio n form ula of sn 2 [12, 21] in a similar manner . 3 Geometric description in terms of the trop ical geom etry It is shown in [5, 25] that the tropical g eometry p rovides a geometric fr amew ork fo r the description o f the ultradiscrete integrable systems. Therefor e it may b e natural to expect that a similar framework also works well for our case. In this section, we sho w that the ultradiscrete Schr ¨ od er map can b e interpreted as the duplication map on the Jacobian of a certain tropical curve. As for the basic notions of the tropical geometry , we refe r to [1, 7, 27]. W e first con sider the elliptic curve  xy − b ( x + y ) + c  2 = 4 d 2 xy , (3.1) 6 parametrize d by ( x , y ) = sn 2 ( u ; k ) cn 2 ( u ; k ) , sn 2 ( u + η ; k ) cn 2 ( u + η ; k ) ! , (3.2) where η is a constant and a , b , d are gi ven by b = 1 k ′ 2 cn 2 ( η ; k ) sn 2 ( η ; k ) , c = 1 k ′ 2 , d = − 1 k ′ 2 dn( η ; k ) sn 2 ( η ; k ) , (3.3) respectively . Eliminating η in (3.3), we see that b and d satisfy the relatio n k ′ 2 d 2 = (1 + k ′ 2 b )(1 + b ) . (3.4) W e m ay regard the Schr ¨ oder map (2.3) as th e projectio n of the dynamics of th e point on the ellip tic curve (3.1) to the x -axis. W e next apply the ultradiscretization to the elliptic curve. Putting x = e X ǫ , y = e Y ǫ , b = e B 2 ǫ , 4 d 2 = e D ǫ , k ′ = e − L 2 ǫ , c = 1 k ′ 2 = e L ǫ , L > 0 , (3.5) and taking the limit ǫ → + 0, (3.1) and (3.4) yield max(2 X + 2 Y , B + 2 X , B + 2 Y , 2 L ) = X + Y + D , (3.6) and − L + D = max  0 , B 2 − L  + m ax  0 , B 2  , (3.7) respectively . The cond ition (3.7) gi ves t he following three cases: (i) B > 2 L > 0 , D = B , (3. 8) (ii) 2 L > B > 0 , D = L + B 2 , (3.9) (iii) 0 > B , D = L . (3.10) For ea ch case, the set of points d efined by (3.6) is (i) a line connecting ( B 2 , B 2 ) and ( L − B 2 , L − B 2 ), (ii) a rectangle with vertices (0 , L − B 2 ), ( L − B 2 , 0), ( L , B 2 ) and ( B 2 , L ), (iii) a line connecting ( B 2 , L − B 2 ) and ( L − B 2 , B 2 ), respectively , as illustrated in figure 5. In the following, we consider only the case (ii) and we denote the rectangle as C . X Y L − B 2 L − B 2 B 2 B 2 X Y L − B 2 L L − B 2 B 2 B 2 L X Y L L L − B 2 L − B 2 B 2 B 2 Figure 5: Ultradiscre tization of the elliptic curve ( 3.1). Left: c ase (i), center: case (ii), righ t: case (iii). Let us recall some notions of the tropical geometry . Th e tropical curve defined by the tropical polynomial Ξ ( X , Y ) = max ( a 1 , a 2 ) ∈A ( λ ( a 1 , a 2 ) + a 1 X + a 2 Y ) , A ∈ Z 2 , (3.11) is a set of p oints ( X , Y ) ∈ R 2 where Ξ is n ot smooth. Here A is a fin ite subset of Z 2 called the sup port, and we deno te as ∆ ( A ) the con vex hull of A . Let Γ d be the triangle in Z 2 with vertices (0 , 0) , ( d , 0), (0 , d ). Th en the degree o f the 7 tropical cu rve is d if ∆ ( A ) is inside Γ d but no t inside Γ d − 1 [40]. The g enus of the trop ical curve is defined as th e first Betti number of the curve, namely the number of its c ycles [1, 18, 19]. W e con sider the tropical polynom ial Ψ ( X , Y ) = max(2 X + 2 Y , B + 2 X , B + 2 Y , 2 L , X + Y + D ) , (3.12) under the co ndition ( 3.9). Let C be the tro pical cu rve de fined b y Ψ , wh ich is illustrated in figure 6. Then the d egree and the genus of C are 4 and 1, respectively . Note that the rectang le C is exactly the c ycle of C . X Y L L − B 2 B 2 B 2 L − B 2 L V 1 V 2 V 3 V 4 A X Y ∆ ( A ) Figure 6: Left: tr opical curve C defined by (3.12). Right: suppor t o f (3.12). V igeland [40] h as successfully introduced th e group law on the tropical elliptic cur ve. Unfo rtunately , howe ver , his d efinition o f tropical elliptic curve is lim ited to “smooth” cu rve of degree 3 an d h ence it does n ot cover ou r ca se. Nev ertheless, it is p ossible to define the tro pical Jacob ian J ( C ) of C [5, 20] and characterize the dynamics o f the ultradiscrete Schr ¨ ode r map (2.1 0) on it in the following manner: let V i and E i ( i = 1 , . . . , 4) be the vertices and edges of C defined by V 1 = O =  0 , L − B 2  , V 2 =  L − B 2 , 0  , V 3 =  L , B 2  , V 4 =  B 2 , L  , (3.13) V 1 V 2 = E 1 , V 2 V 3 = E 2 , V 3 V 4 = E 3 , V 4 V 1 = E 4 , (3. 14) respectively . The length of each edge is giv en as | E 1 | = √ 2  L − B 2  , | E 2 | = √ 2 2 B , | E 3 | = √ 2  L − B 2  , | E 4 | = √ 2 2 B . (3.15) The primitive tang ent vector for each edge is v 1 = ( 1 , − 1) , v 2 = ( 1 , 1) , v 3 = ( − 1 , 1) , v 4 = ( − 1 , − 1) . (3.16) W e introd uce the total lattice length L as the su m of the length of each e dge scaled b y the len gth of correspon ding primitive tang ent vector , which is com puted as L = 4 X i = 1 | E i | | v i | = 2 L . (3.17) Then the tropical Jacobian J ( C ) is d efined by J ( C ) = R / L Z = R / 2 L Z . (3.18) The Abel-Jacobi map µ : C → J ( C ) is d efined as the piecewise lin ear map which is linear on each edge satisfying µ ( V 1 ) = 0 , µ ( V 2 ) = L − B 2 , µ ( V 3 ) = L , µ ( V 4 ) = 2 L − B 2 . (3.19) Let π : C → R b e th e pro jection of the point on C to the X -axis. Let ρ be the map defined by ρ = π ◦ µ − 1 : J ( C ) → R which map s µ ( P ) ( P ∈ C ) to the X -co ordinate of P . Here we no te that π − 1 is 1:2 and we d efine π − 1 ( X ) to be th e point on C whose Y -co ordinate is smaller . In this setting, ρ ( p ) ( p ∈ J ( C )) can be written as ρ ( p ) = ( π ◦ µ − 1 )( p ) =        p 0 ≤ p ≤ L , − p + 2 L L ≤ p ≤ 2 L , (3.20) 8 V 4 V 3 V 2 V 1 = O J ( C ) L L − B 2 B 2 L − B 2 L 2 L − B 2 X ρ 2 L J ( C ) 2 L π π − 1 µ µ − 1 ϕ 2 p ′ n p ′′ n X n X n +1 X L L Φ 2 Figure 7: Left: co rrespon dence betwee n X and J ( C ) by ρ . Right: du plication map ϕ 2 and Φ 2 . as shown in the left of figure 7. Now we define the duplication map ϕ 2 : J ( C ) → J ( C ) by ϕ 2 ( p ) ≡ 2 p (mod L ) , p ∈ J ( C ) , (3.21) and introdu ce Φ 2 : R → R as the con jugation map of ϕ 2 by ρ , Φ 2 = ρ ◦ ϕ 2 ◦ ρ − 1 . (3.22) In order to write down the map Φ 2 explicitly , we introduce p ′ , p ′′ ∈ J ( C ) for P = ( X , Y ) ∈ C b y p ′ = ρ − 1 ( X ) = ( µ ◦ π − 1 )( X ) = X , p ′′ = ϕ 2 ( p ′ ) = 2 p ′ = 2 X . (3.23) Then the map Φ 2 is expressed as follows (th e right of figure 7): (1) For 0 ≤ X ≤ L 2 : since 0 ≤ p ′′ ≤ L , (3.2 0) implies Φ 2 ( X ) = ρ ( p ′′ ) = 2 X . (3.24) (2) For L 2 ≤ X ≤ L : since L ≤ p ′′ ≤ 2 L , (3.20) implies Φ 2 ( X ) = ρ ( p ′′ ) = − 2 X + 2 L . (3.25 ) The dynam ical system X n + 1 = Φ 2 ( X n ) = L 1 − 2      X n L − 1 2      ! = ( 2 X n 0 ≤ X ≤ L 2 , − 2 X n + 2 L L 2 ≤ X ≤ L , (3.26) coinsides with the ultr adiscrete Schr ¨ oder map (2.10). Ther efore we ha ve shown th at the ultrad iscrete Schr ¨ oder map (2.10) can be regarded as the du plication ma p o n the Jacobian J ( C ) of th e tropical curve C defined b y th e tropical polyno mial (3.12). Similarly , we define the triplication map ϕ 3 : J ( C ) → J ( C ) by ϕ 3 ( p ) ≡ 3 p (mod L ) , p ∈ J ( C ) , (3.2 7) and introdu ce Φ 3 : R → R as the con jugation map of ϕ 3 by ρ , Φ 3 = ρ ◦ ϕ 3 ◦ ρ − 1 . (3.28) Then the corresp onding d ynamical system is giv en by X n + 1 = Φ 3 ( X n ) =              3 X n 0 ≤ X n ≤ L 3 , − 3 X n + 2 L L 3 ≤ X ≤ 2 L 3 , 3 X n − 2 L 2 L 3 ≤ X ≤ L = 3 X n − 2 max(0 , 3 X n − L ) + 2 max(0 , 3 X n − 2 L ) , (3.29) which is equiv alent to (2.38) on [0 , L ]. For general m , the m -th multiplicatio n map yields the dynamica l s ystem X n + 1 = Φ m ( X n ) = mX n + 2 m − 1 X i = 1 ( − 1) i max(0 , mX n − i L ) , (3.30) which may be regarded as the ultradiscretization of the map arising from the m -th multiplication formula of sn 2 cn 2 . 9 4 Concludin g Remarks In this article, we have presen ted a ne w relationship between tw o typical chaotic one-dim ensional m aps, the Schr ¨ oder map and the tent map, thro ugh the ultradiscr etization. Although the ultradiscretization ha s bee n de veloped in the theory of in tegrable system s, the results in this article imply th e possibility of ap plying the meth od to wider class o f dynamica l systems. Our results also suggest that the tropical geometry combined with the ultradiscretization provides a powerful tool to stud y a piecewise linear map, since the ultradiscretization translates the geometr ic backgro und of the original rational map into that of the corresp onding piece wise linear map. It would be an interesting problem to study various ultradiscrete or piece wise linear systems, such as ultradiscrete analogues of Painle v ´ e system s, gener alized QR T maps, and higher-dimen sional solvable chaotic maps in this direction. Ackno wledgement The authors would like to express their sincere thanks to Prof. Y utaka Ishii fo r stimulating discu ssions an d v alu- able informatio n. T his work was su pported by the JSPS Grant- in-Aid for Scientific Research 193400 39, 19740086 , 19840 039, and th e Grant for Basic Science Research Projects of the Sumitomo Foundation 071254. Refer ences [1] Gathmann A 2006 T ropical algebraic geometry Pr eprint math .A G / 0601 322 v1 [2] Hatayama G, Hikami K, Inoue R and K uniba A 2001 J. Math. Phys. 42 274- 308 [3] Hikami K, Inoue R and K omori Y 1999 J. Phys. Soc. Jpn. 68 2234-40 [4] Hirota R and T akahashi D 200 5 Glasgo w Math. J. 47 77-85 [5] Inoue R and T akenawa T 2 008 Int. Math. Res. 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