Probability density functions of some skew tent maps

Transcription

1 Chos, Solitons nd Frctls 12 (2001) 365±376 wwwelseviernl/locte/chos Probbility density functions of some skew tent mps L Billings, EM Bollt b, * Nvl Reserch Lbortory, Code 67003, Specil Project in Nonliner Science, Wshington, DC 20375, USA b Deprtment of Mthemtics, US Nvl Acdemy, Ntionl Science Foundtion Grnt DMS , 572 Hollowy Rod, Annpolis, MD , USA Accepted 9 December 1999 Abstrct We consider fmily of chotic skew tent mps The skew tent mp is two-prmeter, piecewise-liner, wekly-unimodl, mp of the intervl F ;b We show tht F ;b is Mrkov for dense set of prmeters in the chotic region, nd we exctly nd the probbility density function (pdf), for ny of these mps It is well known (Boyrsky A, Gor P Lws of chos: invrint mesures nd dynmicl systems in one dimension Boston: Birkhuser, 1997), tht when sequence of trnsformtions hs uniform limit F, nd the corresponding sequence of invrint pdfs hs wek limit, then tht invrint pdf must be F invrint However, we show in the cse of fmily of skew tent mps tht not only does suitble sequence of convergent sequence exist, but they cn be constructed entirely within the fmily of skew tent mps Furthermore, such sequence cn be found mongst the set of Mrkov trnsformtions, for which pdfs re esily nd exctly clculted We then pply these results to exctly integrte Lypunov exponents Ó 2000 Elsevier Science Ltd All rights reserved 1 Introduction Let F ;b denote the two-prmeter piecewise-liner mp on the intervl 0; 1Š stisfying b = x if 0 6 x < ; F ;b x ˆ 1 x = 1 if 6 x 6 1 1:1 with 0 < < 1 nd 0 6 b 6 1 It hs been shown tht [2] in the following region of the prmeter spce, D ˆf ; b : b < 1= 2 nd b < 1 or b P 1 nd b > g; 1:2 F ;b hs chotic dynmics, nd in the prmeter subset D 0 ˆf ; b : b < 1= 2 nd b < 1 g; 1:3 chotic dynmics occur on the entire intervl 0 6 x 6 1 See Bssein [2] for complete clssi ction of the dynmics in prmeter spce, ; b 2 0; 1 0; 1Š We will show tht F ;b is Mrkov for dense set of ; b in D 0 This is interesting becuse the probbility density function for ny Mrkov trnsformtion in this set cn be found exctly to be piecewise-constnt function We cn use these exct results to pproximte the probbility density function (pdf) for ny other trnsformtion in D 0 We lso show tht Lypunov exponents cn be clculted exctly on the Mrkov set, nd therefore, e ciently pproximted on ll of D 0 * Corresponding uthor /00/$ - see front mtter Ó 2000 Elsevier Science Ltd All rights reserved PII: S ( 9 9 )

2 366 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 In Section 2, we review su cient condition for piecewise-liner mp on the intervl to be Mrkov, nd we discuss symbolic dynmics for this type of mp Then, we prove tht Mrkov mps re dense in the prmeter set D 0 Illustrting the simpli ctions of Mrkov mps, we discuss in Section 3 the techniques used to exctly nd the pdf nd the Lypunov exponent We conclude the discussion of Mrkov mps with n explicit exmple in Section 4 In Section 5, we nish with proof tht the properties of the remining mps in the prmeter set D 0 cn be pproximted su ciently by members of the dense set of Mrkov mps We chose the form of F ;b from [2], wherein cn be found concise discussion of the full rnge of topologicl dynmics in the prmeter spce ; b 2 0; 1 0; 1Š The form of the skew tent mp we study, Eq (11), is topologiclly conjugte to the skew tent mp in the topologicl studies of Misiurewicz nd Visinescu [3,4] Piecewise-liner trnsformtions on the intervl hve been widely studied, nd under different nmes s well: broken liner trnsformtions [5], wek unimodl mps [6] Other closely relted res in the study of the chotic behvior of these mps include the stbility of the ssocited Frobenius±Perron opertor [7], the topologicl entropy [4], nd kneding sequences [8] 2 Mrkov trnsformtions In this section, we clssify the Mrkov prtitions which re prominent in clcultions in section 3 In the specil, but importnt, cse tht trnsformtion of the intervl is Mrkov, the symbol dynmics is simply presented s nite directed grph A Mrkov trnsformtion is de ned s follows De nition 21 ([1]) Let I ˆ c; dš nd let s : I! I Let P be prtition of I given by the points c ˆ c 0 < c 1 < < c p ˆ d For i ˆ 1; ; p, let I i ˆ c i 1 ; c i nd denote the restriction of s to I i by s i Ifs i is homeomorphism from I i onto some connected union of intervls of P, then s is sid to be Mrkov The prtition P is sid to be Mrkov prtition with respect to the function s The following result describes set in D 0 for which F ;b is Mrkov Theorem 22 For given ; b 2D 0,ifx 0 ˆ 1 is member of periodic orbit, then F ;b is Mrkov Proof Set F ˆ F ;b Assume x 0 ˆ 1 is member of period-n orbit n > 1 Next, form prtition of 0; 1Š using the n members of the periodic orbit The two endpoints of the intervl re included since 8 ; b 2D 0, F 1 ˆ0 Order these n points so 0 ˆ c 0 < c 1 < < c n 1 ˆ 1, regrdless of the itertion order For i ˆ 1; ; n 1, let I i ˆ c i 1 ; c i nd denote the restriction of F to I i by F i For given I i ˆ c i 1 ; c i, the endpoints c i 1 nd c i will mp exctly to two members of the prtition endpoints by de nition of the periodic orbit Let these points be c j nd c k, with c j < c k nd j; k 2f0; ; n 1g The only turning point of the mp is x ˆ, nd8 ; b 2D 0, F ˆ1 Therefore x ˆ must lwys be prt of the period orbit nd member of the prtition endpoints, implying ech F i is liner nd hence, homeomorphism Also F i I i ˆ c j ; c k, connected union of intervls of the prtition By de nition, F is Mrkov At this point, symbolic nottion becomes useful The point x ˆ is the criticl point t the ``center'' of the intervl, denoted by the letter C All < x 6 1 is right of, represented by R, nd ll 0 6 x < is left of, represented by L Represent ech step of the itertion mp by one of these three symbols All prmeter sets for which F x is Mrkov must hve period-n orbit contining the point x ˆ nd be of the form f; ; F n 1 ; g For exmple, the period-3 orbit hs the form f; 1; 0; g nd occurs for ny prmeter set ; b on the curve b ˆ It repets the pttern CRL, which we cll the kneding sequence K F ;b ˆ CRL 1 If periodic orbit contins the point x ˆ nd 6ˆ b, the point x ˆ b will either be greter or less thn For period-4 orbit with > b, the symbolic sequence must repet CRLL, or if < b, CRLR Therefore, period-4 is found two wys See Fig 1 for cobweb digrm of n exmple period-4 CRLL orbit

3 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365± Fig 1 Numericl clcultion of the PDF (histogrm brs) compred to the exct solution (solid line) The clcultion used itertions nd 50 intervls Repeting this method for period-5 nd higher, we see tht there re 2 n 3 possible combintions of the C±L±R sequences for period-n orbit which includes the criticl point The exponent n 3 re ects the necessry 3-step pre x CRL A full binry tree with 2 n 3 leves on ech tier is possible, ech implying condition on the prmeters nd forming countble set of curves in prmeter spce [4] We cn now restte Theorem 22 in terms of kneding sequences Corollry 23 If K F ;b is periodic, then F ;b is Mrkov We now prove tht the function F ;b x (11) is Mrkov on dense set of curves in D 0 (13) De ne R 2 s the spce of symbol sequences contining the full fmily of kneding sequences for two symbols De ne the kneding sequence r ˆ r 0 r 1 r 2 ; the metric d r; ^r ˆP1 P iˆ0 jr i ^r i j=2 i, nd the norm krk ˆ 1 iˆ0 r i=2 i ˆ r 0 r 1 =2 r 2 =2 2 Lemm 24 Periodic r re dense in R 2 Proof 8e > 0 nd given WOLOG / ˆ / 0 / 1 / 2 which is not periodic, 9N > 0 lrge enough so tht r i ˆ / i, i 6 N nd r ˆ / 0 / 1 / N 1 for kr /k R2 < e: 3 Probbility density function nd Lypunov exponents Another bene t of the set of Mrkov chotic functions is tht the pdfs cn be determined quite esily In fct, expnding piecewise-liner Mrkov trnsformtions hve piecewise-constnt invrint probbility density functions Theorem 31 [1, Theorem 942] Let s : I! I be piecewise-liner Mrkov trnsformtion such tht for some k P 1,

4 368 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 j s k 0 j > 1; wherever the derivtive exists Then s dmits n invrint probbilityš density function which is piecewiseconstnt on the prtition P on which s is Mrkov Using the Frobenius±Perron opertor P, the xed-point function h stis es the de nition P F h ˆ h, implying tht h is the pdf for mesure tht is invrint under F Since F is piecewise-monotone function, the ction of the opertor is simply P F h x ˆ X z2ff 1 x g h z jf 0 z j : The periodic orbit formed by the itertion of x ˆ forms prtition of the domin 0; 1Š on which h is piecewise-constnt On ech intervl I i, cll the corresponding constnt h i ˆ hj Ii The pdf dmits n bsolutely continuous invrint mesure on the Mrkov prtition This mesure cn be used to nd the Lypunov exponent, nd therefore quntify the verge rte of expnsion or contrction for n intervl under itertion Set F ˆ F ;b for some ; b 2D 0 nd form prtition of 0; 1Š using the n members of the periodic orbit, so 0 ˆ c 0 < c 1 < < c n 1 ˆ 1 Assume c k ˆ, for some k 2f1; ; n 2g Note jf 0 x j ˆ = if x <, nd jf 0 x j ˆ 1= 1 if x > K ;b ˆ ˆ Z 1 0 Z c1 c 0 ˆ ln ln jf 0 x jh x dx ln jf 0 x jh 1 dx X k iˆ1 Z cn 1 c n 2 1 c i c i 1 h i ln 1 ln jf 0 x jh n 1 dx X n 1 iˆk 1 c i c i 1 h i : 3:1 4 Following the left brnch As n exmple, in this section we will derive the probbility density function for F x when it is Mrkov nd hs periodic orbit of the pttern CRL; CRLL; CRLLL;, following the left brnch of the binry tree In the discussion below, x ˆ is prt of periodic orbit of period p ˆ n 3 nd n represents the number of Ls fter the initil CRL in the symbolic representtion of tht periodic orbit p z } { f; 1; 0; b; F b ; F 2 b ; g)crl LLL {z } n 4:1 Proposition 41 The generl reltion of nd b in the period p ˆ n 3 in Eq (41) is n 1 ˆ 1 : b 4:2 Proof Renme the brnches of the piecewise-de ned function, f L on 0 6 x < nd f R on 6 x 6 1 Using the geometric series expnsion, express fl n b s f n Xn L b ˆ b k kˆ0 b 1 = n 1 ˆ : 4:3 1 =

5 The prmeter sets for ech of these periodic orbits is found by solving fl n b ˆ, where the period is n 3 By using Eq (43), we hve the implicit reltion (42) It now seems quite nturl for the CRL orbit to occur on the prmeter set b ˆ The equtions relting to b for the di erent periodic orbits form pttern in prmeter spce, nd limit to b ˆ 0 On these curves in the chotic region, n invrint density function h cn be found in closed form It is piecewise-de ned function on its ssocited Mrkov prtitions Therefore, h x hs the form 8 h 1 if 0 < x < b; >< h 2 if b < x < F b ; h x ˆ : 4:4 >: h p 1 if < x < 1: Then the ction of the Frobenius±Perron opertor is 1 h fl P f h x ˆ x j fl 0 f L 1 x jv ;1Š x hf 1 R x fr 0 f R 1 j x j ˆ hf 1 L x v ;1Š x 1 h f 1 R x : 4:5 We use Eq (45) to construct system of equtions to solve for h x This system hs the following form: h 1 ˆ 1 h p 1 ; h 2 ˆ h 1 1 h p 1 ; h p 1 ˆ L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365± h p 2 1 h p 1 : For proof, see the geometry of the inverse function Note the recursion; the constnt h 1 depends only on h p 1 From the monotonicity of the inverse of f L nd f R, ech constnt, h k, depends on h k 1 nd h p 1 Proposition 42 The system of Eq (46) is underdetermined for the vribles h 1 ; h 2 ; ; h p 1 The determinnt is 0 precisely when the prmeters stisfy Eq (42) Proof This is proved by induction Bse cse: n ˆ 0ndp ˆ 3: h 1 ˆ 1 h 2 ; 4:7 h 2 ˆ h 1 1 h 2 : 4:8 This set of equtions cn be rewritten s the homogeneous system 1 1 h1 ˆ 0 : = :9 h 2 To nd if there is unique solution, nd the determinnt M 0 : 1 1 M 0 ˆ = 1 1 ˆ : 4:10 Substituting Eq (42) for n ˆ 0, b ˆ 0 implies tht M 0 ˆ 0 To prove M j 1 ˆ 0, use the reltion 4:6 M j ˆ 1 1 Xj kˆ0 k; 4:11

6 370 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 nd ssume Eq (42) is true for n ˆ j Therefore, M j 1 ˆ 1 1 Xj 1 k kˆ0! 1 = j 2 ˆ = 1 j 1 ˆ 1 b b 1 : 4:12 Substituting Eq (42) for n ˆ j, M j 1 ˆ 0: The freedom implied by Proposition 42 is expected, since the density function must be normlized Set the re under the curve to 1: Z 1 0 h x dx ˆ 1: 4:13 Using the prtition, 0 ˆ c 0 < c 1 < < c p 1 ˆ 1, we know c n 2 ˆ nd c k ˆ f k Xk i L 0 ˆb for k ˆ 1; ; p 3: 4:14 iˆ0 Therefore, Eq (413) simpli es to the following: c 1 c 0 h 1 c p 1 c p 2 h p 1 ˆ 1; 4:15 b Xp 3 iˆ0 i h i 1 1 h p 1 ˆ 1: 4:16 With this dditionl constrint, the system of equtions hs dimension p p h h 2 SB A ˆ B 0 A ; 1 h p 1 4:17 de ning the coe cient mtrix S of Eq (417) s the mostly bnded mtrix = 1 1 = 1 1 S ˆ : 4:18 = 1 1 B 0 = A b b = b = p 3 1 The next step is to prove tht this new system hs rnk p Nme the rows in the coe cient mtrix S s R 1 ; R 2 ; ; R p Using the de nition of liner dependence, there must exist set of constnts tht multiply the rows of the mtrix so tht they sum to 0: R 1 k 1 R 2 k 2 R p k p ˆ ~0: 4:19

7 Solving for these constnts, we nd i k p i ˆ k p 1 kp b i 1 i p 2 for i ˆ 2; ; p 1;! k p 1 ˆ 1 X p 2 k i 1 : iˆ1 There is only one degree of freedom nd the system hs rnk p 1 There is unique solution to the system, if one exists The closed forms for the constnts h i in the probbility density function re i h i ˆ 1 for i ˆ 1; ; p 2; 4:20 b p 1 1 b h p 1 ˆ 1 b p 1 : 4:21 We show this clcultion in Appendix A Exmple The CRLL fmily illustrtes the previous results The Mrkov prtition is formed by the period-4 orbit 0; b; ; 1 See Fig 1 for cobweb digrm of speci c period-4 orbit In prmeter spce, the fmily occurs on the curve ˆ f L b, or b ˆ 1 p : 4:22 2 Note tht the other brnch of the solution occurs outside the prmeter domin, ; b 2 0; 1 0; 1Š Eq (45) produces the system of equtions h 1 ˆ 1 h 3 ; h 2 ˆ h 3 ˆ h 1 1 h 3 ; h 2 1 h 3 : To normlize h, set the re under the curve to 1, bh 1 b h 2 1 h 3 ˆ 1: L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365± Hence, the invrint density function is 8 1 b if 0 6 x < b; >< 1 b 3b 1 b h x ˆ 2 2 if b 6 x < ; 1 b 1 b 3b >: 1 b if 6 x 6 1: 1 1 b 3b See Fig 1 for comprison of this exct result to histogrm generted by ``typicl'' orbit 4:23 4:24 Of prticulr use, the invrint mesure cn be used to exctly clculte the Lypunov exponent for ny set of prmeters ; b long the curves described by Eq (42) in D 0 WOLOG, ssign c k ˆ nd k ˆ p 2 Use Eqs (31), (420), nd (421) to derive the following: K ;b ˆ ln X k 1 X c i c i 1 h i ln p 1 1 c i c i 1 h i iˆ1 iˆk 1 ˆ ln 1 1 h 1 p 1 ln 1 1 h p 1 ˆ ln ln 1 1 h p 1 ˆ ln ln 1 1 b b p 1 : 4:25

8 372 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 Fig 2 The Lypunov exponent s function of the prmeter, where x ˆ 0 is member of period-4 orbit Exmple Continuing the period-4 CRLL exmple from bove, we derive the Lypunov exponent exctly Set p ˆ 4 nd in Eq (425) K ;b ˆ ln ln 1 b 1 : 4:26 3b In prmeter p spce, the fmily occurs on the curve derived in Eq (422) Therefore, set b ˆ =2, which mkes the Lypunov exponent function of one prmeter, See Fig 2 for grph of Eq (426) s function of Becuse the Lypunov exponent is positive for 0 < < 1, F ;b must be chotic on this entire curve, not just in the region D 0 of Eq (13) (see Fig 3) 5 Non-Mrkov trnsformtions In this section, we prove tht Mrkov mps re dense in D 0 Then, reclling results concerning wek limits of invrint mesures of sequence of trnsformtions, we note tht the Mrkov techniques cn be pplied, in limiting sense, to describe sttisticl properties for ll F ;b with ; b 2 D 0 We begin by noting previous work on the mp The following is the proof tht the mp f k;l studied in [4] is conjugte to the piecewise-liner, intervl mp F ;b we re studying f k;l x ˆ 1 kx if x < 0; 1 lx if x P 0: with k 6 1, l > 1, 0 < < 1, nd 0 6 b 6 1 Using the conjugcy # x ˆ x = 1, show tht # 1 f k;l # x ˆF ;b x Since # x is liner function of x, it is homeomorphism nd # 1 is uniquely de ned s # 1 x ˆ 1 x # 1 f # x ˆ 1 k kx if x < ; 1 l lx if x P : 5:1 Let k ˆ = nd l ˆ 1= 1 Then 1 k ˆ 1 ˆ b; 5:2

9 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365± Fig 3 The line through prmeter spce ssocited with the numericl pproximtion of the pdf shown in Fig 4, ˆ 0:9 b The points re the exct prmeters for the period-3 through period-8 Mrkov mps 1 l ˆ 1 l 1 1 ˆ 1 1 : 5:3 Therefore, # 1 f k;l # x ˆF ;b x by the homeomorphism # nd the two mps F ;b nd f k;l re topologiclly conjugte Cll M the clss of sequences M which occur s kneding sequences of F ; 1 2 = 1 for 0 < 6 1=2, lso known s the primry sequences Misiurewicz nd Visinescu proved the following theorems: Theorem 51 [4, Theorem A] If ; b, 0 ; b 0 2D nd ; b > 0 ; b 0 then K ; b > K 0 ; b 0 Theorem 52 [4, Theorem B] If ; b 2D then K ; b 2M Theorem 53 [6, Intermedite vlue theorem for kneding sequences] If one-prmeter fmily G t of continuous unimodl mps depends continuously on t nd h G t > 0 for ll t then if K G t0 < K < K G t1 nd K 2 M then there exists t between t 0 nd t 1 with K G t ˆK Using these previous results, we cn prove the following: Theorem 54 8 ; b 2D 0, one of the following is true: 1 F ;b is Mrkov 2 9 ; b D 0 such tht F ;b uniformly pproximtes F ;b

10 374 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 Proof If F ;b is Mrkov, we re done Otherwise, choose F 0 ;b 0 such tht K F 0 ;b 0 is non-periodic Given smll e > 0, let 0 ˆ nd b 1 ˆ b 0 e By Theorem 51, K F ;b 0 < K F ;b 1 nd by Theorem 52, K F ;b 0, K F ;b 1 2M WOLOG, we choose indices of b 0 nd b 1 to crete this ordering Recll by Lemm 24, tht periodic sequences re dense in R 2 Therefore, we my choose sequence M 2 M such tht K F ;b 0 < M < K F ;b 1 Since F ;b does vry continuously with prmeters nd b, then Theorem 53 implies n intermedite vlue b such tht b 0 < b < b 1, nd this intermedite mp hs the deciml kneding K F ;b ˆM Therefore, in ny given neighborhood of non-mrkov mp in D 0, there exists Mrkov mp M Hence, we cn construct sequence, in D 0, of Mrkov mps tht converges to ny F ;b with ; b 2D 0 Considering our Theorem 54, nd the following result [1], we conclude tht ny trnsformtion F ;b with ; b 2D 0 is either member of the Mrkov set which we constructed, nd the invrint density function cn be clculted directly s described erlier in this pper, or if F ;b is not in tht set, then sequence of uniformly convergent Mrkov trnsformtions, F i;bi! F ;b nd i ; b i! ; b, ech hve esily clculted invrint densities which converge to the invrint density of F ;b De ne Q be the set fc 0 ; c 1 ; ; c p 1 g nd P be the prtition of I into closed intervls with endpoints belonging to Q : I 1 ˆ c 0 ; c 1 Š; ; I p 1 ˆ c p 2 ; c p 1 Š Theorem 55 [1, Theorem 1032] Let f : I! I be piecewise-expnding trnsformtion, nd let ff n g n P 1 be fmily of Mrkov trnsformtions ssocited with f Note Q 0 ˆ Q, nd Q k ˆ [k jˆ0 f j Q 0 ; k ˆ 1; 2; Moreover, we ssume tht f n! f uniformly on the set In [ Q k k P 0 nd f 0 n! f 0 in L 1 s n!1 Any f n hs n invrint density h n nd fh n g n P 1 is precompct set in L 1 We clim tht ny limit point of fh n g n P 1 is n invrint density of f 6 Conclusion For the two-prmeter fmily of skew tent mps, we hve shown tht there is dense set of prmeters in the chotic region for which the mps re Mrkov nd we exctly nd the pdf for ny of these mps It is known, [1], tht when sequence of trnsformtions hs uniform limit, nd the corresponding sequence of invrint probbility density functions hs wek limit, then the limit of the pdfs is invrint under the limit of the trnsformtions We construct such sequence entirely within the fmily of skew tent mps mongst the set of Mrkov trnsformtions Numericl evidence tht this is possible cn be seen in Fig 4 The grph represents n pproximtion of how the pdf (over the intervl 0; 1Š) chnges s function of the prmeter b, when ˆ 0:9 b The ``jumps'' or ``steps'' vry continuously with the prmeter, nd we hve highlighted the exmple symbolic pttern CRL; CRLL; CRLLL;, for which the results were derived exctly in Section 4 We lso presented n ppliction of this work in the exct clcultion of Lypunov exponents Acknowledgements We would like to thnk J Curry for his suggestion of this topic EB is supported by the Ntionl Science Foundtion under grnt DMS

11 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365± Fig 4 A numericl pproximtion of the probbility density function s function of the prmeter b with ˆ 0:9 b The red lines re the exct solutions for the period-3 through period-8 orbits Appendix A Probbility density function clcultion From Eq (46), we see the pttern h i ˆ h p 1 1 Xi 1 j for i ˆ 1; ; p 2 jˆ0 1 = i ˆ h p = 1 b 1 1 = i ˆ h p 1 : A:1 1 b Substitute this expression for h i in Eq (416) b Xp 3 i h i 1 1 h p 1 ˆ 1; iˆ0 X p 3 1 i 1 i bh p h p 1 ˆ 1; 1 b iˆ0 b 1 X p 3 i! h p 1 1 h p 1 ˆ 1; 1 b iˆ0 1 X p 3 h p 1 b i! Xp 3 b 1 h p 1 ˆ 1; A:2 1 b iˆ0 iˆ0 1 b p 2 h p 1 1 h p 1 ˆ 1; 1 b 1 b p 2 h p 1 1 h p 1 ˆ 1: 1 b

12 376 L Billings, EM Bollt / Chos, Solitons nd Frctls 12 (2001) 365±376 Solving this eqution for h p 1, we nd 1 b p 2 h p 1 ˆ 1 b b ˆ 1 b p 2 1 b 1 b ˆ 1 b p 1 : A:3 Therefore, h i cn be expressed s i h i ˆ 1 b p 1 for i ˆ 1; ; p 2: A:4 References [1] Boyrsky A, Gor P Lws of chos: invrint mesures nd dynmicl systems in one dimension Boston: Birkhuser, 1997 [2] Bssein S Dynmics of fmily of one-dimensionl mps Amer Mth Monthly 1998;118±30 [3] Mrcurd JC, Visinescu E Monotonicity properties of some skew tent mps Ann Inst Henri Poincre 1992;28(1):1±29 [4] Misiurewicz M, Visinescu E Kneding sequences of skew tent mps Ann Inst Henri Poincre 1991;27(1):125±40 [5] Gervois A, Meht ML Broken liner trnsformtions J Mth Phys 1977;18:1476±9 [6] Misiurewicz M Jumps of entropy in one dimension Fund Mth 1989;132:215±26 [7] Lsot A, Mckey M Chos, frctls, nd noise, vol 97 Appl Mth Sci, New York: Springer, 1994 [8] Collet P, Eckmnn J-P Iterted mps on the intervl s dynmicl systems Boston: Birkhuser; 1980