2 L. BILLINGS AND E.M. BOLLT 2 s well roken liner trnsformtions [5], wek unimodl mps [6]. Other closely relted res in the study the chotic ehvior of t

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1 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS L. BILLINGS AND E.M. BOLLT 2 Astrct. We consider fmily of chotic skew tent mps. The skew tent mp is two-prmeter, piecewise-liner, wekly-unimodl, mp of the intervl F ;. We show tht F ; is Mrkov for dense set of prmeters in the chotic region, nd we exctly nd the proility density function (pdf), for ny of these mps. It is well known, [], tht when sequence of trnsformtions hs uniform limit F, nd the corresponding sequence of invrint pdf's hs wek limit, then tht invrint pdf must e F invrint. However, we show in the cse of fmily of skew tent mps tht not only does suitle sequence of convergent sequence exist, ut they cn e constructed entirely within the fmily of skew tent mps. Furthermore, such sequence cn e found mongst the set of Mrkov trnsformtions, for which pdf's re esily nd exctly clculted. We then pply these results to exctly integrte Lypunov exponents.. Introduction Let F ; denote the two-prmeter piecewise-liner mp on the intervl [; ] stisfying ρ + x if» x< (.) F ; (x) x if» x» with << nd»». It hs een shown tht [2] in the following region of the prmeter spce, (.2) D f(; ) <(2 ) nd (( < ) or ( nd >))g ; F ; hs chotic dynmics, nd in the prmeter suset (.3) D f(; ) <(2 ) nd ( < )g; chotic dynmics occur on the entire intervl» x». See Bssein [2] for complete clssiction of the dynmics in prmeter spce, (; ) 2 (; ) [; ]. We will show tht F ; is Mrkov for dense set of (; ) ind. This is interesting ecuse the proility density function for ny Mrkov trnsformtion in this set cn e found exctly to e piecewise constnt function. We cn use these exct results to pproximte the proility density function for ny other trnsformtion in D. We lso show tht Lypunov exponents cn e clculted exctly on the Mrkov set, nd therefore, efciently pproximted on ll of D. In Section 2 we review sufcient condition for piecewise-liner mp on the intervl to e Mrkov, nd we discuss symolic dynmics for this type of mp. Then, we prove tht Mrkov mps re dense in the prmeter set D. Illustrting the simplictions of Mrkov mps, we discuss in Section 3 the techniques used to exctly nd the proility density function 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 cn e pproximted sufciently y memers of the dense set of Mrkov mps. Wechose the form of F ; from [2], wherein cn e found concise discussion of the full rnge of topologicl dynmics in the prmeter spce (; ) 2 (; ) [; ]. The form of the skew tent mp we study, eqution (.), 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 een widely studied, nd under different nmes Deprtment of Mthemticl Sciences, University of Delwre. 2 Deprtment of Mthemtics, U.S. Nvl Acdemy, Ntionl Science Foundtion Grnt DMS

2 2 L. BILLINGS AND E.M. BOLLT 2 s well roken liner trnsformtions [5], wek unimodl mps [6]. Other closely relted res in the study the chotic ehvior of these mps include the stility of the ssocited Froenius-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 the next section. In the specil, ut importnt, cse tht trnsformtion of the intervl is Mrkov, the symol dynmic is simply presented s nite directed grph. A Mrkov trnsformtion is dened s follows. Denition 2. ([]). Let I [c; d] nd let I! I. Let P e prtition of I given y the points c c <c <<c p d. For i ; ;p, let I i (c i ;c i ) nd denote the restriction of to I i y i. If i is homeomorphism from I i onto some connected union of intervls of P, then is sid to e Mrkov. The prtition P is sid to e Mrkov prtition with respect to the function. The following result descries set in D for which F ; is Mrkov. Theorem 2.2. For given (; ) 2 D,ifx is memer of periodic orit, then F ; is Mrkov. Proof. Set F F ;. Assume x is memer of period n orit (n >). Next, form prtition of [; ] using the n memers of the periodic orit. The two endpoints of the intervl re included since 8(; ) 2 D, F (). Order these n points so c <c < < c n, regrdless of the itertion order. For i ; ;n, let I i (c i ;c i ) nd denote the restriction of F to I i y F i. For given I i (c i ;c i ), the endpoints c i nd c i will mp exctly to two memers of the prtition endpoints y denition of the periodic orit. Let these points e c j nd c k, with c j < c k nd j; k 2 f; ;n g. The only turning point of the mp is x, nd 8(; ) 2 D, F (). Therefore x must lwys e prt of the period orit nd memer of the prtition endpoints, implying ech F i is liner nd hence, homeomorphism. Also F i (I i ) (c j ;c k ), connected union of intervls of the prtition. By denition, F is Mrkov. At this point, symolic nottion ecomes useful. The point x is the criticl point t the center" of the intervl, denoted y the letter C. All <x» is right of,represented y R, nd ll» x< is left of, represented y L. Represent ech step of the itertion mp y one of these three symols. All prmeter sets for which F (x) is Mrkov must hve period-n orit contining the point x nd e of the form n ; ;F n (); o. For exmple, the period three orit hs the form Φ ; ; ; Ψ nd occurs for ny prmeter set (; ) on the curve. It repets the pttern CRL, which we cll the kneding sequence K(F ; )(CRL). If periodic orit contins the point x nd 6, the point x will either e greter or less thn. For period four orit with >, the symolic sequence must repet CRLL, or if <, CRLR. Therefore, period four is found two wys. See Figure for cowe digrm of n exmple period 4 CRLL orit. Repeting this method for period ve nd higher, we see tht there re 2 n 3 possile comintions of the C-L-R sequences for period n orit which includes the criticl point. The exponent n 3 reflects the necessry 3-step prex CRL. A full inry tree with 2 n 3 leves on ech tier is possile, ech implying condition on the prmeters nd forming countle set of curves in prmeter spce [4]. We cn now restte Theorem 2.2 in terms of kneding sequences. Corollry 2.3. If K(F ; ) is periodic, then F ; is Mrkov. We now prove tht the function F ; (x) (.) is Mrkov on dense set of curves in D (.3). Dene ± 2 s the spce of symol sequences contining the full fmily of P kneding sequences for two symols. Dene the kneding sequence ff ff ff ff 2, the metric d(ff; ^ff) i jf ^fj 2, nd the norm jjffjj P i i f 2 i ff + ff 2 + ff Lemm 2.4. Periodic ff re dense in ± 2.

3 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 3 Proof. 8" > nd given WOLOG f f f f 2 which is not periodic, 9N > lrge enough so tht ff i f i, i» N nd ff (f f f N ) for jjff fjj ±2 <". 3. Proility density functions nd Lypunov exponents Another enet of the set of Mrkov chotic functions is tht the proility density functions cn e determined quite esily. In fct, expnding piecewise liner Mrkov trnsformtions hve piecewise constnt invrint proility density functions. Theorem 3. ([], Theorem 9.4.2). Let I! I e piecewise liner Mrkov trnsformtion such tht for some k, j( k ) j > ; wherever the derivtive exists. Then dmits n invrint [proility] density function which is piecewise constnt on the prtition P on which is Mrkov. Using the Froenius-Perron opertor P, the xed-point function h stises the denition P F h h, implying tht h is the proility density function for mesure tht is invrint under F. Since F is piecewise monotone function, the ction of the opertor is simply P F h(x) X z2ff (x)g h(z) jf (z)j The periodic orit formed y the itertion of x forms prtition of the domin [; ] on which h is piecewise-constnt. On ech intervl I i, cll the corresponding constnt h i hj Ii. The proility density function dmits n solutely continuous invrint mesure on the Mrkov prtition. This mesure cn e used to nd the Lypunov exponent, nd therefore quntify the verge rte of expnsion or contrction for n intervl under itertion. Set F F ; for some (; ) 2 D nd form prtition of [; ] using the n memers of the periodic orit so c <c <<c n. Assume c k, for some k 2f; ;n 2g. Note jf (x)j if x<, nd jf (x)j if x>. (3.) Λ ; Z Z c ln jf (x)j h(x) dx c ln jf (x)j h dx + + ln kx i (c i c i )h i +ln Z cn c n 2 4. Following the left rnch ln jf (x)j h n dx n X (c i c i )h i ik+ As n exmple, in this section we will derive the proility density function for F (x) when it is Mrkov nd hs periodic orit of the pttern CRL, CRLL, CRLLL,, following the left rnch of the inry tree. In the discussion elow, x is prt of periodic orit of period p n +3 nd n represents the numer of L's fter the initil CRL in the symolic representtion of tht periodic orit. (4.) Φ Ψ p z } ; ; ;;F();F 2 (); ) CRL LLL z } n Proposition 4.. The generl reltion of nd in the period p n +3 in eqution (4.) is, n+ (4.2)

4 4 L. BILLINGS AND E.M. BOLLT 2 Proof. Renme the rnches of the piecewise dened function, f L on» x<nd f R on» x». Using the geometric series expnsion, express f n L () s nx k (4.3) fl() n k n+ The prmeter sets for ech of these periodic orits is found y solving f n L (), where the period is n +3. By using eqution (4.3), we hve the implicit reltion (4.2). It now seems quite nturl for the CRL orit to occur on the prmeter set. The equtions relting to for the different periodic orits form pttern in prmeter spce, nd limit to. On these curves in the chotic region, n invrint density function h cn e found in closed form. It is piecewise dened function on its ssocited Mrkov prtitions. Therefore, h(x) hs the form (4.4) (4.5) h(x) 8 >< > h h 2. h p Then the ction of the Froenius-Perron opertor is P f h(x) h(f (x)) L if <x< if <x<f() if <x< jf L (f (x))j χ [;](x)+ h(f (x)) R L jf R (f (x))j R h(f L (x))χ [;](x)+( )h(f R (x)) We use eqution (4.5) to construct system of equtions to solve for h(x). This system hs the following form (4.6) h ( )h p h 2. h p h +( )h p h p 2 +( )h p For proof, see the geometry of the inverse function. Note the recursion; the constnt h depends only on h p. From the monotonicity of the inverse of f L nd f R, ech constnt, h k, depends on h k nd h p. Proposition 4.2. The system of Equtions (4.6) is underdetermined for the vriles h ;h 2 ; ;h p. The determinnt is zero precisely when the prmeters stisfy eqution (4.2). Proof. This is proved y induction. Bse cse n nd p 3 (4.7) h ( )h 2 (4.8) h 2 h +( )h 2 This set of equtions cn e rewritten s the homogeneous system ( ) h (4.9) h 2 ( )

5 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 5 To nd if there is unique solution, nd the determinnt M (4.) M ( ) ( ) ( ) ( ) Sustituting eqution (4.2) for n, implies tht M. To prove M j+, use the reltion jx k (4.) M j ( ) ; k nd ssume eqution (4.2) is true for n j. Therefore, Xj+ k (4.2) M j+ ( ) ( ) Sustituting eqution (4.2) for n j, M j+. k j+2 C A ( ) ψ j+! The freedom implied y Proposition 4.2 is expected since the density function must e normlized. Set the re under the curve to one (4.3) Z h(x)dx Using the prtition, c <c <<c p,we know c n 2 nd kx i (4.4) c k f k l () for k ; ;p 3 i Therefore, eqution (4.3) simplies to the following (4.5) (4.6) (c c )h + +(c p c p 2 )h p Xp 3 i h i+ +( )h p i With this dditionl constrint, the system of equtions hs dimension p (p ). (4.7) S h h 2. h p C A C A ;

6 6 L. BILLINGS AND E.M. BOLLT 2 dening the coefcient mtrix S of eqution (4.7) s the mostly nded mtrix (4.8) S ( ). ( ) ( ) ( ) p 3 ( ) The next step is to prove tht this new system hs rnk p. Nme the rows in the coefcient mtrix S s R, R 2,, R p. Using the denition of liner dependence, there must exist set of constnts tht multiply the rows of the mtrix so tht they sum to zero, (4.9) Solving for these constnts, we nd k p i k p k p R k + R 2 k 2 + +R p k p ~ i + kp (i ) ψ p 2! X i k i + C A i p+2 for i 2; ;p There is only one degree of freedom nd the system hs rnk p. There is unique solution to the system, if one exists. The closed form for the constnts h i in the proility density function re (4.2) (4.2) h i h p (p ) ψ ( )( (p )) i! for i ; ;p 2 We show this clcultion in Appendix A. Exmple The CRLL fmily illustrtes the previous results. The Mrkov prtition is formed y the period four orit (;;;). See Figure for cowe digrm of specic period four orit. In prmeter spce, the fmily occurs on the curve f L (), or (4.22) + p Note tht the other rnch of the solution occurs outside the prmeter domin, (; ) 2 (; ) [; ]. Eqution (4.5) produces the system of equtions (4.23) h ( ) h 3 h 2 h 3 To normlize h, set the re under the curve to one, (4.24) h +( )h 3 h 2 +( )h 3 h +( )h 2 +( )h 3

7 Hence, the invrint density function is PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 7 h(x) 8 >< > 3 ( ) 2 2 ( )( 3) if» x< if» x < ( )( 3) if» x» See Figure for comprison of this exct result to histogrm generted y typicl" orit. Of prticulr use, the invrint mesure cn e used to exctly clculte the Lypunov exponent for ny set of prmeters (; ) long the curves descried y eqution (4.2) in D. WOLOG, ssign c k nd k p 2. Use equtions (3.), (4.2), nd (4.2) to derive the following (4.25) Λ ; ln ln ln ln P k i (c i c i )h i +ln ( ( )hp )+ln +ln +ln Λ ; ln ( )( ) ( )( ) ( )h p P p ik+ (c i c i )h i ( )h p (p ) Exmple Continuing the period four CRLL exmple from ove, we derive the Lypunov exponent exctly. Set p 4 nd in eqution (4.25), +ln (4.26) ( )( ) 3 In prmeter spce, the fmily occurs on the curve derived in eqution (4.22). Therefore, set (+ p )2, which mke mkes the Lypunov exponent function of one prmeter,. See Figure 2 for grph of eqution (4.26) s function of. Becuse the Lypunov exponent is positive for <<, F ; must e chotic on this entire curve, not just in the region D of eqution (.3). 5. Non-Mrkov Trnsformtions In this section, we prove tht Mrkov mps re dense in D. Then, reclling results concerning wek limits of invrint mesures of sequence of trnsformtions, we note tht the Mrkov techniques cn e pplied, in limiting sense, to descrie sttisticl properties for ll F ; with ; 2 D. We egin y noting previous work on the mp. The following is the proof tht the mp f ;μ studied in [4] is conjugte to the piecewise liner, intervl mp F ; we re studying. ρ + x if x f ;μ (x)» μx if x with», μ>, <<, nd»». Using the conjugcy #(x) (x )( ), show tht # f f ;μ f #(x) F ; (x). Since #(x) is liner function of x, it is homeomorphism nd # is uniquely dened s # (x) ( )x+. (5.) Let ( ) nd μ ( ). Then (5.2) (5.3) ρ # ( (f(#(x))) )+ x if x» ( + μ) μx if x +μ μ Therefore, # f f ;μ f #(x) F ; (x) y the homeomorphism # nd the two mps F ; nd f ;μ re topologiclly conjugte.

8 8 L. BILLINGS AND E.M. BOLLT 2 Cll M the clss of sequences M which occur s kneding sequences of F ; 2 known s the primry sequences. Misiurewicz nd Visinescu proved the following theorems for <» 2, lso Theorem 5. ([4], Theorem A). If (; ), ( ; )2D nd (; ) > ( ; ) then K(; ) >K( ; ). Theorem 5.2 ([4], Theorem B). If (; ) 2 D then K(; ) 2M. Theorem 5.3 ([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 ) > for ll t then if K(G t ) <K<K(G t ) nd K 2Mthen there exists t etween t nd t with K(G t )K. Using these previous results, we cn prove the following, Theorem (; ) 2 D, one of the following is true. F ; is Mrkov. 2. 9( Λ ; Λ )D such tht F Λ ; Λ uniformly pproximtes F ;. Proof. If F ; is Mrkov, we re done. Otherwise, choose F ; such tht K(F ; ) is nonperiodic. Given smll ">, let Λ nd + ". By Theorem 5., K(F Λ ; ) <K(F Λ ; ) nd y Theorem 5.2, K(F Λ ; ), K(F Λ ; ) 2 M. WOLOG, we choose indices of nd to crete this ordering. Recll y Lemm 2.4, tht periodic sequences re dense in ± 2. Therefore, we mychoose sequence M 2Msuch tht K(F Λ ; ) <M<K(F Λ ; ). Since F ; does vry continuously with prmeters nd, then Theorem 5.3 implies n intermedite vlue Λ such tht < Λ <, nd this intermedite mp hs the deciml kneding K(F Λ ; Λ)M. Therefore, in ny given neighorhood of non-mrkov mp in D, there exists Mrkov mp M. Hence, we cn construct sequence, in D, of Mrkov mps tht converges to ny F ; with (; ) 2 D. Considering our Theorem 5.4, nd the following result [], we conclude tht ny trnsformtion F ; with (; ) 2 D is either memer of the Mrkov set which we constructed, nd the invrint density function cn e clculted directly s descried erlier in this pper, or if F ; is not in tht set, then sequence of uniformly convergent Mrkov trnsformtions, F i; i! F ; nd ( i ; i )!(; ), ech hve esily clculted invrint densities which converge to the invrint density off ;. Dene Q e the set fc ;c ; ;c p g nd P e the prtition of I into closed intervls with endpoints elonging to Q I [c ;c ]; ;I p [c p 2 ;c p ]. Theorem 5.5 ([], Theorem.3.2). Let f I! I e piecewise expnding trnsformtion, nd let ff n g n e fmily of Mrkov trnsformtions ssocited with f. Note Q () Q, nd k[ Q (k) f j (Q () ); k ;2; j Moreover, we ssume tht f n! f uniformly on the set I n [ Q (k) k nd f n! f in L s n!. Any f n hs n invrint density h n nd fh n g n isprecompct set in L. We clim tht ny limit point of fh n g n 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 proility density function for ny of these mps. It is known, [], tht when sequence of trnsformtions hs uniform limit, nd the corresponding sequence of invrint proility density functions hs wek limit, then the limit of the proility density functions is invrint under the limit of the trnsformtions. We construct such sequence

9 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 9 entirely within the fmily of skew tent mps mongst the set of Mrkov trnsformtions. Numericl evidence tht this is possile cn e seen in Figure 4. The grph represents n pproximtion of how the proility density function (over the intervl [; ]) chnges s function of the prmeter, when 9. The jumps" or steps" vry continuously with the prmeter, nd we hve highlighted the exmple symolic 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. 7. Acknowledgements We would like to thnk J. Curry for his suggestion of this topic. E.B. is supported y the Ntionl Science Foundtion under grnt DMS

10 L. BILLINGS AND E.M. BOLLT 2 (A.) Appendix A. Proility density function clcultion From eqution (4.6), we see the pttern h i h p ( ) h p ( ) Xi j j h p ( )() Sustitute this expression for h i in eqution 4.6. (A.2) p 3 X h p i ( )() ( ) ( )() h p ( ) h p ( )() ( ) ψ i ψ p 3 X i i ( ) X p 3 i p 3 X ψ h p ( )() ( ) for i ; ;p 2 C A i i+! i i h i+ +( )h p i p 3 X i (p 2) i!! h p ( )( (p 2)) ( ) +( )h p +( )h p +( )h p +( )h p Solving this eqution for h p,we nd ( )( (p 2)) (A.3) h p +( ) ( ) ( ) ( )(( (p 2)) + ( )) ( ) ( )( (p )) Therefore, h i cn e expressed s (A.4) h i (p ) ψ i! +( )h p for i ; ;p 2 References [] A. Boyrsky nd P. Gór, Lws of chos invrint mesures nd dynmicl systems in one dimension, Birkhuser, Boston, 997. [2] S. Bssein, Dynmics of fmily of one-dimensionl mps, Amer. Mth. Monthly Fe. (998), 8 3.

11 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS [3] J.C. Mrcurd nd E. Visinescu, Monotonicity properties of some skew tent mps, Ann. Inst. Henri Poincre 28 (992), no., 29. [4] M. Misiurewicz nd E. Visinescu, Kneding sequences of skew tent mps, Ann. Inst. Henri Poincre 27 (99), no., [5] A. Gervois nd M.L. Meht, Broken liner trnsformtions, J. Mth. Phys. 8 (977), [6] M. Misiurewicz, Jumps of entropy in one dimension, Fund. Mth. 32 (989), [7] A. Lsot nd M. Mckey, Chos, frctls, nd noise, Applied Mthemticl Sciences, vol. 97, Springer-Verlg, New York, 994. [8] P. Collet nd J.-P. Eckmnn, Iterted mps on the intervl s dynmicl systems, Birkhuser, Boston, 98.

12 2 L. BILLINGS AND E.M. BOLLT 2 Histogrm PDF for.4, 5 itertions, 5 ins.8 p(x) x Figure. Numericl clcultion of the PDF (histogrm rs) compred to the exct solution (solid line). The clcultion used 5, itertions nd 5 intervls.

13 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 3.7 Lypunov Exponent Period Four Orit Λ, Figure 2. The Lypunov Exponent s function of the prmeter where x is memer of period four orit.

14 4 L. BILLINGS AND E.M. BOLLT 2 Stle Period Two Chos on Suintervls Chotic on the Entire Intervl Mixed Region.9- P8 Figure 3. The line through prmeter spce ssocited with the numericl pproximtion of the proility density function shown in Figure 4, 9. The points re the exct prmeters for the period three through period eight Mrkov mps.

15 PROBABILITY DENSITY FUNCTIONS OF SOME SKEW TENT MAPS 5 Proility Density s Function of Prmeter 2.5 pdf.5. p6 p7 p8.2 p5.3 p4.4 p x.6.8 Figure 4. A numericl pproximtion of the proility density function s function of the prmeter with 9. The red lines re the exct solutions for the period three through period eight orits.