Bobok, Jozef

Kuchta, Milan

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJozef Bobok; Milan KuchtaTopological entropy and variation for transitive mapsMathematica Slovaca, Vol. 43 (1993), No. 3, 345--355Persistent URL: http://dml.cz/dmlcz/131432Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©1993 . . . _ . . _ / 4 / , A . x . . _ _ . _ _ _ _ Mathematical Institute Math. SlOVaCa, 43 (1993), NO. 3, 345-355 Slovák Academy of Sciences TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS J O Z E F B O B O K * ) — MILAN K U C H T A * * ) (Communicated by Milan Medvěd') ABSTRACT. We s tudy the continuous functions which m a p a compact real inter­val back into itself. We investigate the relations between two important concepts of the dynamical systems and real analysis for transitive functions, topological entropy and variation. 0. Introduct ion This paper is concerned with investigation of relations between the topolog­ical entropy and variation for transitive maps. Topological entropy, denoted ent(-), is a numerical conjugacy invariant of continuous maps. Variation of a function / on the interval 7, denoted Var(/, 7) , is a length of a way of a point f(x) if a point x goes through the interval 7. A continuous map is transitive if some point has a dense orbit. Let 7 = [0,1] be the closed unit interval and C(I, I) be the set of all con­tinuous functions which map the interval 7 back into itself. MAIN THEOREM. Let (x, y) be a pair of numbers. Then there exists a tran­sitive function f E (7(7,7) such that (x,y) = (Var(/, 7), ent(/)) if and only if (x,y) e ((l,oo] x (logx/2,oo]) U ((1,2) x { l o g v ^ } ) . In section 1 we give the definitions and some known results. In section 2 we define three transitive maps with prescribed topological entropy and in section 3 we show that all of the cases for a pair of numbers (Var(/, 7), ent(/)) from Main Theorem are possible. In section 4 we prove that, up to conjugacy, there A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 58F03, 58F13. K e y w o r d s : Topological entropy, Variation, Topological conjugacy. 345 JOZEF BOBOK — MILAN KUCHTA is only one transitive function Q G C(I, I) such that ent(Q) = l o g v " . The proof of the Main Theorem is a straightforward combination of Lemma 1.2 (6), Lemma 1.3, Lemma 3.1, Lemma 3.2 and Corollary 4.2. 1. Background Let I = [0,1] be the closed unit interval and C(I, I) be the set of all con-tinuous functions which map the interval I back into itself. For / G C(I, I) we define fn inductively by / ° (x ) = x and (for n > 1) fn(x) = f{fn~l(x)) . fn is called the ra-th iterate of / . For x G I the orbit of x under / is {fn(x)}n . A point x is said to be periodic with period n if fn(x) = x and fl(x) ^ x for 0 < i < n. A fixed point is a periodic point with period 1 and Fix( / ) is the set of all fixed points of / . A map / is called piecewise monotone if there exist N > 0 and 0 = d0 < di < • • • < djv+i = 1 such that / is strictly monotone on [dfc, dfc+i] for each k = 0 , . . . , N. If / G C(I, I) is piecewise monotone, then a point w G (0,1) is called a turning point of / if / is not monotone in any neighbourhood of w. The critical points of / are the turning points of / and the endpoints of the interval I. A set X C I is an invariant set under / if f(x)cx. If *P = {pi < • • • < pn} is a finite subset of I, let /<p be the map de-fined on [pi ,pn] which agrees with / on *-p and which is linear on [pt-P*+i] (i = l , . . . , n - l ) . A map f € C(I, I) is transitive if the only closed invariant subset of I with non-empty interior is I itself. Let variation of the / G C(I, I) at the interval [a, 6] be Vax ( / , [a, b] ) = sup < ] T | / (a*+i) - f(xk)\, a = x0 < ^i < • • • < x n = 6 > , l k = o J Var ( / , | J Ii) = ] T Var(/ , It), where h n Ij = 0 for i^j. iGK iEK In particular, if / is piecewise monotone with critical points 0 = do < d\ < N • • • < dN < div+i = 1, then clearly Var(/, I ) = £ \f(dk+i) ~ f(dk)\ • k=0 A map / G C(I, I) is Lipschitz with constant L > 0 if for every pair x ^ y G I w e h a v e f ^ ~ f ^ <L. x-y The basic equivalence relation of dynamical systems is topological conjugacy. Two functions / , g G C(I, I) are topologically conjugate if there is a homeomor-phism h G C(I, I) such that g = h o / o b,-1 . The topological entropy denoted ent(-) is a conjugacy invariant of continuous maps. 346 TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS Now we recall the definition of topological entropy and some useful facts. DEFINITION 1.1. Let f:M->M, n G N and e > 0 . Let S(f,n,e) C M be the set with maximal possible number of points with property that for every X,V £ S(f,n,e), x 7-- y there is i G { 0 , 1 , . . . , n} such that \fl(x) — fl(y)\ > e. Then ent ( / ) = lim lim sup — log card 5 ( / , n, e). £—•0 n _ > o o n LEMMA 1.2. Le£ / , g G C(I,I). Then the following conditions hold. (1) 7/ / 25 a Lipschitz with constant L, then ent(f) < l o g L ; (2) en t ( / n ) = n e n t ( / ) ( n > 0 ) ; (3) if f(J) C J and J C I is closed, then ent(f) > ent (f\j) ; (4) if f,g are topologically conjugate, then ent(f) = ent(g) ; (5) if f,g are topologically conjugate and f is transitive, then g is tran-sitive; (6) if f is a transitive, then Var(/ , I) > 1 . LEMMA 1.3. (A . M . B 1 o k h [4]) Let / G C(I,I) be a transitive map. Then ent(f) > log \ / 2 . LEMMA 1.4. (L . B l o c k , J . G u c k e n h e i m e r , M . M i s i u r e w i c z , L . S . Y o u n g [3]) Let f G C(I, I). If a function f has a periodic orbit of period n = 2rnp, p > 1 is odd, then ent(f) > ------- log Xp, where Xp is the unique positive root of the polynomial Xp — 2XP~2 — 1 . It is easy to verify that for any odd p > 1 we have Xp > \[2. LEMMA 1.5. ( M . B a r g e , J . M a r t i n [1]) If f G C(I,I) is transitive but not so f2, then there exist the intervals Jf, Kf such that I = Jf U Kf, Jf DKf = {pf}, f(Jf) = Kf, f(Kf) = Jf and f2\j , f2\xf are transitive on Jf, Kf respectively. LEMMA 1.6. ( M . B a r g e , J . M a r t i n [2]) Let f G C(I, I). If f2 is a transitive map, then f has a point of odd period greater than 1. LEMMA 1.7. ( E . M . C o v e n , M . C . H i d a l g o [5]) Let f G C(7,1) be a transitive map. If f is not piecewise monotone and has at least two fixed points, then ent(f) > log 2 . LEMMA 1.8. ( E . M . C o v e n , M . C . H i d a l g o [5]) Let f G C(I, I) be a transitive map and let ty be a finite invariant set. Then ent(f) > ent(/<p), with equality if and only if f is piecewise monotone and ^3 contains the critical points of f. 347 JOZEF BOBOK — MILAN KUCHTA LEMMA 1.9. ( W . P a r r y [7]) Let f E C(I,I) be a piecewise monotone transitive function with ent( / ) = logcr. Then f is topologically conjugate to a piecewise linear function whose linear pieces have slopes +a . LEMMA 1.10. (M. M i s i u r e w i c z , W . S z l e n k [6]) Let f e C(I, I) be a piecewise monotone map. Then lim — log Var ( / n , 7 ) = ent ( / ) . n—• oo n 2. Maps with prescribed entropy Entropy equal to l og \ /2 . Let Q(x) = xV2 + 2-y/2 if i G [ 0 , l - f ] , Q(x) = y/2 - xy/2 if x G [l - ^ , l] . R e m a r k . Q is a unimodal map with a constant slope ± \ / 2 (see Figure 1). PROPOSITION 2.1. Function Q has the following properties: (i) QeC(lj), (ii) Var(Q,J) = >/2, (iii) Q is transitive, (iv) ent(Q) = l o g v/ 2 . P r o o f . The properties (i), (ii) are clear from definition of Q and (iv) fol-lows from Lemma 1.10. Hence it suffices to show that for any interval J C I there is a number n E N such that [0,1 — -^- ] C Qn(J) • We have two possible cases. Either \Q2(J)\ = 2\J\ or {0,1} n Q2(J) ^ 0. Hence there is m e N such that {0,1} n Q2rn(J) ^ 0. Then the fixed point ^ e Q2rn+2(J) and we are easily done. D Finite entropy greater than \ogy/2. Let a > y/2. We will define a map F with infinitely many pieces where it is linear with a slope +a. Moreover, F will have only one limit point of the critical points and any critical point will be mapped after finite time into a periodic point (see Figure 2). 348 TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS Figure 1. Function Q. Figure 2. Function F for a = 2 . 349 JOZEF BOBOK — MILAN KUCHTA Here is a formal definition of F. Let *o = l , F ( x 0 ) = 0, X l = X̂T ~ ( \u ' F ^ ) = X ' a + 1 a ( a + l) 2 x2 _ a l p( \ _ a - a + 1 a ( a + 1) ' V ' <* + 1 We have that x 2 > 0. Let na G {0,1,2,...} be the smallest one such that i 2nQ (_l)i -, a ( a + l ) a » - < **• L e t F ( 0 ) = S V " ' * ~ = a(a+l)a»- " " " ' Now let p — 1 and we will define the set { # i } S 3 and F(xi) using induction from n = 1. 2z? p If x 2 n -,————• < XQO , t h e n we set p = —— , else let a(a + 1) a X2 n + 1 ~ Ж2n " a(a + l) ' F ^ 2 n + l ) ~ a П + ӢЛ ' 2p a ( a + l ) ' * 2 „ + 2 = X2n - •——- , F ( x 2 n + 2 ) = - - - — . Finally let F be linear on the complementary intervals to the points {0,Xoo} U PROPOSITION 2.2. Function F has the following properties: (i) FeC(I,I); (ii) Xi > x -̂j-i /o r i > 0 . and lim x^ — Xoo ; 2 — • O O (in) F has slopes _ a on [0,XQO] and [x-;+i,Xi] /o r i > 0 ; (ivj Var(F, / ) = a ; (v) tfie orbit {0, F ( 0 ) , F 2 ( 0 ) , . . . , F2n«+1(0)} is periodic with period 2na + 2 ; (vi) /o r every x*, either there is a ki G N suc/i £/ia£ Fki(xi) — 0 (z 25 odd), or F(xi) = ——— and ———- is a fixed point; (vii) F is transitive but not so F2 ; (viii) ent(F) = log a . P r o o f . The properties (i) - (vi) are clear from the construction of the func-tion F. Since F ( [ 0 , ^ ] ) = [ -£- . ! ] and F 2 ( [ 0 , ^ ] ) = [0, - £ - ] , F 2 can-not be transitive. In order to prove a transitivity of F let us show that for each interval J C / there is n > 0 such that Fn(J) D [0, -^h] . It is clear if 350 TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS ^ n ( { x 0 0 , ^ I } u { x 2 i } ^ 1 ) ^ © . A s s u m e that Jn({x00,-gI}\J{x2i}?l1) = 0 and J C [ 0 , ^ j ] . Then the definition of F implies that F2(J) c [0, ^ ] and | F 2 ( J ) | > ^ - . Hence F2™(J) n ( { x ^ , ^ } U { s 2 i } £ i ) 7- 0 for some ra > 0 and (vii) is proved. Finally, it thus only remains to show (viii). By (i), (Hi) and Lemma 1.2(1) we have that ent(F) < logo;. Denote tyn = { 0 , x 0 , x i , . . . , x n _ i , x n } . By (v) and (vi) there is an m G N such that the finite set *£n = \J F*(<pn) is F-in-2=0 variant. Since ?$n \ %$n C [xi,x0] we have that F~ = F<pn . By Lemma 1.8 ent(F) > ent(F<pn). Hence it is sufficient to show that lim ent(Fp n ) = log a . n—>cx) However, by (Hi) the slope of F p n on each interval [x{, Xi-\], i G { 1 , 2 , . . . , n} , is equal to ±a and if we denote /3n the slope of F p n on [0, xn], then from (ii) we have that lim (3n = —a. Now using Lemma 1.10 one can show that n—KX> ent(Fpn) > log |/3n | and the proof is finished. • Figure 3. Function G . 351 JOZEF BOBOK — MILAN KUCHTA Infinite entropy. I x r t x o - l . x , - } - - ^ - - f o r i = 1,2,... a n d / , ( 2 ( 2 .1+ 1 ) ) = - l , fi (2f+r) = °a n d fi ( 1{+T) = l for j = °'ls • • •' * •Now l e t fi b e l i n e a r on the complementary intervals to this points and h^(x) = x(x^ — x;_i) -f xi-1 . Now let G(0) = 1, G ( | ) = 0, G ( | ) = | , G(§) = § , G(l) = 0, G be linear on the intervals [0, ^] , [|, | ] , [|, l] and for x _ [x^-i,^] let G(x) = / l i C / i ^ r 1 ^ ) ) ) ( i = 1,2,... ) (see Figure 3). PROPO'SITION 2.3. Function G has the following properties: (i) GeC(lj), (ii) Var(G,J) < o o , (iii) G is transitive, (iv) ent(G) = oo . P r o o f . The property (i) is clear from the construction of G and for the variation we have oo Var(G.I) = Var (G, [0, | ] ) + Var (G, [|, l ] ) + ^ V a r (G, [_._!,_,]) Ż - 1 2 + V - + І --- 3-2 г І = l 2i + 3 ^ < 0 0 . Now we will prove (iii). Let J C I be an interval. We will show that there is m <E N such that G m (J ) = I. If J n Fix(G) n (_», x i + 1 ) = 0 for all i > 0, then 3|J| 1^(^)1 > o • S° we can assume that J n Fix(G) H (xi0,Xi0+i) 7- 0 for some 3|J| io > 0. Now, if x0 <£ G(J), then |G(J) | > —-r— . Hence we can simply assume that x0 G G(J ) . Now it is easy to see that G2 io+3(J) = I. Finally, it thus only remains to show (iv). Denote ( x n - x n _ i ) i \2 ^ + i 4 5 n - | x „ _ 1 + 2 n + 1 j j = Q . Now by Lemma 1.8 we have that ent(G) > ent(G<rjJ and by Lemma 1.10 we have that ent(G<pj = log(2n + 1). So we have ent(G) _cx) . • 352 TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS 3. How to get any variation Let /, h G C(I,I) and ft be a homeomorphism. Then we have that Var(ft o / o ft-1, /) = Var(ft o /, / ) . (It is the same to count variation through all dissections {di}™=0 or through all dissections {h~1(di)}i=Q .) If we compare the graphs of / and ft o / , then we can see that homeomorphism ft in the second case only vertically deforms graph of / . We will use this mechanism to change variation of the functions given above to get all the possible variations. LEMMA 3 .1 . For any K G (1,2) there is a transitive map f G C(7,1) such that Var(/, I) = K and ent(f) = log \[~. P r o o f . Let ft G C(J, /) be a homeomorphism such that ft (ч/2 + J 2 — K and let / = ft o Q oft x . Then we have Var(/, I) = K and the rest by Lemma 1.2 (4)(5) and Proposition 2.1. • LEMMA 3.2. Let (K, a) G (1, oo] x (\/2, oo]. Then there is a transitive function f G C(I,I) such that (if, log a ) = (Var(/,/) ,ent(/)) . P r o o f . Let H = F if a is finite or H = G if a = oo. The map H is such that we can divide I into the sequence of closed disjoint (except boundary points) intervals {J^}g 1 such that H~x(Ji) is a union of Ui closed disjoint (except boundary points) intervals, each of them mapped homeomorphically by H onto Ji. Moreover n\ = 1 and lim Ui = oo. Then for every sequence {a i}^i i—>oo oo of positive numbers with ~~ ai = 1 there is a homeomorphism ft G C(I, I) such i=i that \h(Ji)\ = ai. Let / = ft o H o ft"1. We have Var(/, I) = Var(ft o H, 7) = ~~ UiOi. Clearly, with the suitable choice of the sequence {ai\*?Lx we can get 2 = 1 OO ~2 niai = K - Lemma 1.2(4)(5), Proposition 2.2 and Proposition 2.3 complete i = i the proof. • 4. Maps with entropy equal to log \/2 THEOREM 4 . 1 . Let f be transitive with ent(/) = log y/~. Then f is topolog­ical^ conjugate to the function Q . P r o o f . Assume that / G C(I, I) is transitive and ent(/) = log y/2. Then by Lemma 1.4 and Lemma 1.6 we have that / 2 is not transitive. Hence by Lemma 1.5 there exist the intervals J/ , Kf such that I = Jf U Kf , Jf D Kf 353 JOZEF BOBOK — MILAN KUCHTA = {Pf}> f(Jf) = Kf> f(Kf) = Jf a n d f2\jf > f2\Kf a r e t r a n s i t i v e o n Jf > Kf respectively. Now we distinguish two cases: Case J. Suppose that / is piecewise monotone. Then by Lemma 1.9 / is topologically conjugate to a piecewise linear transitive function q G C(I,I) whose linear pieces have slopes ± \ / 2 . Note there is a fixed point pq G I such thcit <l([0,pg]) = [pg, 1] and g([pg , l ]) = [0,Pg]. Prom transitivity we have that there is a point a ^ pq such that q(a) = pq. We can assume that a € [0,pg] (if not we can use topological conjugacy by h(x) = 1 — x ) . Now let J = fl,Pg+ (pg~°)^2 n J. It is easy to see that q(J) C J and from transitivity ny we have that a = 0 and pg = — — . And finally, because q is piecewise 1 + v 2 linear with slopes ± \ / 2 , and g([0,pq]) = [pg, 1], q([pq, 1]) = [0,Pg], it is very easy to see that q = Q. Case JJ. Now let / be not piecewise monotone. Set g = f2\ j . Then g is transitive not piecewise monotone and g(pf) = p / . So g has at least two fixed points in Jf (transitivity). Thus by Lemma 1.7 we have ent(^) > log 2 and by Lemma 1.2(2)(3)(4) we have ent(/) > log\/2 which is a contradiction. • COROLLARY 4.2. Let f€C(I,I) be a transitive function and e n t ( / ) = log \ /2 . Then Var(/,J) G (1,2) . REFERENCES [1] BARGE, M .—MARTIN, J.: Chaos, periodicity and snakelike continua, Trans. Amer. Math. Soc. 2 8 9 (1986), 355-365. [2] BARGE, M.—MARTIN, J . : Dense orЫts on the interval, Michigan M a t h . J. 3 4 (1987), 3-11. [3] BLOCK, L . — G U C K E N H E I M E R , J . — M I S Ш R E W I C Z , M . — Y O U N G , L. S.: Periodic points and topological entropy of one-dimensional maps. In: Global Theoгy of Dynamical Systems (Proc. Internat. Conf., Noгthwesteгn Univ., Evanston, 111., 1979). Lecture Notes in M a t h . 812, Spгinger, Beгlin, 1980, pp. 18-34. [4] BLOKH, A. M . : Sensitive mappings of an interval. (Russian), Uspekhi Mat . Nauk 3 7 (1982), 189-190. [5] COVEN, E. M . — H I D A L G O , M. C : On the topological entropy of transitive maps of the interval, Preprint (1990). [6] MISIUREWICZ, M .—SZLENK, W . : Entropy of piecewise monotone mappings, Studia Math. 6 7 (1980), 45-63. 354 TOPOLOGICAL ENTROPY AND VARIATION FOR TRANSITIVE MAPS [7] PARRY, W . : Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math . Soc. 122 (1966), 368-378 . Received December 18, 1991 Revised July 9, 1992 * ) Department of Mathematics Faculty of Civil Engineering CVUT Thdkurova 7 166 29 Praha 6 Czech Republic **) Mathematical Institute Slovak Academy of Sciences Stefánikova 49 814 73 Bratislava Czechoslovakia Present address: Department of Mathematics University of Warwick Coventry, CV4 7AL Great Britain 355 

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Topological entropy and variation for transitive maps

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Topological entropy and variation for transitive maps

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