AN ARC-LIKE CONTINUUM THAT ADMITS A HOMEOMORPHISM WITH ENTROPY FOR ANY GIVEN VALUE

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1 AN ARC-LIKE CONTINUUM THAT ADMITS A HOMEOMORPHISM WITH ENTROPY FOR ANY GIVEN VALUE CHRISTOPHER MOURON Abstract. A arc-like cotiuum X is costructed with the followig properties: () for every ɛ [0, ] there exists a homeomorphism g ɛ : X X such that the etropy of g ɛ is ɛ ad (2) X does ot cotai a pseudo-arc. This aswers a questio by W. Lewis.. Itroductio Let f : X X be a cotiuous fuctio. The topological etropy of f is a umber i [0, ] that measures the rate of expasio ad mixig of f. For more o etropy see [], [8] ad [9]. The purpose of this work is to costruct a arc-like cotiuum X such that for every ɛ [0, ] there exists a homeomorphism g ɛ : X X such that the etropy of g ɛ is ɛ ad such that X cotais o pseudo-arc. This aswers a questio by Waye Lewis i the egative (see [4]). A cotiuum is a compact, coected metric space. A cotiuum is arc-like if it is the iverse limit of arcs. Arc-like cotiua are also called chaiable. A cotiuum X is decomposable if there exist proper subcotiua A ad B such that X = A B. A cotiuum is idecomposable if it is ot decomposable. A cotiuum is hereditarily idecomposable if every subcotiuum is idecomposable. The pseudo-arc is a arc-like hereditarily idecomposable cotiuum. For more o the pseudo-arc see [4]. It has bee show i [6] that every arc-like cotiuum that admits a homeomorphism with positive etropy must cotai a idecomposable subcotiuum. Also, the pseudo-arc admits a cotiuum-wise expasive homeomorphism which implies that it admits a ifiite etropy homeomorphism (see [3] ad [7]). However, it is ukow to whether the pseudo-arc admits a positive but fiite etropy homeomorphism. 2. Prelimiaries o Etropy ad Iverse Limits U is a cover of X if X U U U. U is a ope cover if every elemet of U is ope. Let X be a compact metric space, f : X X be a map, ad U be a fiite cover of X. Defie N(U, X) be the umber of sets i a subcover of U o X with smallest cardiality. If U ad V are two ope covers of X, let U V = {U V U U, V V} ad f (U) = {f (U) U U}. Also, defie f i (U) = U f (U)... f + (U), where f 0 = id ad Et(f, U) = lim (/) log N( f i (U), X). The the topological etropy of f is defied as Et(f) = sup{et(f, U, X) : U is a ope cover of X} Mathematics Subject Classificatio. Primary: 54H20, 54F50, Secodary: 54E40. Key words ad phrases. etropy, iverse limit, idecomposable cotiuum.

2 2 C. MOURON If U ad V are fiite covers of X such that for every V V there exists a U U such that V U, the V refies U. If for every V V there exists a U U such that V U, the V closure refies U. The followig propositios are well kow ad ca be foud i several texts such as [8]. Propositio. If V, U are covers of X such that V refies U, the Et(f, V) Et(f, U). Propositio 2. For each positive iteger k, Et(f k ) = ket(f). Let T : [0, ] [0, ] be defied by T (x) = { 2x if x [0, /2] 2 2x if x (/2, ], T (x) is ofte called the tet map. The followig theorem is well-kow. For further details see []: Theorem 3. Et(T ) = log(2). Theorem 4. Suppose that f, g, ad h are cotiuous fuctios such that the followig diagram commutes: the Et(g) Et(f). f X X h h Y g Y Proof. Follows from the fact that if U is a fiite ope cover of Y the Et(f, h (U)) = Et(g, U). Corollary 5. Suppose that f ad g are cotiuous fuctios ad h is a homeomorphism such that the followig diagram commutes: the Et(g) = Et(f). f X X h h Y g Y Let I be the uit iterval ad f i : I I be a cotiuous fuctio called a bodig map. The collectio {I, f i } i= is called a iverse system. Each iterval I is called a factor space of the iverse system. If each bodig map is the same map f, the the iverse system ca be writte as {I, f} i=. Every iverse system {I, f i} i= determies a topological space X called the iverse limit of the system ad is writte X = lim{i, f i } i=. The space X is the subspace of the Cartesia product i= I give by X = lim{i, f i } i= = { x i i= i= I f i(x i+ ) = x i }. X has the subspace topology iduced o it by i= I. If x = (x i) i= ad y = (y i) i= are two poits of the iverse limit, we defie distace to be d(x, y) = x i y i i= 2. i

3 ENTROPY HOMEOMORPHISMS 3 For more o the iverse limit arcs see [2]. If each bodig map f i is the same map f the there is a shift homeomorphism f : X X iduced by f: f(x) = f(x, x 2, x 3...) = (f(x ), f(x 2 ), f(x 3 ),...) = (f(x ), x, x 2,..). Also, otice that f (x, x 2, x 3...) = (x 2, x 3, x 4,...). The followig theorem aids i the computatio of etropy of fuctios o iverse limits ad is foud i [9]: Theorem 6. Let Y be a compact Hausdorff space, f : Y Y be a cotiuous oto fuctio ad X = lim{y, f} i=. Suppose that g : Y Y is cotiuous ad satisfies g f = f g. Additioally, suppose that ĝ : X X is defied by ĝ( x i i= ) = g(x i) i=. The Et(ĝ) =Et(g). The followig theorem by Mahavier [5] gives a sufficiet coditio for whe each subcotiuum i the iverse limit space must cotai a arc. Theorem 7. If f is a piecewise mootoe fuctio from [0, ] oto [0, ], the each subcotiuum of X = lim{[0, ], f} i= must cotai a arc. We will use the followig corollary to show that are cotiuum i the mai result cotais o pseudo-arc. Corollary 8. If f is a piecewise mootoe fuctio from [0, ] oto [0, ], the X = lim {[0, ], f} i= cotais o pseudo-arc. Proof. Suppose that X cotais a pseudo-arc P. The by Theorem 7, P must cotai a arc. However, arcs are decomposable, so P is ot hereditarily idecomposable ad hece ot a pseudoarc. 3. Mai Result I this sectio the mai result is costructed. U(Y ) = {U Y U U ad U Y }. If U is a cover of X ad Y X the defie Lemma 9. Suppose that h : X X is a cotiuous fuctio ad Y X such that h(y ) = Y. The Et(h Y ) Et(h). Proof. Let U be a fiite ope cover of Y (relative to the subspace topology iduced by X). The there exists a fiite ope cover V of X such that V(Y ) = U. Hece for each, (h Y ) i (U), Y ) = N( (h Y ) i (V(Y )), Y ) N( h i (V), X). N( Thus, Et(h Y, U) Et(h, V) ad the result follows. Lemma 0. For a (0, ) let f : [0, a] [0, a] ad g : [a, ] [a, ] be cotiuous fuctios such that f(a) = g(a). Defie { f(x) if x [0, a] h(x) = g(x) if x [a, ], the Et(h) = max{et(f), Et(g)}.

4 4 C. MOURON Proof. Let U be ay fiite ope cover of [0, ]. The there exists fiite ope covers V 0 ad V of [0, a] ad [a, ] respectively such that V 0 V refies U. The So (h) i (V 0 V ), [0, ]) = N( (f) i (V 0 ), [0, a]) + N( (g) i (V )[a, ]) N( Et(h, U) Et(h, V 0 V ) = lim sup lim sup 2 max{n( (f) i (V 0 ), [0, a]), N( log N( (h) i (V 0 V ), [0, ]) (g) i (V ), [a, ])}. log(2 max{n( (f) i (V 0 ), [0, a]), N( (g) i (V ), [a, ])}) log(max{n( (f) i (V 0 ), [0, a]), N( (g) i (V ), [a, ])}) = lim sup = max{et(f, V 0 ), Et(g, V )}. Thus Et(h) max{et(f), Et(g)}. It follows from Lemma 9 that Et(h) = max{et(f), Et(g)}. Theorem. Suppose that {a } =0 is a strictly icreasig sequece i [0, ) that coverges at where a 0 = 0. Let f : [a, a + ] [a, a + ], such that f (a + ) = f + (a + ) for every 0. Defie f : [0, ] [0, ] by { f (x) if x [a f(x) =, a + ] if x =, The Et(f) = sup{et(f )}. Proof. By Lemma 0, it follows that Et(f) = max{{et(f [0,a])}, Et(f [a,]} = max{{et(f i )}, Et(f [a,]} = sup{et(f )}. Let f be a cotiuous fuctio o the iterval I. Defie the extesio of f by { /2f(2x) if x [0, /2] E(f)(x) = (2 f())x + f() if x (/2, ], ad the two extesio of f by (f(0) + )x if x [0, /3) E 2 (f)(x) = /3f(3(x /3)) + /3 if x [/3, 2/3] (2 f())x + f() if x (2/3, ].

5 ENTROPY HOMEOMORPHISMS Figure. E(T )(x) ad E 2 (T )(x) (See Figure.) Let {g i } be such that the domai of g i is the rage of g i+. The defie g i+ i g i+k i = g i g i+... g i+k. Theorem 2. Et(E 2 (f)) = Et(E(f)) = Et(f). = g i ad Proof. Proof will be for Et(E(f)) = Et(f). Proof that Et(E 2 (f)) = Et(f) is similar. Let a = E(f)(/2) There are 2 Cases: Case : a < /2. First let a 0 = /2, a = (E(f)) (a 0 ) (/2, ] ad the iductively defie a = (E(f)) (a ). (Note that E(f) is - o [a, ]). Also otice that a as icreases. Next let g : [0, /2] [0, /2], g 0 : [/2, a ] [a, /2], g : [a, a 2 ] [/2, a ],... g k : [a k, a k+ ] [a k, a k ], where g(x) = E(f)(x) ad g i (x) = E(f) [ai,a i+](x). Notice that for each i 0, g i is - ad oto. Let V 0 be a ope cover of [0, /2] the defie V = {g 0 (V [a, /2]) V V 0} = {(E(f)) (V ) [/2, a ] V V 0 }. Cotiuig iductively defie V k = {g k (V ) V V k } = {(E(f)) k (V ) [a k, a k ] V V 0 }. Clearly, V k is a ope cover of [a k, a k ]. Next defie W (V 0 ) = ( V k ) {(a, ]}. k=0

6 6 C. MOURON It will be show that Et(E(f), W (V 0 )) = Et(g, V 0 ) for ay 0. First otice that () V 0 is ivariat uder g (2) g0 V 0 V = V (3) g i (V i ) = V i+ for i (4) (E(f)) ((a, ]) = (a +, ], Next let ad + W (V 0 ) g (W (V 0 )) = (V 0 (E(f)) (V 0 )) V k {(a +, ]}. The it follows that for m < ad for m that Q k = {U [0, /2] U Q k = {U [a, /2] U i= k= k g i (V 0 )} k g i (V 0 )}. m m m+ (E(f)) i (W (V 0 )) = Q m (g0) i (Q m i) V i {(a +m, ]}, i= i=m m m+ (E(f)) i (W (V 0 )) = Q m (g0) i (Q m i) V i {(a +m, ]}. So for m m N( (E(f)) i (W (V 0 )), [0, ]) = m N( Q m, [0, /2]) + Hece, Et(E(f), W (V 0 )) = lim sup m i= i=m m ( + m + )N( Q m, [0, /2]) +. lim sup m = lim sup m = Et(f, V 0 ). +m N((g0) i (Q m i), [a i, a i ]) + N(V i, [a i, a i ]) + log N( m (E(f)) i (W ), [0, ]) m + log(( + m + )N( m Q m, [0, /2]) + ) m + log N( m Q m, [0, /2]) m + Now let U be ay fiite ope cover of [0, ]. The there exists a fiite ope cover V of [0, /2] with sufficietly small mesh ad a iteger sufficietly large such that W (V) refies U. Hece Et(E(f), U) Et(E(f), W (V)) Et(f, V). i=m

7 ENTROPY HOMEOMORPHISMS Figure 2. D(T )(x) ad H(T )(x) Thus Et(E(f)) Et(f). Sice both 2x ad (/2)x are homeomorphisms, it follows from Corollary 5 that Et(f) = Et(/2f(2x)). Also from Lemma 9 it follows that Et(/2f(2x)) Et(E(f)). Hece, Et(E(f)) = Et(f). Case 2 a = /2. The E(f) is the idetity o [/2, ]. Thus it also follows from Lemma 9 ad Corollary 5 that Et(E(f)) = Et(f). Defie the halvig of f by ad the doublig f by H(f)(x) = { /2E(f)(2x) if x [0, /2] x if x [/2, ], D(f)(x) = { /2E(f)(2x) if x [0, /2] /2E(f)(2 2x) if x [/2, ]. (See Figure 2.) Defie H 2 (f) = H(H(f)) ad cotiuig iductively defie H (f) = H(H (f)). O the other had (H(f)) 2 = H(f) H(f). Notice that (H(f)) 2 = D(f). Theorem 3. Et(D(f)) = Et(f). Proof. Notice that h (x) = (/2)x ad h 2 (x) = (/2)x are homeomorphisms. Thus by Corollary 5, Et(E(f)) = Et(h E(f) h ) = Et(/2E(f)(2x)) ad Et(E(f)) = Et(h 2 E(f) h 2 ) = Et( /2E(f)(2 2x)). So by Lemma 0 ad Theorem 2, Et(D(f)) = max{et(/2e(f)(2x)), Et( /2E(f)(2 2x))} = Et(E(f)) = Et(f).

8 8 C. MOURON Theorem 4. Et(H (f)) = 2 Et(f). Proof. First, sice (H(f)) 2 = D(f), it follows by Propositio 2 ad Theorem 3 that Et(H(f)) = 2 Et(D(f)) = 2 Et(f). Cotiuig iductively suppose that Et(H (f)) = 2 Et(f). The it follows that Et(H (f)) = Et(H(H (f))) = 2 Et(H (f)) = 2 Et(f). The ext theorem is the mai result of this paper: Theorem 5. There exists a arc-like cotiuum X such that for every ɛ [0, ] there exists a homomorphism g ɛ : X X such that Et(g ɛ ) = ɛ. Furthermore, X does ot cotai a pseudo-arc. Proof. Now for the costructio of the arc-like cotiuum. For each iteger let be defied by s : [ 2 2, ] [ 2 2, ] s (x) = E 2(H (T ))(2 (x 2 2 ). The by Theorems 3, 2 ad 4, Et(s ) = 2 log(2). Next defie { s (x) if x [ g(x) = 2 if x =. 2, ] for = 0,, 2,... Let X = lim{[0, ], g} i= ad X = lim{[ 2 2 ], s + } i= for = 0,, 2,... The sice s is composed of a fiite umber of mootoe pieces, each X does ot cotai a pseudo-arc by Corollary 8. Thus, X = { i= } =0 X does ot cotai a pseudo-arc (see Figure 3). Give ɛ > 0, there exist positive itegers {c } =0 such that c c 2 log(2) ɛ ad lim 2 log(2) = ɛ. For ɛ = 0 let c = 0 ad for ɛ = let c = 2 2. The take 2, 2+ { s c g ɛ (x) = (x) if x [ 2 if x =. 2, ] for = 0,, 2,... So by Propositio 2, Et(s c (x)) = c 2 log(2). Thus, it follows from Theorem that Et(g ɛ ) = ɛ. Also sice s s c = s c s, it follows that g g ɛ = g ɛ g. Let ĝ ɛ : X X be defied by ĝ ɛ ( x i ) = g ɛ (x i ). The by Theorem 6, Et(ĝ ɛ ) = Et(g ɛ ) = ɛ. Also, otice that ĝ ɛ [ 2 is a shift homeomorphism o X. Thus, it follows that ĝ ɛ is a homeomorphism. 2, 2+ = ŝc 2+ ] which

9 ENTROPY HOMEOMORPHISMS 9 Figure 3. Cotiuum X. Questio. If h : P P is a homeomorphism o the pseudo-arc P, must Et(P ) = 0 or Et(P ) =? Ad if so, the cosider the followig questio: Questio 2. If h : X X is a homeomorphism o a hereditarily idecomposable cotiuum X, must Et(h) = 0 or Et(h) =? 4. Refereces () R. L. Adler ad M. H. McAdrew, The etropy of Chebyshev Polyomials, Tras. A.M.S., Vol. 2 No. (966), p (2) W. T. Igram, Iverse limits, Aportacioes Matemáticas: Ivestigació [Mathematical Cotributios: Research], 5. Sociedad Matemática Mexicaa, México, (3) H. Kato, Cotiuum-wise expasive homeomorphisms, Ca. J. Math. 45(993), o. 3, (4) Waye Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicaa (3) 5 (999), o., p (5) William Mahavier, Arcs i iverse limits o [0, ] with oly oe bodig map, Proc. A.M.S., Vol. 2, No.3 (969), p (6) Christopher G. Mouro, Positive etropy homemorphisms o chaiable cotiua, preprit. (7) Christopher G. Mouro, Hereditarily idecomposable compacta do ot admit expasive homemorphisms, preprit. (8) P. Walters, A Itroductio to Ergodic Theory, Graduate Texts i Math. 79, Spriger, New York, 982. (9) Xiagdog Ye, Topological etropy of the iduced maps of the iverse limits with bodig maps, Top. Appl., 67 (995) p Departmet of Mathematics, The Uiversity of Alabama at Birmigham, Birmigham, AL Departmet of Mathematics ad Computer Sciece, Rhodes College, Memphis, TN 382 address: mouroc@rhodes.edu