ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS

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1 ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1,

2 1. Symbolic Dyamics Defiitio 1.1. The sequece space o two symbols is the set Σ = {(s 0, s 1, s,...) : s j = 0 or 1} Propositio 1.. The distace d o Σ is defied by d(s, t) = Σ s i t i i. Proof. Let s = (s 0, s 1, s,...), t = (t 0, t 1, t,...) ad u = (u 0, u 1, u,...). Clearly, d(s, t) 0 ad d(s, t) = 0 if ad oly if s = t. Sice s i t i = t i s i, it follows that d(s, t) = d(t, s). Fially, for ay three real umbers s i, t i, u i we have the usual triagle iequality from which we deduce that This completes the proof. s i t i + t i u i s i u i d(s, t) + d(t, u) d(s, u). Theorem 1.3. Let s, t Σ ad suppose s i = t i for i=0,1,,... The d(s, t) 1 Coversely, if d(s, t) < 1, the s i = t i for i. Proof. If s i = t i for i, the d(s, t) = Σ s i t i i=+1 i Σ 1 i=+1 i = 1 O the other had, if s i t i for some i, the we must have d(s, t) 1 1 i. Cosequetly if d(s, t) < 1, the s i = t i for i = 0, 1,,... Defiitio 1.4. The shift map σ : Σ Σ is defied by σ(s 0, s 1, s,...) = (s 1, s, s 3,...). Our first observatio about this map is that the subset of Σ that cosist of all periodic poits i Σ is a dese subset. To see why this is true, we must show that, give ay poit s = (s 0, s 1, s,...) i Σ, we ca fid a periodic poit arbitrarily close to s. So let ɛ > 0. Let s choose a iteger umber so that 1 < ɛ. We may ow write dow a explicit periodic poit withi 1 uits of s. Let t = (s 0, s 1,..., s, s 0, s 1,..., s ). The first + 1 etries of s ad t are the same. By Theorem 1.3 this meas that d(s, t ) 1 < ɛ. Clearly t is a periodic poit of period + 1 for σ : Σ Σ. Sice ɛ ad s were arbitrary, we have succeeded i fidig a periodic poit arbitrarily close to ay poit of Σ. Note that the sequece (of sequeces) {t } coverges to s i Σ as. A secod ad eve more iterestig property of σ is that there is a poit whose orbit is dese i Σ. That is we ca fid a orbit which comes arbitrarily close to ay poit of Σ. Clearly, this kid of orbit is far from periodic or evetually periodic. As above, we ca write dow such a orbit explicitly for σ. Cosider the poit ŝ = ( ). I other words, ŝ is the sequece which cosists of all possible blocks of 0 s ad 1 s of legth, followed by all such blocks of legth, the legth 3, ad so forth. The poit ŝ has a orbit that forms a dese subset of Σ. To see this, we agai choose a arbitrary s = (s 0, s 1, s,...) Σ ad a ɛ > 0. Agai choose so that 1 < ɛ. Now we show that the orbit of ŝ comes withi 1 uits of s. Far to the right i the expressio for ŝ, there is a block of legth + 1 that cosists of the digits s 0 s 1...s. Suppose the etry s 0 is at the k-th.

3 place i the sequece. Now apply the shift map k times to ŝ. The the first +1 etries of σ k (ŝ) are precisely s 0 s 1...s. So by Theorem 1.3 we get d(σ k (ŝ), s) 1 < ɛ. There is a dyamical otio that is itimately related to the property of havig a dese orbit. This is the cocept of sesitivity. Defiitio 1.5. A metric dyamical system ((X, d), F ) depeds sesitively o iitial coditios, if there is a β > 0 such that for ay x X ad ay ɛ > 0 there is k N ad y X with d(x, y) < ɛ such that d(f k (x), F k (y) β 3 To see that shift map depeds sesitivity o iitial coditios, we choose β = 1. For ay s Σ ad ɛ > 0 oe ca agai pick N so that 1 < ɛ. Suppose t Σ satisfies d(s, t) < 1 but t s. The we kow that t i = s i for i = 0, 1,, 3,... However, sice t s there is k > such that s k t k. So s k t k = 1. Now cosider the sequece σ k (s) ad σ k (t). The iitial etries of each of these sequeces are differet, so we have d(σ k (s), σ k (t)) s k t k 0 + Σ 0 i = 1. This proves sesitivity for the shift.. Chaos by R.L.Devaey Defiitio.1. A dyamical system F : X X is chaotic if P 1 Periodic poits for F are dese. P F is trasitive. P 3 F depeds sesitivity o iitial coditios. Theorem.. The shift map σ : Σ Σ is a chaotic dyamical system. Theorem.3. The doublig map f is chaotic o the uit circle. Proof. Let S 1 be the uit circle {(x, y) : x + y = 1}. I the complex aalysis S 1 = {e iθ : θ R}. 1).Let e iθ S 1 ad U is a ope eighborhood of e iθ. Let A be a ope arc i U cotaiig e iθ, too. Note that f () (A) is a arc -th loger as A. There exists N such that f () (a) is a cover of S 1. Deote this iteratio by N. There are two poits x, y with d(f (N) (x), d(f (N) (y)) = 1. ). Let U, V are ope sets i S 1. If we precede as above, the for large eough we got that f () (U) covers S 1 ad therefore itersects V. 3). For poits of the form e iθ with period, the followig equatio holds e iθ. This meas that there periodic poits are uit roots with order 1. The set of all there poits is dese i S 1. Theorem.4. Let s suppose that g T = T f, ad T is cotiuous ad subjective map. If f is trasitive or periodic o X, the g is also trasitive or periodic o Y. Theorem.5. The fuctio g(x) = x chaotic order the iterval [, ]

4 4 Proof. Let T : S 1 [, ] be defied as T (e iθ ) = cosθ. It s clear to see that T is cotiuous ad subjective ad g T (e iθ ) = T f(e iθ ) = cosθ. f is chaotic o S 1, the by theorem.5, the fuctio g is chaotic too. The followig theorem which relates to chaos defiitio was published i 199. Theorem.6. Let (X, d) be a metric space that is iclude a ifiite set of poits. If the mappig f : X Y is cotiuous ad trasitive ad if a set of periodic poits is dese i X, the f is sesitive depedet o iitial coditios. Proof. Let choose two periodic poits q 1, q such that O(q 1 ) O(q ) =. Let δ 0 = d(o(q 1 ), O(q )). We ll show that the sesitive depedet o the iitial coditios holds whe δ = δ 0 8. Notice that δ 0 > 0, ad for every x X, other d(x, O(q 1 )) > δ 0 or d(x, O(q )) > δ 0. Let x X ad U be a ope set that icludes x. Let B δ (x) be a ope sphere with radius δ ad ceter x. Let p be a periodic poit i W = U B δ (x) with period. From this we coclude that oe of the poits q 1, q (deoted by q) has a orbit, for which d(x, O(q)) > 4δ. Let s defie V = i=0f ( i) (B δ (f (i) (q))). The set V is o empty, because q V ad V is ope. From the trasitivity of f exists a poit y W ad iteger umber k such that f (k) (y) V. Let j be a iteger part of k + 1. Cosequetly, k + 1 = j + r, whe r is the rest, 0 r < 1. Clearly, j k = r. It follows that 0 j k. By costructio, Let f (j) (y) = f (j k) (f (k) ) f (j k) (V ) B δ (f (j k) (q)). a = f (j) (y), b = f (j k) (q). Note that d(a, b) < δ. Let us use the triagle iequality for poits p, a, b ad x, p, b: Whe or By costructio or d(p, b) d(p, a) + d(a, b), d(x, b) d(x, p) + d(p, b). d(x, b) d(x, p) + d(p, a) + d(a, b), d(p, a) d(x, b) d(x, p) d(a, b). d(x, b) = d(x, f (j k) (q)) d(x, O(q)) 4δ. Sice p B δ (x), the d(x, p) < δ. From this it follows that d(a, b) > 4δ δ δ, d(f (j) (p), f (j) (y)) > δ.

5 Applyig the triagle iequality to the followig poits f (j) (x), f (j) (p), f (j) (y), we get that: d(f (j) (x), f (j) (p)) > δ or d(f (j) (x), f (j) (y)) > δ. 3. Topological trasitivity, miimality 5 Give a dyamical system (X, S) we let sx deote the image of x X uder the homeomorphism correspodig to the elemet s S. Let O S x be the orbit of x; i.e. the set {sx : s S}. O S x will deote the orbit closure of x. If (X, S)is a system ad Y a closed S-ivariat subset, the we say that (Y, S), the restricted actio, a subsystem of (X, S). Defiitio 3.1. The dyamical system (X, S) is called topologically trasitive or just trasitive, if for every pair of o-empty ope sets U, V i X there exists s S with su V. Defiitio 3.. The dyamical system (X, S) us called poit trasitive, if there exists poit x 0 X with O S x = X. Such x 0 is called a trasitive poit. Example 3.3. If X = S 1 ad S = {f () : N, f(z) = z }, the this dyamical system is trasitive. Defiitio 3.4. We dyamical system (X, S) is called miimal, if O S x = X for every x X. Defiitio 3.5. A poit x i dyamical system (X, S) is called miimal(or almost periodic), if the subsystem O S x is miimal. Defiitio 3.6. If set of miimal poits is dese i X, we say that (X, S) satisfies the Brostei coditio. If, i additio, the system (X, S) is trasitive, we say that it is a M-system. Defiitio 3.7. A poit x X is a periodic poit, if O S x is fiite set. If (X,S) is a trasitive system ad set of periodic poit is dese i X, the we say that it is a P - system. For a system (X, S) ad subsets A, X X, we use the followig atatio N(A, B) = {s S : sa B }. I particular, for A = {x} we write N(x, B) = {s S : sx B}. Defiitio 3.8. A subset P S is (left) sydetic, if there exists a fiite set F S such that F P = S. Theorem 3.9. The followig are equivalet: 1 (X, S) is miimal. (X, S) is a trasitive ad for every x X ad eighborhood U of x, the set N(x, U) = {s S : sx B} is sydetic i S. Proof. (a) = (b): If (X, S) is a miimal system the for every o-empty set U i X there exists a fiite subset F = {s 1, s,..., s k } i S with s i U = X. The j N(x, s i U) := S. But N(x, s j U) = s j N(x, U), the N(x, U) us sydetic. (a) = (b): For every x X ad every eighborhood U a set s j U is covered X. The for all ope set V exists s i S such that s i U V. Because X is a metric space we get that (X, S) is a miimal system.

6 6 4. Equicotiuity ad almost equicotiuity Defiitio 4.1. The system (X, S) is equicotiuous if the semigroup S acts equicotiuously o X; for every ɛ > 0 there exists δ > 0 such that d(x 1, x ) < δ implies d(sx 1, sx ) < ɛ, for every s S. Example 4.. Every isometric system is equicotiuous. It s clear to see that we must take ɛ = δ. Defiitio 4.3. Let (X, S) be a dyamical system. A poit x 0 X is called a equicotiuity poit if for every ɛ > 0 there exists a eighborhood U of x 0 such that for every y U ad every s S, d(sx 0, sy) ɛ. Propositio 4.4. A system (X,S) is equicotiuous iff every x X is a equicotiuity poit. Proof. : It s clearly. : Give ɛ > 0, let I := {U x : x X} be a collectio of eighborhoods as i the defiitio of equicotiuity poits. Ay Lebesgue umber δ for the ope cover I will serve for the equicotiuity coditio. Defiitio 4.5. The dyamical system (X, S) is called almost equicotiuous (or is a AE-system) if the subset EQ(X) of equicotiuity poits is a dese subset of X. Propositio 4.6. A miimal almost equicotiuous system is equicotiuous. Proof. Let x 0 be a trasitive poit ad x EQ(X). We shall show that also x 0 EQ(X). Give ɛ > 0 there exists δ > 0 such that for all x,, x,, B δ (x), d(x,, x,, ) < ɛ for all s S. Sice x 0 is a trasitive poit, there exists s, S ad η > 0 such that s, B η (x 0 ) B δ (x). Thus for every z B η (x 0 ) ad every s S we have d(ss, z, ss, x 0 ) < ɛ foe all s S; i.e. x 0 EQ(X) ad we coclude that the set of trasitive poits is cotaied i EQ(X). The a miimal almost equicotiuous system is equicotiuous. 5. Sesitive dyamical system Defiitio 5.1. We shell say that a system (X, S) is sesitive if it satisfies the followig coditio(sesitive depedece o iitial coditio): there exists a ɛ > 0 such that for all x X ad all δ > 0 there are some y B δ (x) ad s S with d(sx, sy) > ɛ. We say that (X, S) is o-sesitive otherwise. Propositio 5.. A trasitive dyamical system is almost equicotiuous iff it is osesitive. Proof. Clearly a almost equicotiuous system is o-sesitive. Coversely, beig osesitive meas that for every ɛ > 0 there exists x ɛ X ad δ ɛ > 0 such that for all y B δɛ (x ɛ ) ad every s S, d(sx ɛ, sy) < ɛ. For m N set V 1/m = B δ1/m (x 1/m ), U m = SV 1/m ad let R = m N U m. Suppose x R ad ɛ > 0. Choose m so that /m < ɛ, the x U m implies the exists s 0 S ad x 0 V 1/m such that s 0 x 0 = x. Put V = s 0 V 1/m. We ow see that for all y V ad every s S d(sx, sy) = d(ss 0 x 0, ss 0 y 0 ) < /m < ɛ, y 0 V 1/m. Thus the dese G δ set R cosists of equicotiuity poits. I this propositio we have see that miimality ad almost equicotiuity imply equicotiuity. We easily get a stroger result.

7 Theorem 5.3. A almost equicotiuous M-system (X, S) is miimal ad equicotiuous. Thus a M-system (hece also P -system) which is ot miimal equicotiuous is sesitive. Proof. Every trasitive poit is a equicotiuity poit. Let x 0 X be a equicotiuity ad trasitive poit. Give ɛ > 0 there exists a 0 < δ < ɛ such that x B δ (x 0 ) implies d(sx 0, sx) < ɛ for every s S. Let x, B δ (x 0 ) be a miimal poit. It that follows that T = {s S : d(sx 0, sx) ɛ} is a sydetic subset of S. Collectig these estimatios we get, for every s S, d(sx 0, x 0 ) d(sx 0, sx, ) + d(sx,, x 0 ) ɛ. Thus for each ɛ > 0 the set N(x 0, B ɛ (x 0 ) is a sydetic, whece x 0 is a miimal. It follows that X is a miimal, hace also equicotiuous by Propositio 4.6. Now, by propositio 5. we have that dyamical system which ot miimal or equicotiuous is sesitive. 7

8 8 Refereces 1. R.M. Crowover, Itroductio to Fractals ad Chaos, Joes ad Barlett Publishers, R.L. Devaey, A first course i chaotic dyamical systems. Theory ad experimet, Joes ad Barlett Publishers, R.M Crowover, Itroductio to Fractals ad Chaos, Joes ad Barlett Publishers, J.Baks, J.Brooks G.Gairs,G.Davis,P Stacey, O Devaey s defiitio of chaos,amer.math.mothly 99, E. Glaser, Joiigs ad Ergodyc Theory, 00. Departmet of Mathematics, Bar-Ila Uiversity, 5900 Ramat-Ga, Israel address: bizia@walla.co.il URL: katore