yunping jiang- asymptotic differentiable structure on cantor set

8/3/2019 Yunping Jiang- Asymptotic Differentiable Structure on Cantor Set

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C o m m u n . M a th . Phys . 155, 503- 509 (1993) Communications Ma t hema t i c a lPhysics Springer-Verlag 1993

Asymptotic Differentiable Structure on Cantor SetYunping JiangD e p a r t m e n t of M a t he m a t ic s, Qu een s College of C U N Y , Flushing, NY 11367, USAReceived Janua ry 17, 1992; in revised form N o v e m b e r 3, 1992

Abstract. We study hyperbolic maps depending on a parameter . Each of them hasan invariant Cantor set. As tends to zero, the map approaches the boundary ofhyperbolicity. We analyze the asymptotics of scaling function of the invariant Cantorset as goes to zero. We show that there is a limiting scaling function of the limitingmap and this scaling function has dense jump discontinuities because the limiting mapis not expanding. Removing these discontinuities by continuous extension, we showt h a t we obtain the scaling function of the limiting map with respect to a Ulam- vonN e u m a n n type metric.0. IntroductionA differentiable structure on the topological Cantor set determined by the intrinsicscaling function of a geometric Cantor set on the line has been introduced by Sullivan[4]. Consider a family of geometric Cantor sets on the line. What will happen to suchdifferentiable structures if all gaps of geometric Cantor sets finally close to form aninterval (a one dimensional differentiable manifold)? It is the question we will discussin this paper. The main results (Theorems 1 and 2) say that for a family of geometricC a n t o r sets on the line generated by a family of nonlinear folding maps, the scalingfunctions tend to the scaling function of a sequence of partitions on the interval[2] as all gaps finally close. And moreover, the scaling function of the sequence ofpartitions on the interval has countably many jump discontinuities, and is uniformlyc o n t i n u o u s at the rest of the points on the dual Cantor set. If we replace all thediscontinuities by the continuous extension of the values on the rest of the points,we get a Holder continuous function on the dual Cantor set. This function is thescaling function of a sequence of partitions with respect to a Ulam- von Neumanntype metric [6] (we call this metric an orbifold metric (with respect to the Lebesguem e t r i c ) ) . In the other words, if we consider the interval with this Ulam- von Neumanntype metric as a Riemannian manifold (an orbifold, compare to [5]), the limit ofdifferentiable structures on the topological Cantor set is the differentiable structure ofthis Riemannian manifold.


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504 Y. JiangThis paper is organized as follows: in Sect. 1, we recall a short definition of a

scaling function for both the geometric Cantor set on the line and a sequence ofpartitions on the line. Following the definitions we state the main results (Theorem 1an d 2). In Sect. 2, we show two uniform estimates of the nonlinearities of longcompositions of any map in a family (Lemma 2 and 3) and the proofs of the mainresults.

1. Statements of Main ResultsSuppose / is the interval [1,1] and / is a C 1 folding map from / to the real linewith a unique critical point 0. We say 0 is a power law critical point if there is an u m b e r 7 > 1 such that the limits of f ' ( x ) / \ x \ ~ exist and equal nonzero numbersas x tends to zero from above and below. The number 7 is called an exponent of / .Hen c efo r t h , we will assume that / is increasing on [1,0] and decreasing on [0,1],an d / (0) > 1, / ( - I ) = / ( I ) = - 1 , and that 0 is the power law critical point of / .

Let g0 from / to [1,0] and g from / to [0,1] be the left and right inversebranches of / . For a sequence w = i0 ... in of symbols {0,1}, let gwgiQ" 9inbe the composition of gi to gin and I2 = gw(I) be the image of / under gw. We use nto denote the collection of intervals Iw for all w of length n + 1 and n = max \ IW\to denote the maximum of the lengths of the intervals in n. iwe nDefinition 1. We say / is in J&, the space of hyperbolic maps, if (a) / (0) > 1, (b) /is C+a for some 0 < a < 1, and (c) there are constants K > 0 and 0 < < 1 sucht h a t n < Kn for all integers n > 0. + 00Suppose / is a map in 3$. Then Cf = f] f~ n( I) , the maximal invariant set of / ,

= 0is a ge o m e t r ic C a n t o r set on the l i ne (thep r o o f is to use the n a i ve d i st o r t i o n l e m m a ) .T h e s c a l i n g f u n c t i o n of t h i s g e o m e t r i c C a n t o r set has b e e n d e f i n e d in [4] as fo l lows:s u p p o s e wn = in ... iji 0 is a f i n i t e s equence of 0 and 1. T a k e vn_ = in .. x anddefine s(wn) = \ I W v\ l \ I Vn _ x\ > w h e r e IWn = gW n( I ) , IVn__ = 9 Vn _ {I ) and | | is th eLebesgue measure. Suppose

is the dual Cantor set. If lim s(wn) exists for every * = . . . wn in C*, we definea function s( *) = lim s( uO on C*. This function is the scaling function of Cf. H+ + OO JI t is a C1- invariant (in other words, it is the same for both / and hofoh~ wheneverh is a C^diffeomorphism of / ) .Theorem A (Sullivan [4]). Suppose f is a map in 3$. The scaling function s of C^exists and is Holder {see [1] for the definition of a Holder continuous function on asymbolic space).

N o w we consider a map / with / (0) = 1, which maps / into itself. In this case,th e interval / is the maximal invariant set of / and we define an orbifold metric on/ (associated with / ) as

, dxay = .(1 - x1) 7


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Asymptotic Differentiable Structure on Cantor Set 505

Remark 1. The orbifold metric dy can be thought of as a singular metric withrespect to the Lebesgue metric dx and the corresponding change of coordinate is

th (x ) = 1 + 6 / dt/{\ - t2p- Vh, where b is determined by h(l) = 1.- 1Lemma 1. Suppose f hy o / oh~ {it is f: (/, dy) (/ , dy)). Then f is continuousand / |[- 1 , 0] and / |[0,1] are C - embeddings.Proof. It is clearly that / is continuous and f'(x) 0 for x in (0,1). By the definitionof the power law critical point in this paper, it is easy to check that the one- sidedderivatives of f(x) at 0, 1 and 1 are not zero either, and that f(x) is continuouson [- 1 ,0] and [0,1].Definition 2. We say / is on JBH, the boundary of hyperbolicity, if (1) / (0) = 1,(2) / |[- l, 0] and / |[0,1] are C 1 + embeddings for some 0 < a < 1, and (3) thereare constants K > 0 and 0 < < I such that n < Kn for all integers n > 0.Remark!. Let r__{x) and r+(x) be f ' (x ) / \x \ ~ x on [1,0] and [0,1], respectively.T h e n (2) is equivalent to say that / , r_ and r + are C+a for some 0 < o < 1. Someexamples of maps on JSH can be found in [3].

Although the maximal invariant set of / on J$H is the interval / , we can stilldefine the scaling function of the sequence of partitions of / similarly: supposeV ^nlnS) i s m e sequence of partitions of / determined by / . For everyfinite sequence wn fcn Vo f ^ an


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506 Y. Jiang

Suppose / is on J9H and = { n}^0 is the sequence of partitions of /determined by / . Let s(wn) = | | / W n | | / | | / ^ _ 1 | | , where wn = vn_xiQ and || || isth e measure on / induced by the orbifold m et r i c dy. If s( *) = lim s(w ) existsfor every * = . . . wn in C *, t hen s is defined as the scaling function of u n d e r dy.Corollary 1. Suppose f is on J8H. The scaling function s of under the orbifoldmetric dy exists and is Holder, and moreover, it is the continuous extension of s\ I* toC * (see Fig. 1) .

A family {/ }0< < i in 3%U JBE with / (0) = 1 + is said to be regular if(i) F(x, ) = f (x) is a C 1 function of [- 1,1] x [0,1] and f(x) are uniformly - H ld er co n tin u o us functions of x,(ii) f has the same exponent 7 at 0 for every in [0,1] and r_ (x, ) f' ( x ) / \ x \ ~ on [1,0] x [0,1] and r+{x, ) = f' ( x ) / \ x \ ~ on [0,1] x [0,1] are c o n t i n u o u s andr + ( x , ) and r_(x, ) are uniformly - H o ld er c on t in u o u s functions of x, and(iii) t he r e are two constants K > 0 and 0 < < 1 such t h a t n < Kn for alln > 0 and 0 < < 1.

Theorem 2. Suppose {/ }0< < i in 3@U ^ ^ is regular and { 0 swc/i that for every in [ 0 , 1 ] , if f(x) andf (y ) are in the same connected component ofV for i = 0,1, . . . , n 1,

f It is the naive distortion lemma since f (x) 0 for x in V .Lemma 3. There is a constant K > 0 such that for every in [ 0 , 1 ] , every pair x andy in an interval of n and every 0 < n < m, if f n(x) and f n(y ) are in U , then

' " n V "' < exp(K|/ (x) - f ivT).\ ( f n) ' (y) \


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Asymptotic Differentiable Structure on C antor Set 507

Proof. Let x % an d y % be the images of x an d y under / \ Then x i an d y i lie in asame interval of r]^ ra_ ) . F or every positive integer n < ra ,

e ) ( y ) f jThis product can be factored into two products,TT A> a n d M .1 1 f/ > \ 1 1 / /(> ^

Since (i) and (iii) in the definition of a regular family, there is a constant K x > 0such that l/(*t,yeve J {

To estimate the second product, leth {x) = - l+b _ i ((1 + ) - t2) i

be the family of changes of coordinates on 7 (where h 0 is the orbifold metric wementioned in Sect. 1). Let

Then we can factor the second product into three factors,

J k K W Jk h < Jk & h& 2> JkFurther, the last one can be factored into

n { l + i y ' ) ) t - d (1 - M ))t xt,viUe (l + fe(Xi))y i,yieu ( i _ / e ( ^ ) ) VSince h ' (x ) an d f' (x) are uniformly a'- H older continuous functions on each ofJ 0 1 an d In(U = I 0l U I n ), there is a constant K 2 > 0 so that

j ^ l < ^ and ^ ^I t is clearly that there is a constant K 2 > 0 such that

( l + ( y i ) ) ^ ! < g 3To fin ish the proof we write

- / e(a r f = 1


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508 Y. Jiang

F r o m the facts that / ( l ) = / ( - l ) = - 1 and foj(%i) is in U for some i < j 0such that

( 1 - / , ( % ) ) VT h e n K KK^K^K^ is a constant which satisfies the lemma.A straightforward calculation from Lemma 2 and 3 to the cases that * = . . . wnis in O* and is in / *, respectively, implies that there are constants K = K(a*) > 0

and 0 < a < 1 and a subsequence {n j of the integers such that for all m > k > ni9and and ' in [0,1],

\ s {wk) - s ,(wm)\ < K{\I W n JProof of Theorem 1. The existence of the scaling function s of the sequence ofpartition follows from the inequality (*) as ' 0.

The scaling function s is continuous at every point * in 7* and the restrictions|I * is uniformly continuous follow from the inequality (*) and the fact thatsup K(a*) < oo too.

Suppose * = (. .. OOwi)is in O*. Let 0 n be the finite string of zeroes of lengthn . Then the interval IOnW is eventually in the left interval 700

We use bn and an tondenote the lengths of nW and IOnWi, respectively (where IOnU)i C I On W ). Let cn bet h e distance from IOnW to - 1 (see Fig. 3).

- 1I I I

quasilm e a r

Fig. 3quasilinear

Using the naive distortion lemma, we can show

a*) = lim > 2(a*) = lim


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Asymptotic Differentiable Structure on Can tor Set 509

exist. These implies that the limit of the sequence {s(j 10 nwi)}^L { (where j is either0 or 1) exists and equals

I( i + T i ( * ) ) 7 _F o r a point 6* = (.. . j10n ;i) in C*, because Ij\onW is in C/ we can use Lemma 3to show that the error of s(J10ni) to s(6*) can be estimated by \IjiOnWl\ a, that is,t he r e is a positive constant K such that

F u r t h e r , lim 5(6*) = lim s(J10nwi).Since s(a*) = 2(a*)/ (a*), we get

lim 5(6*) 5(6*) ^ s ( * ) ,in the other words, s has jump discontinuity at *.Proof of Corollary 1. It follows from the naive distortion lemma and the fact that

Proof of Theorem 2. (I) is also from the naive distortion lemma. (II) follows fromth e inequality (*). (Ill) follows from the inequality (*) and that 0 < max K(a*)< + 00. * e l *

Acknowledgement. I would like to thank Dennis Sullivan for valuable suggestions and remarks.

References1. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Berlin,

Heidelberg, New York: Springer2. Jiang, Y.: Ratio geometry in dynamical systems. Preprint , June, 1989, IHES3. Jiang, Y.: G eneralized Ulam- von Neum ann transformat ions. Thesis, Graduate School of CU NY,

1990 and UMI dissertation information service4. Sullivan, D.: Differentiable Structure on Fractal Like Sets Determined by In trinsic Scaling

F u n c t i o n s on Dual Cantor Sets. The Proceedings of Symposia in Pure Mathematics, Vol. 48,1988

5. Thurston, W.: The geometry and topology of three- manifolds. Preprint, Princeton University 19826. Ulam, S.M., von N eumann , J.: On the Combinations of Stochastic and Deterministic Processes.Bull. Am. Math. Soc. 53, 1120 (1947)

C o m m u n i c a t e d by J.- P. Eckmann


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yunping jiang- asymptotic differentiable structure on cantor set

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