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Scalings in Circle Maps III

arXiv:math/9202209v1 [math.DS] 3 Feb 1992Scalings in Circle Maps IIIJ. GraczykG.

´Swi¸atekF.M. TangermanJ.J.P.

VeermanNovember 20, 2017AbstractCircle maps with a flat spot are studied which are differentiable,even on the boundary of the flat spot. Estimates on the Lebesguemeasure and the Hausdorffdimension of the non-wandering set areobtained.

Also, a sharp transition is found from degenerate geometrysimilar to what was found earlier for non-differentiable maps with aflat spot to bounded geometry as in critical maps without a flat spot.1Introduction1.1Maps with a flat spotWe consider degree one weakly order-preserving circle endomorphisms whichare constant on precisely one arc (called the flat spot.) Maps of this kindappear naturally in the study of Cherry flows on the torus (see [1]) as wellas “truncations” of smooth non-invertible circle endomorphisms (see[5]).They have been less thoroughly researched than homeomorphisms.Topologically, one nice thing about maps with a flat spot is that they stillhave a rotation number.

If F is a map with a flat spot, and f is its lifting,the rotation number ρ(F) is the limitlimn→∞f n(x)n(mod1)which turns out to exist for every x and its value is independent of x. Thedynamics is most interesting if the rotation number is irrational.1

We study first the topology of the non-wandering set, then its geometry.Where the geometry is concerned, we discover a dichotomy. Some of ourmaps show a “degenerate universality” akin to what was found in a similarcase considered by[2] and[10], while others seem to be subject to the“bounded geometry” regime, very much like critical homeomorphisms, i.e.maps which instead of the flat spot have just a critical point.Before we can explain our results more precisely, it is necessary to defineour class and fix some notations.Almost smooth maps with a flat spot.We consider the class of con-tinuous circle endomorphisms F of degree one for which an arc U exists sothat the following properties hold:1.

The image of U is one point.2. The restriction of F to S1 \ U is a C2- diffeomorphism onto its image.3.

Consider a lifting to the real line, and let (a, b) be a preimage of U,while the lifting of F itself is denoted with f. On some right-sidedneighborhood of b, f can be represented ashr((x −b)pr)for pr ≥1 with hr which extends as a C2-diffeomorphism beyond b.Analogously, in a left-sided neighborhood of a, f ishl((a −x)pl) .The ordered pair (pl, pr) will be called the critical exponent of the map.If pl = pr the map will be referred to as symmetric.In the future, we will deal exclusively with maps from this class. More-over, from now on we restrict our attention to maps with an irrational rota-tion number.Basic notations.The critical orbit is of paramount importance in study-ing any one-dimensional system, thus we will introduce a simplified notationfor backward and forward images of U.

Instead of F i(U) we will simply write2

i. This convention will also apply to more complex expressions.

For exam-ple, F −3qn−20(0) will be abbreviated to −3qn −20. This is certainly differentfrom F −3qn(0) −20 where −20 means an element of the group T1.

In ournotation, this difference is marked by 20 not being underlined, i.e. −3qn−20.An underlined complicated expression should be evaluated as a single imageof 0.

Thus, underlined positive integers are points, and non-positive ones areintervals.Let qn denote the closest returns of the rotation number ρ(F) (see [10]for the definition).Next, we define a sequence of scalingsσ(n) :=dist(0, qn)dist(0, qn−2) .A summary of previous related results.Maps with the critical expo-nent (1, 1) were studied first. The most complete account can be found in[9].

They turn out to be expanding apart from the flat spot. Therefore,the geometry can be studied relatively easily.

One of the results is that thescalings σ(n) tend to 0 fast.Next, critical exponents (1, ν) or (ν, 1) were investigated for ν > 1 inde-pendently in [2] and [10]. The main result was that σ(n) still tend to 0.

Thiswas shown to lead to “degenerate universality” of the first return map on(qn−1, qn). Namely, as n grows, the branches of this map become at least C1close to either affine strongly expanding maps, or a composition of x →xνwith such maps.Finally, we need to be aware of the results for critical maps where U isa point and the singularity is symmetric.

The scalings can still be definedby the same formula, but they certainly do not tend to 0 (cite [3] and [7]).Moreover, if the rotation number is golden mean, then they are believed totend to a universal limit (see [4].) This is an example of bounded geometry,and conjectured “non-degenerate” universality.In this context, we are ready to present our results.1.2Statement of resultsWe investigate symmetric almost smooth maps with a flat spot with thecritical exponent (ν, ν) ,ν > 1.First, we get results about the non-3

wandering set which are true for any ν. Also, we permanently assume thatthe rotation number is irrational.Theorem 1For any F with the critical exponent (ν, ν) , ν > 1, the setS1 \S∞i=0 f −i(U) has zero Lebesgue measure.

Moreover, if the rotation num-ber is of bounded type (i.e. qn/qn−1 are uniformly bounded), the Hausdorffdimension of the non-wandering set is strictly less than 1.Corollary.There are no wandering intervals and any two maps fromour class with the same irrational rotation number are topologically conju-gate.Theorem 2Again, we assume that the critical exponent is (ν, ν) withν > 1.

Then, we have a dichotomy in the asymptotic behavior of scalings. Ifν ≤2, the scalings σ(n) tend to 0.

If ν > 2, and the rotation number is ofbounded type, then lim infn→∞σ(n) > 0.Comments.Thus, Theorem 2 shows that a transition occurs from the“degenerate universality” case to the “bounded geometry” case as the ex-ponent passes through 2. This is the first discovery of bounded geometrybehavior in maps with a flat spot (which was conjectured in [10].

)Numerical findings.A natural question appears whether bounded geom-etry, when it occurs, is accompanied by non-trivial universal geometry. Moreprecisely, we have two conjectures:Conjecture 1For a map F from our class with the golden mean rota-tion number, the scalings σ(n) tend to a limit.We found this conjecture supported numerically, albeit only for one mapconsidered.

Moreover, the rate of convergence appears to be exponential.The reader is referred to Appendix B for a detailed description of our exper-iment.There is a much bolder conjecture:4

Conjecture 2Consider two maps from our class with the same criticalexponent larger than 2 and the same irrational (bounded type?noble? )rotation number.

Then, the conjugacy between them is differentiable at 1(the critical value according to our convention. )This conjecture is motivated by the analogy with the critical case.

Thesame analogy (see [6]) makes us expect that Conjecture 2 would be impliedby Conjecture 1 if the convergence in Conjecture 1 is exponential and thelimit is independent of F.Parameter scalingsConsider a smooth one parameter family ft of circlemaps in our class with constant critical exponent (ν, ν) for which d/dt ft >0. Assume that f0 has golden mean rotation number.Denote by In theinterval of parameters t for which ft has as rotation number pn/qn, the n−thcontinued fraction approximant to the golden mean.

The length |In| of theinterval In tends to zero as n tends to infinity. Define the parameter scalingδn as:δn = |In|/|In−2|When 1 ≤ν ≤2 the arguments in [11] yield an asymptotically exactrelation between parameter scalings and geometric scalings for f0:δn = σ(n −1)νWe conjecture that when ν > 2, the parameter scalings tend to a universallimit only depending on ν.

In fact, the same relation between parameterscalings and geometric scalings appears to hold.1.3Technical toolsDenote by (a, b) = (b, a) the open shortest arc between a and b regardlessof the order of these two points. If the distance from a to b is exactly 1/2,choose the arc which contains some right neighborhood of 0.

The distancebetween two sets X and Y is defined asdist(X, Y ) = inf{dist(x, y) : x ∈X, y ∈Y } .We shall write l(I) and r(I) appropriately for the left and the right endpointof interval I. In particular we set l = l(U) and r = r(U).5

The cross-ratio inequality.Suppose we have four points a, b, c, d ar-ranged according to the standard orientation of the circle so that a < b

Each point of the circle belongs to at most k among intervals (ai, di).2. Intervals bi, ci do not intersect UthennYi=0DCr(ai, bi, ci, di) ≤Ckand the constant Ck does not depend on the set of triples.In this paper all sets of triples will be formed by taking iterations ofan initial quadruple.

Therefore we will only indicate the initial quadrupletogether with the number of iterations one performs.This inequality was introduced and proved in [8].Lemma 1.1 There is a constant K so that for any two points y, z, if f is adiffeomorphism on (y, z), the following inequality holds:| f(y, z) |dist(f(y), f(U)) ≤K | (y, z) |dist(y, U)provided dist(z, U) ≤dist(y, U).Proof:It is a simple calculation.✷6

The Distortion Lemma.We use the following lemma which can be con-sidered a variant of the “Koebe lemma” which was the basis of estimates in[10].Let f be a lifting of an almost smooth map with a flat spot, and consider asequence of intervals Ij with 0 ≤j ≤n so that Ij+1 = f −1(Ij) and U ∩Ij = ∅for 0 ≤j < n. Choose an interval (a, b) ⊂I0 and let A be the orientation-preserving affine map from [0, 1] onto I0. Then, we define the “rescaled” map˜f := f −n ◦A.

So, ˜f maps [0, 1] onto In.The nonlinearity of ˜f satisfies the following estimate:˜f ′′˜f ′ ≤KCr(0, A−1(a), A−1(b), 1)where K is a uniform continuous function of Pn−1j=0 |Ij| only.proofThe lemma follows directly from the “Uniform Bounded Distor-tion Lemma” of [7].2Estimates valid for any critical exponent2.1Geometric boundsLemma 2.1 The sequence dist(qn, U) tends to zero at least exponentiallyfast.Proof:The orbit of U for 0 ≤i ≤qn+1+qn−1 together with open arcs lying betweensuccessive points of the orbit constitute a partition of the circle. Let I bethe shortest arc belonging to the setA := {(qn + i, i) : 0 ≤i ≤qn+1} .Denote the ratio| (3qn, qn) |dist(qn, U)by Γ(n).

We will show that Γ(n) is bounded away from zero. Lemma 1.1implies that| (3qn + 1, qn + 1) || (3qn + 1, 1) |≤K | (3qn, qn) |dist(3qn, U) .7

If I coincides with (qn, ∂U) then clearly Γ(n) ≥1/2. Otherwise we caniterate i times, mapping the interval (qn + 1, 1) onto I.

Note that intervals(3qn + 1 + i, qn + 1 + i) and (1 + i, −qn + qn+1 + 1 + i)cover two adjacent intervals to I from the set A.Now we write the cross-ratio inequality for{3qn + 1, qn + 1, 1, −qn + qn+1 + 1}and the number of iterations equal to i. We obtain the following estimate:| (3qn + 1, qn + 1) || (3qn + 1, 1) || (1, −qn + qn+1 + 1) || (qn + 1, −qn + qn+1 + 1) | ≥4/C3 .ThusΓ(n) ≥4/C3Kand dist(qn, U) ≤(1/(1 + Γ))dist(3qn, U).

The ordering of the orbit of Uimplies the next inequalitydist(qn, U) ≤(Γ/(1 + Γ))dist(qn−4, U)which completes the proof.✷Proposition 11. The sequence {σ(n)} is bounded away from 1.2.

The sequence| −qn−1 || (qn, qn−2) |is bounded away from zero.Proof:Let Un be the n-th partition of the circle given by all qn+1 + qn −1 preimagesof U, Jn = {−i : O ≤i ≤qn+1 + qn −1}, together with the holes betweensuccessive preimages of U. It is easy to see that the holes are given by thefollowing formula:8

1. −qn is on the left side of U. Set✷ni := f −i(r(−qn), l(U)) and ⃝nj := f −j(r(U), l(−qn+1)) .where j ranges from 0 to qn, and i is between 0 and qn+1.2.

−qn is on the right side of U. Set✷ni := f −i(r(U), l(−qn)) and ⃝nj := f −j(r(−qn+1), l(U)) .with i ranging from 0 to qn+1 and j from 0 to qn.ThenUn \ Jn = {✷ni , 0 ≤i < qn+1} ∪{⃝nj , 0 ≤j < qn} .Note that ⃝n−1j= ✷njTake two successive preimages of U which belong to the n-th partitionUn, say −i and −j. We may assume that −i lies to the left of −j.

Take asthe initial quadruple the endpoints of the considered preimages of U. We caniterate the quadruple{l(−i), r(−i), l(−j), r(−j)}until we hit U .

The cross-ratio inequality gives the following estimate:Cr(l(−i), r(−i), l(−j), r(−j)) ≥≥(| U | /C1)| −|i −j| || −|i −j| | +dist(−|i −j|, U)where |i −j| is equal to either qn or qn+1. Thanks to lemma 1.1 we knowthat the ratio of lengths of intervals adjacent to the plateau can be changedonly by a bounded amount.| −|i −j| + 1 || −|i −j| + 1 | +dist(−|i −j| + 1, 1) ≤≤K| −|i −j| || −|i −j| | +dist(−|i −j|, U) .9

Now we form a new quadruple from the endpoints of −|i −j| + 1 and twoadditional points: |i −j| and 1. To obtain the next estimate we write thecross-ratio inequality for the quadruple and the number of iterates equal to|i −j|.

Let us recall that we proved in lemma 2.1 that | (3|i −j|, |i −j|) |was big with comparison to dist(|i −j|, U). Hence| −|i −j| + 1 || −|i −j| + 1 | +dist(−|i −j| + 1, 1)≥Γ | U | /C3 .Combining all above inequalities we get| −i || (l(−i), l(−j)) || −j || r(−i), r(−j) | ≥≥Γ | U | /C3C1 .To finish the proof note that interval (qn−2, qn) contains exactly one preim-age of U which belong to Un−2, namely −qn−1.✷Lemma 2.2 The lengths of intervals ✷ni and ⃝nj tend to zero uniformlyexponentially fast with n.Proof:An interval ✷ni is subdivided into preimages of the flat spot and intervals ofthe form ✷n+1jand ⃝n+1k.

We will argue that a certain proportion of measureis lost in the preimages of U. To this end, apply to the cross-ratio inequalityto a quadruple given by the endpoints of two neighboring preimages of U inthe subdivision.

By Proposition 1, this cross-ratio is bounded away from 0.✷10

2.2Proof of Theorem 1The first claim of the Theorem follows directly from Lemma 2.2.The claim concerning the Hausdorffdimension requires a bit longer ar-gument. Suppose that the rotation number is of bounded type.

Take then −1-th partition of the circle S1. The elements of the next partition subdi-vide the holes of latter one in the following way:✷n−1i⊂an+1[j=0✷ni+qn+jqn+1 ∪⃝ni ,⃝n−1i= ✷n+1i.We estimateX(| ✷ni |α + | ⃝ni |αwhereP means the sum over all holes of n -th partition.

By Proposition 1follows that there is a constant β < 1 so thatan+1Xj=0| ✷ni+qn+jqn+1 |≤β | ✷n−1i|holds for all ’long’ holes ✷ni+qn+jqn+1 of n-th partition. In particular it meansthat the holes of n-th partition decrease uniformly and exponentially fast tozero while n tends to infinity.

We use concavity of function xα to obtain thatan+1Xj=0| ✷ni+qn+jqn+1 |α≤≤| an+1 + 1 |1−α βα | ✷n−1i|α≤≤| ✷n−1i|αif only α is close to 1. Hence the sum over all holes at power α of n -thpartition is a decreasing function of n. Consequently, the sum is less than1.

The only remaining point is to prove that for a given ε the holes of n -thpartition constitute an ε -cover of Ωif only n is large enought. But this is sosince the length of the holes of n -th partition goes to zero uniformly.

Thiscompletes the proof.11

3Controlled Geometry: recursion on the scal-ings3.1Proof of Theorem 2The strategy of the proof of the first part of this theorem is to establishrecursion relations between scalings (proposition 3.1), similar to what wasdone in [10]. A close study of these relations then implies the first part oftheorem 2: when ν > 2, these scalings are bounded away from zero.We will give the derivation of the recursion relation between scalings.Since this derivation is in many respects analogous to what was done inchapter 4 of [10], (in fact the only difference in the proofs is the change of thephrase ”essentially linear” to ”a priori bounded nonlinearity”), the discussionwill be somewhat sketchy.

The basic strategy is that closest returns factoras a composition of a power law and a map of a priori bounded distortion.This allows one to control ratio’s of lengths of dynamically defined intervals.Let f be a map satisfying the assumptions of theorem 2: the critical expo-nent is (ν, ν) and the rotation number is of bounded type. Then proposition2.1 supplies us with a priori bounds.In the sequel it is convenient to introduce a symbol (≈) for approximateequality.

Let { α(n) } and { β(n) } be two positive sequences. The notationα(n) ≈β(n)means that there exists a constant K ≥1 depending only on the a prioribounds and the type of the rotation number so that for all n:1K ≤α(n)β(n) ≤KProposition 2.1 (a priori bounds) implies that|−qn| ≈|qn−1| (3.1)The interval [qn, qn−2] contains the interval −qn−1 as well as its inverse im-ages: f −iqn(−qn−1) (i = 1,..., an −1).

Each interval [i qn, (i + 1) qn] containsone such inverse image. The distortion lemma (see introduction), the assump-tion that the singularity is a power law (with power ν), and the assumption12

that an is bounded imply (see also [10], chapter 4]:|((i −1) qn, i qn)| ≈|i qn|; for i = 2, .., an (3.2)This relation immediately implies:|(f((i −1) qn), f(i qn))| ≈|f(i qn)| (3.3)|((i −1) qn, i qn)||f(i qn)|Df(i qn) ≈ν (3.4)This last relation is the analogue of:x . ν xν−1xν= νDefine scalings σ(n, i) asσ(n, i) = |((i −1) qn, i qn)||(i qn, (i + 1) qn)|, for i = 1, .., an −1σ(n, an) = |((an −1) qn, an qn)||(an qn, qn−2)|Remark: σ(n) can not quite be expressed in these scalings.

However onehas:σ(n) =|(0, qn)||(0, qn−2)| ≈|(0, qn)||(an qn, qn−2)| = σ(n, 1) · . .

. · σ(n, an)We now show that the various scalings are related, through suitable deriva-tives of iterates at the critical value.

An application of the chain rule willfinally yield an interesting recursion relation. These recursion relations werefirst discovered in section 4 of [10], under the additional assumption thatscalings tended to zero.

Denote by { D(n) } the sequence of derivatives ofiterates at the critical value:D(n) = Df n(1)13

Of particular interest are those derivatives for closest returns.We nowpresent the relations of interest. As remarked before, their proofs are es-sentially the same as in [10] if one replaces the phrase ”essentially linear” to”a priori bounded nonlinearity”.

As in lemma 4.8 [10] we have:If an > 1 D(qn) ≈νσ(n, 1) (3.5a) :For i = 2, .., an −1 σ(n, i) ≈σ(n, i −1)ν (3.5b)The last relation implies that σ(n, i) can be expressed in terms of σ(n, 1):σ(n, i) ≈σ(n, 1)νi−1 (3.6)As in Theorem 4.6 [10] we have that:if an = 1 D(qn) ≈νan−1 νσ(n, 1)(3.7)if an > 1 D(qn) ≈νan−1 νσ(n, an) Πan−1i=1 σ(n, i)ν−1 (3.8)Equations 3.5 a,b and 3.6,8 imply that when an > 1 (but bounded by thetype of the rotation number)σ(n, an) ≈νan−1 σ(n, 1)νan−1 (3.9)The previous relations imply that every σ(n, i) can be expressed in termsof σ(n, 1). Consequently, D(qn) can be expressed in terms of σ(n, 1).

Thechain rule will finally yield a recursion relation between scalings at variouslevels. As in proposition 4.5 [10] we have that:D(qn+1) ≈D(qn)an Πani=2Df(iqn)Df(qn) D(qn−1) Df(qn+1)Df(qn−1)Expressing this relation in terms of σ(n + 1, 1), σ(n, 1) and σ(n −1, 1) oneobtains the following simple recursion relation.Proposition 3.1:σ(n + 1, 1)νan+1 ≈νp σ(n, 1)1−νan1−ν σ(n −1, 1)14

The power p only depends on the values of an, an−1 .Remark: 1. The quantity σ(n, 1)νan has a geometric interpretation as:σ(n, 1)νan ≈|(1, 1 + an qn)||(1, 1 + i qn−2)|2.

We have thatσ(n) ≈σ(n, 1)1−νan1 −νProof of the first part of Theorem 2 : If ν > 2 then lim inf σ(n) > 0Proof:By the second part of the last remark, it suffices to show that lim inf σ(n, 1) >0. Define the quantitys(n) = −νan ln(σ(n, 1))Proposition 3.1 implies that we have the recursion inequality:|s(n + 1) −1 −ν−anν −1s(n) −ν−an−1 s(n −1)| ≤boundHere the quantity bound only depends on the apriori bounds, the power ν andthe type of the rotation number.

It now suffices to show that the sequence{ s(n) } is bounded.Define the sequence of vectors { ζ(n) } as:ζ(n) =s(n)s(n −1)and the sequence of matrices { B(n) } as:B(n) = 1−ν−anν−1ν−an−110Then the recursion inequality implies that||ζ(n + 1) −B(n) ζ(n)|| ≤boundHere ||.|| denotes the Euclidean distance on the plane.15

We study long compositions of these matrices in appendix A. Since ν > 2,lemma A.2 in the appendix implies the existence of an integer N so that forany n, the compositionB(n + N) ◦... ◦B(n)uniformly contracts the Euclidean metric by a factor less than .8.Therefore the sequence of lengths {||ζ(n)||} is bounded.

Consequentlythe sequence { s(n) } is bounded and the sequences { σ(n, 1) } and { σ(n) }are bounded away from zero.✷Proof of the second part of Theorem 2: If ν ≤2 then limn→∞σ(n) =0The main idea is that when the power ν is close to 1, the map is actuallynot very non-linear.Consider the configuration of intervals described infigure 3.1. The intervals are: A = [0, an qn] and B = [an qn, qn−2] Applyqn−1 iterates to A ∪B.

Then A maps to A′ and B maps to B′. Note that B′contains U (is asymptotically equal to it) and is therefore large.

In particularthe ratio of lengths |A′||B′| is very small. Therefore, if the qthn−1 iterate of f onA ∪B is not very non-linear, one should expect that the initial ratio|A||B|is also small.

Consequently, the scalings tend to zero. The details for thisargument are found in the proof of proposition 3.2 below.An important observation is that the intervals { f i(∂U, qn−2)}i=1,..,qn−1do not intersect.We will need the following lemma.Lemma 3.1: Let a,b and z be positive reals:0 < a < b < zLet S be the map S : x →xν.

Then:|S(b) −S(a)||S(z) −S(b)| ≤(ab)ν−1 |b −a||z −b|16

A’B’U0ABUfq n-1qn-2qn-1q n+1q na nFigure 1:Proof:Consider the quotient r(z) of ratio’s:r(z) = bν −aνb −az −bzν −bνFix a and b and take the supremum over z:supz∈(a, ∞) r(z) = limz ↓b r(z) =bν −aνb −a1ν bν−1 ≤(ab)ν−1✷Proposition 3.2: For ν ≤2,limn →∞σ(n, an) σ(n + 1, an+1) = 0Proof:We will find an upper bound for the following quantity, measuring the non-linearity:Rn = ln |f qn−1(B)|/|f(B)||f qn−1(A)|/|f(A)|17

x lrxrx+Urightleftx l -MMFigure 2:Decompose the complement of the flat spot U in three overlapping parts (seefigure 2):An interval M = (xr, xl) in which f has bounded non-linearity.An interval right = (∂U, xr + ǫ) to the right of U where f is the com-position of x →|x|ν and a diffeomorphism.An interval left = (xl −ǫ, ∂U) to the left of U on which f on which f isthe composition of the x →−|x|ν and a diffeomorphism.We again remark that the interval f(A ∪B) and its qn−1−2 images underf are disjoint and do not land in the interval [qn−1, qn−2] containing the flatspot U.Denoting the ith forward image by a subscript i, we then have for i ∈{ 1, .., qn−1 −1}:Rn =XAi ∪Bi ⊂Mln |f(Bi)|/|Bi||f(Ai)|/|Ai| +XAi ∪Bi ⊂rightln |f(Bi)|/|Bi||f(Ai)|/|Ai| +XAi ∪Bi ⊂leftln |f(Bi)|/|Bi||f(Ai)|/|Ai| =:XM+Xright+XleftIn order to avoid over-counting, any couple of intervals which is strictlycontained in M, is included in the first sum. We now estimate each of thethree contributions separately.For the intervals that land in M, there are points ζi ∈Bi and ηi ∈Aisuch that (nf(x) =D2f(x)Df(x) ):|XM| = |Xln Df(ζi)Df(ηi)| ≤18

X|Z ζiηinf(x) dx| ≤ZM |nf(x)| dxwhich is bounded by say CM.In right f is a composition of the power law map x →xν and a diffeomor-phism. Therefore we may assume that nf(x) > 0 and equals ν −1x+ O(x).Since the intervals avoid the interval (∂U, qn−2), an estimate similar to theone above yields:Xright≤Z xr + ǫqn−2nf(x) dx ≤Z xr + ǫqn−2ν −1x+ O(x) dx ≤ln (|qn−2|1 −ν) + CrightIn left, we may assume that the nonlinearity n(f) is negative.

This impliesthat if (Ai ∪Bi) ⊂left, the ratio of lengths decreases when f is applied.For n very large, there are on the order of n2 times when (Ai ∪Bi) ⊂left,for which moreover (−qn−k) ⊂Bi for some k < n. An application of lemma3.3, (reverse the orientation) yields an estimate of the amount of decrease ofthe ratio. Since for the indices i under consideration (−qn−k) ⊂Bi for somek < n, we obtain that the ratio is uniformly decreased.

Namely, there existsδ ∈(0, 1) such that:Xleft< n2 ln δ < 0Putting the three estimates together, we obtain that:|f qn−1(B)|/|f(B)||f qn−1(A)|/|f(A)| = eRn ≤δn2 eCM + Cright |qn−2|1 −νBut|f qn−1(B)|/|f(B)||f qn−1(A)|/|f(A)| = |U|/(|qn−2|ν −|an qn|ν)|qn−1 −qn+1|/|an qn|ν≈|an qn|ν(|qn−2|ν −|an qn|ν)|U||qn−1|Since|an qn||qn−2| ≈σ(n, an), the previous implies that there exists a constantK so thatσ(n, an)ν ≤K δn2 |qn−2|1 −ν |qn−1|19

Multiplying this inequality by the analogous inequality for σ(n −1 , an−1)yields:σ(n −1, an−1)ν σ(n, an)ν ≤K2δ δn |qn−2|2−ν |qn−3|1 −ν |qn−1| < K2δ δn |qn−2|2−ν |qn−3|2 −νThis goes to zero whenever ν ≤2.✷The proof of the second claim of Theorem 2 is now nearly finished. Proof:Fix ǫ > 0.

We want to show that when n is large enough, σ(n, 1) is less thanǫ. Proposition 3.2 implies that we can choose n large enough so that at leastone of the scalings σ(n, an) and σ(n −1, an−1) is much smaller than ǫ. Bychoosing n still larger, we can arrange that also one of the scalings σ(n, 1)and σ(n −1, 1) is much smaller than ǫ (using equation (3.6)).

We need toshow that σ(n, 1) is smaller than ǫ. By the previous we only have to considerthe case when we only know that σ(n −1, 1) is very small.

Then however,the recursion relation in Proposition 3.1 (applied to n-1) shows that also thenσ(n, 1) is small. This finishes the proof of the second claim of Theorem 2.✷We remark that as the scalings tend to zero, the recursion relations in Propo-sition 3.1 converge to recursion equations.

This case was studied in [10].20

Appendix AFix ν > 1. Let { b(n) } be a sequence of positive numbers which arebounded from above by 1ν.

Define the sequence of matrices { Bν(n) } as:Bν(n) = 1−b(n)ν−1b(n −1)10!Lemma A.1: Assume that ν ≥2. Then the sequence { B◦nν } defined as:B◦nν= Bν(n) ◦· · · ◦Bν(1)is relatively compact.Proof:Each B◦nνis non-negative and we have that B◦n2−B◦nνis non-negative also.It therefore suffices to consider the case when ν = 2.B◦n2can be written in the form:B◦n2=α(n)β(n)α(n −1)β(n −1)One proves by induction that:α(n) = 1 −b(n) + b(n) b(n −1) · · · + (−1)n b(n).b(n −1) · · ·b(1)β(n) = b(0) (1 −b(n) + b(n) b(n −1) · · · + (−1)n−1 b(n).b(n −1) · · ·b(2))Therefore α(n) ≤1 + 12 · · · +12n ≤2 and β(n) ≤1.✷Lemma A.2: When ν > 2, there exists an integer N only depending on thebound 1ν for each b(n) so that when n ≥N, each B◦nνcontracts the Euclideanmetric on the plane by a factor smaller than .8.Proof:Each B◦nνcan be expressed in the form:B◦nν=α(n, ν)β(n, ν)α(n −1, ν)β(n −1, ν)21

Here α(n, ν) and β(n, ν) are polynomials of degree n in the variable1ν−1:α(n, ν) =Xαi(n)1ν −1iβ(n, ν) =Xβi(n)1ν −1iThe coefficients αi(n) and βi(n) only depend on the sequence b(n). We have(Lemma A.1) the estimate:α(n, ν) ≤α(n, 2) ≤2β(n, ν) ≤β(n, 2) ≤1Fix N1 so that1ν −1N1≤110One proves by induction that as n tends to infinity, the finitely manycoefficients{ α0(n) · · ·αN1(n), β0(n) · · ·βN1(n)}tend to zero exponentially fast.

Consequently, there exists N so that when nis bigger than N, each of these coefficients are all smaller than.2N1.Therefore: for n ≥Nα(n, ν) ≤110nXi=N1+1αi(n) + N1.2N1≤.4and β(n, ν) ≤.4. Consequently all the entries in B◦nνare less than or equalto .4.

Therefore the Euclidean metric is contracted by a factor less than .8.✷22

Appendix BDescription of the procedure.A numerical experiment was performedin order to check Conjecture 1 of the introduction. To this end, a family ofalmost smooth maps with a flat spot was considered given by the formulax →(x −1b)3(1 −3x + b −1b+ 6(x + b −1b)2 −10(x + b −1b)3 + (x −1)3)+t(mod1) .These are symmetric maps with the critical exponent (3, 3).

The parameterb controls the length of the flat spot, while t must be adjusted to get thedesired rotation number.In our experiment, b was chosen to be 0.5, which corresponds to theflat spot of the same length. By binary search, a value tAu was found whichapproximated the parameter value corresponding to the golden mean rotationnumber√5−12.

Next, the forward orbit of the flat spot was studied and theresults are given in the table below.It should finally be noted that the experiment presents serious numericaldifficulties as nearest returns to the critical value tend to 0 very quickly sothat the double precision is insufficient when one wants to see more than15 nearest returns. This problem was avoided, at a considerable expenseof computing time, by the use of an experimental package which allows forfloating-point calculations to be carried out with arbitrarily prescribed pre-cision.Results.Below the results are presented.The column yi is defined byyi := dist(0, qi).

The µi is given by µ := σ(i+2)−σ(i+1)σ(i+1)−σ(i) .23

nynσ(n)µn103.010 · 10−3.2637.5869111.544 · 10−3.24501.68312.7044 · 10−3.2340.452713.3328 · 10−3.21561.77514.1460 · 10−3.2072.50791564.04 · 10−6.19241.2851626.99 · 10−6.1849.63961711.22 · 10−6.1752.9773184.562 · 10−6.1690.7485191.829 · 10−6.1630.863420.7229 · 10−6.1585.801521.2826 · 10−6.1546.830722.1095 · 10−6.1514.81912342.07 · 10−9.1488.82432416.06 · 10−9.1467.8241256.097 · 10−9.1449.8172262.305 · 10−9.1435.9982271.070 · 10−9.1423−2.5428.8677 · 10−9.1411−25.929.3252 · 10−9.1441−10.9Interpretation.The most interesting is the third column which showsthe scalings. They seem to decrease monotonically.

The last column attemptsto measure the exponential rate at which the differences between consecutivescalings change. Here, the last three numbers are obviously out of line which,however, is explained by the fact that tAu is just an approximation of theparameter value which generates the golden mean dynamics.Other thanthat, the numbers from the last column seems to be firmly below 1, whichindicates geometric convergence.

If 0.82 is accepted as the limit rate, thisprojects to the scalings limit of about 0.137 which consistent with roughtheoretical estimates of [10].Thus, we conclude that Conjecture 1 has a numerical confirmation.24

References[1] Boyd, C.: The structure of Cherry fields, Erg. Th.

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5(1985), pp. 27-46[2] Graczyk, J.:Ph.D. thesis, Math Department of Warsaw University(1990); also:Dynamics of nondegenerate upper maps, preprint ofQueen’s University at Kingston, Canada (1991)[3] Herman, M.: Conjugaison quasi sym´etrique des hom´eomorphismes anal-itique de cercle `a des rotations, a manuscript[4] Mestel B.: Ph.D. dissertation, Math Department, Warwick University(1985)[5] Misiurewicz, M.: Rotation interval for a class of maps of the real lineinto itself, Erg.

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The C1+α case., Nonlinearity 1 (1988), pp. 181-202[7] Swiatek, G.: Bounded distortion properties of one-dimensional maps,preprint SUNY, Stony Brook, IMS no.

1990/10; also One-dimensionalmaps and Poincar´e metric, to appear in Nonlinearity[8] Swiatek, G.: Rational rotation numbers for maps of the circle, Commun.in Math. Phys., 119, 109-128 (1988)[9] Veerman J.J.P.

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1990/11; to appear in Commun.in Math. Phys.25

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