ATTRACTORS FOR UNIMODAL - Mathematical Sciencesmmisiure/sna.pdf · 2007. 10. 21. · ATTRACTORS FOR...

ATTRACTORS FOR UNIMODALQUASIPERIODICALLY FORCED MAPS

LLUIS ALSEDA AND MICHA L MISIUREWICZ

Dedicated to Gerry Ladas on the occasion of his 70th Birthday

Abstract. We consider unimodal quasiperiodically forced maps, that is, skewproducts with irrational rotations of the circle in the base and unimodal inter-val maps in the fibers: the map in the fiber over θ is a unimodal map f of theinterval [0, 1] onto itself multiplied by g(θ), where g is a continuous function fromthe circle to [0, 1]. Here we consider the “pinched” case, when g attains the value0. This case is similar to the one considered by Gerhard Keller, except that thefunction f in his case is increasing. Since in our case f is unimodal, the basic toolsfrom the Keller’s paper do not work in general.

We prove that under some additional assumptions on the system there existsa strange nonchaotic attractor. It is the graph of a measurable function from thecircle to [0, 1], which is invariant, discontinuous almost everywhere and attractsalmost all trajectories. Moreover, both Lyapunov exponents on this attractor arenonpositive. There are also cases when the dynamics is completely different, becauseone can apply the results of Jerome Buzzi implying the existence of an invariantmeasure absolutely continuous with respect to the Lebesgue measure (and then theattractor is some region in S1 × [0, 1]), and the maximal Lyapunov exponent ispositive. Finally, there are cases when we can only guess what the behavior is byperforming computer experiments.

1. Introduction

In this paper we deal with nonautonomous quasiperiodically forced dynamical sys-tems given by a map F : S1 ×X → S1 ×X:

(1.1) F (θ, x) = (R(θ), ψ(θ, x)),

where X is a topological space X, S1 = R/Z denotes the unit circle and R : S1 → S1

denotes the rotation of the circle by angle ω (that is, R(θ) = θ + ω (mod 1)), whereω is an irrational number.

Such a system is called a skew product. Then, the map R : S1 → S1 is called thebase map and {θ} ×X is called the fiber over θ. For a fixed θ, the map

ψ(θ, ·) : {θ} ×X → {R(θ)} ×X

Date: October 22, 2007.2000 Mathematics Subject Classification. 37E05, 37E99, 37B55, 37C70, 37H15.Key words and phrases. Quasiperiodically forced system, Strange Nonchaotic Attractor, Logistic

map, Lyapunov exponents.The first author was partially supported by MEC grant number MTM2005-021329 and the second

author by NSF grant DMS 0456526.1

2 LLUIS ALSEDA AND MICHA L MISIUREWICZ

is a continuous map from the fiber over θ into the fiber over R(θ). Looking at thosemaps rather than at F justifies the use of the term nonautonomous, because whichmap will be applied at time n depends on Rn(θ).

A particular instance of system (1.1) is when the map F takes the form:

(1.2) F (θ, x) = (R(θ), f(x) · g(θ)),where g : S1 → X and f : X → X are continuous functions. In this case we say thatg quasiperiodically drives f . Then the fiber maps ψ(θ, ·) are of the form f(·)g(θ), andwhen g(θ) = 0 then they are just the constant map 0. To simplify notation, whenwe write a formula for g, we will often consider [0, 1] instead of S1 as its domain. Tomake sure that g is continuous on the circle, not only on the interval, we have toassure that g(0) = g(1).

In this kind of models a lot of effort has been devoted to finding nonchaotic at-tracting invariant graphs (of functions) with strange geometry and which attractalmost every orbit in the space. They are usually called Strange Nonchaotic Attrac-tors (SNA’s). In [4] (see also [1] for a similar approach to this problem) the followingparticular instance of system (1.2) is considered. In what follows we will call it theKeller’s model. It is obtained by taking X = R+ = [0,∞) in (1.2) and choosingfunctions:

(1) f : R+ → R+ of class C1, bounded, strictly increasing, strictly concave withf(0) = 0 (observe that then the circle x = 0 will be invariant by the system),

(2) g : S1 → R+ bounded and continuous.

Keller’s model is a generalization of the system

(1.3)

{θn+1 = θn + ω (mod 1),

xn+1 = 2σ tanh(xn) cos(2πθn)

where x ∈ R, θ ∈ S1, ω is the golden mean, and σ > 1. This system was first studied in[3] with an approach that, at some steps, relies exclusively on numerical computations.The term Strange Nonchaotic Attractor, which is associated to attractors of modelsof the type (1.1), was introduced in that paper.

To have a better intuition on the Keller’s model, in Figure 1 we depict the attractorof the particular instance of that model obtained by setting f(x) = tanh(x), g(θ) =3 |cos(2πθ)|, with ω equal to the golden mean.

The dynamics of the Keller’s model, the asymptotic properties of its orbits and thetopological properties of their attractors are characterized by the following theoremdue to Keller [4]. Before stating it we need to introduce a constant σ as follows. Notethat

I(g) :=

∫S1

log g(θ)dθ ≥ −∞

is well defined because g is bounded. Then we set,

σ :=

{f ′(0) exp(I(g)) if I(g) > −∞,

0 otherwise.

Observe that, since the circle x = 0 is invariant and in view of the Birkhoff ErgodicTheorem, log σ is the vertical Lyapunov exponent on the circle x = 0. Here by thevertical Lyapunov exponent we mean the exponent for the vertical vectors (in the fiber

ATTRACTORS FOR UNIMODAL QUASIPERIODICALLY FORCED MAPS 3

Figure 1. The attractor of the Keller’s model obtained with f(x) =tanh(x), g(θ) = 3 |cos(2πθ)| and ω equals the golden mean. This figurewas obtained by starting from the point (θ, x) = (0.629678 . . . , 5.0),discarding its first 105 images and plotting the next 106 ones.

direction). Note that F maps fibers to fibers, so its derivative maps vertical vectorsto vertical vectors. As the measure for which we compute the vertical Lyapunovexponent we take the only invariant probability measure on the circle x = 0, that isthe normed Lebesgue measure.

Keller’s Theorem. There exists an upper semicontinuous map ϕ : S1 → R+ whosegraph is invariant under Keller’s model. Moreover,

(1) The Lebesgue measure on the circle, lifted to the graph of ϕ is a Sinai-Ruelle-Bowen measure (that is,

limn→∞

1

n

n−1∑k=0

ϑ(F k(θ, x)) =

∫S1

ϑ(θ, ϕ(θ))dθ

for every ϑ ∈ C0(S1 × R+,R) and Lebesgue almost every (θ, x) ∈ S1 × R+),(2) if σ ≤ 1 then ϕ ≡ 0,(3) if σ > 1 then ϕ(θ) > 0 for almost every θ,(4) if σ > 1 and g(θ0) = 0 for some θ0 then the set {θ : ϕ(θ) > 0} is meager and

ϕ is almost everywhere discontinuous,(5) if σ > 1 and g > 0 then ϕ is positive and continuous; if g is C1 then so is ϕ,(6) if σ 6= 1 then |xn − ϕ(θn)| → 0 exponentially fast for almost every θ and every

x > 0.


Let PK be the space of all functions (not necessarily continuous) from S1 to R. Ifwe look for a functional version of the system (1.2) (or the iterates of F ) in PK thenwe have to define the transfer operator TK : PK → PK as:

(TKψ)(θ) = f(ψ(R−1(θ))) · g(R−1(θ))

(the graph of TKψ is the image under F of the graph of ψ). Observe that ϕ is invariantif and only if TKϕ = ϕ.

To obtain ϕ, Keller takes a sufficiently large constant function, applies to it theiterates of the transfer operator TK and takes their limit (which is the infimum). Wehave to emphasize that this process heavily relies on the monotonicity of f and, inparticular, this is the reason that it works.

On the other hand, there are big differences between cases (4) and (5) in Keller’sTheorem. We call case (4), when g takes the value 0 at some point the pinchedcase. It is the most difficult situation. Note that then any TK–invariant functionϕ : S1 → R+ has to be 0 on a dense set.

The aim of this paper is to extend Keller’s Theorem to the case when f is notmonotone on the fibers. We will stay in the simplest case when f is nonmonotone(that is, when f is unimodal) and the most interesting case (for us): the pinched one.Thus, we consider the system given by the map F : S1 × [0, 1] → S1 × [0, 1]:

(1.4) F (θ, x) = (R(θ), f(x) · g(θ)),where g : S1 → [0, 1] is a continuous function which takes the value 0 at some pointand f : [0, 1] → [0, 1] is a unimodal map. By this we mean that f is continuous,f(0) = f(1) = 0, and there exists c ∈ (0, 1) such that f is strictly increasing on [0, c]and strictly decreasing on [c, 1]. Moreover, we assume that f is concave.

Note that, since g is an arbitrary continuous function from S1 to [0, 1], by multi-plying f by a positive constant and dividing g by the same constant if necessary, wealso may assume that f(c) = 1.

To fix ideas we may think on the very particular case when f is a logistic map,f(x) = 4x(1− x), and g is also logistic: g(θ) = aθ(1− θ). In Figure 2 we depict theattractor of this model obtained by setting a = 4 and ω equal to the golden mean.

Since in our case the map f is not monotone in the fibers, to prove the existence ofan invariant function for the transfer operator is more difficult than for the Keller’smodel. The strategy we have chosen is to construct a sequence of functions thatconverges to a function ξ+ whose graph bounds the attractor from above. We do thiswith the help of an operator that we call a semitransfer operator. Then, by using athird operator called quartertransfer operator, under suitable technical conditions wecan prove the existence of the invariant graph for the transfer operator.

Note that the limit behaviors of the iterates of the transfer operator and of themap are different problems and we have to provide separate proofs for the resultsdescribing them. When iterating the transfer operator, we first go back along theorbit of a point on the circle and only then we return forward using the full map.To illustrate the difference of this procedure and just taking forward iterates, let usthink of a two-sided orbit (xn)∞n=−∞ and a function ψ that is 1 at xn with n ≥ 0 and−1 at xn with n < 0. Then the averages of ψ computed along the orbit converge to 1,but if we make computations in the way we use when applying the transfer operator,then the averages will tend to −1.


Figure 2. The attractor obtained with f(x) = 4x(1 − x), g(θ) =4θ(1−θ) and ω equal to the golden mean. The starting point is (θ, x) =(0.604223 . . . , 0.345199 . . .). In the plot we have discarded the first 105

images and plotted the next 106 ones.

This paper is organized as follows. In Section 2 we define the transfer and semi-transfer operators for our system and we obtain bounds for the region of the spacewhere the interesting dynamics occurs. In particular the function ξ+ is obtained andstudied. Section 3 is devoted to the relation between the vertical Lyapunov expo-nent on the circle x = 0, and the dynamics of the system including some topologicalproperties of the function ξ+. In Section 4 we show the existence of the invariantgraph and we study the convergence properties of the orbits of the system to thisgraph such as exponential convergence speed, convergence almost everywhere, etc.Section 5 describes the invariant measures of the system and their dynamical proper-ties. In Section 6 we show that on this invariant graph the Lyapunov exponents arenonpositive. In Section 7 we discuss some examples of our results. In particular weshow some special cases where Keller’s Theorem does not apply whereas our resultsdo. Finally, in Section 8 we list some intriguing questions and open problems.

2. Bounds

In this section we provide upper and lower bounds for a set where all interestingdynamics occurs.

Let P be the space of all functions ψ : S1 → [0, 1]. We will be interested in invariantgraphs of functions from P . Here “invariant” refers to the invariance for F (a set Y


is invariant for F if F (Y ) ⊂ Y ), so a graph A is invariant if F (A) = A. Observe thatsince f(0) = 0, the graph of the zero function is invariant.

We define two operators from P to itself. The first one is the well known transferoperator T . It is defined as

(2.1) (Tψ)(θ) = f(ψ(R−1(θ))) · g(R−1(θ)).

This means that F (R−1(θ), ψ(R−1(θ))) = (θ, (Tψ)(θ)), so the image under F of thegraph of ψ is the graph of Tψ. In particular, the graph of a function ψ is invariantif and only if ψ is a fixed point of T . The second operator, as explained in theintroduction, takes into account the nonmonotone character of the map F in thefibers. We call it a semitransfer operator and denote it by S. It is defined as

(2.2) (Sψ)(θ) = f(min{ψ(R−1(θ)), c}

)· g(R−1(θ)).

This means that F(R−1(θ),min{ψ(R−1(θ)), c}

)= (θ, (Sψ)(θ)), so the image under

F of the graph of the minimum of ψ and c is the graph of Sψ.

Remark 2.1. Let ψ ∈ P and θ ∈ S1. If ψ(R−1(θ)) ≤ c then (Sψ)(θ) = (Tψ)(θ).Otherwise, min{ψ(R−1(θ)), c} = c and f(c) = 1 ≥ f(ψ(R−1(θ))). Consequently,Tψ ≤ Sψ. �

To simplify notation we will denote the constant function with the value a also bya.

Lemma 2.2. The following statements hold.

(a) Assume that ϕ, ψ ∈ P and ϕ ≤ ψ. Then Sϕ ≤ Sψ.(b) The sequence of functions (Sn1)∞n=0 is nonincreasing.

Proof. We start proving (a). Since ϕ ≤ ψ, it follows that

min{ϕ(R−1(θ)), c} ≤ min{ψ(R−1(θ)), c} ≤ c,

and since f is increasing on [0, c], we get

f(min{ϕ(R−1(θ)), c}

)≤ f

(min{ψ(R−1(θ)), c}

).

Therefore, by (2.2), Sϕ ≤ Sψ.Now, 1 ≥ S1 and from (a) we get

1 ≥ S1 ≥ S21 ≥ S31 ≥ · · · .This proves (b). �

By the above lemma, the sequence (Sn1)∞n=0 converges pointwise to a function fromP . We will denote this function ξ+.

Remark 2.3. As a limit of a nonincreasing sequence of continuous functions, ξ+ isupper semicontinuous. Moreover, by (2.2) and continuity of f , the function ξ+ is afixed point of S. Consequently, for each θ ∈ S1

(2.3) ξ+(R(θ)) = f(min{ξ+(θ), c}

)· g(θ).

Therefore, if ξ+(θ) = 0 then ξ+(R(θ)) = 0. Together with the ergodicity of R,this implies that either ξ+ is zero a.e., or it is positive a.e. Here a.e. means almosteverywhere with respect to the Lebesgue measure, and, in general, if we speak of themeasure on S1 then we mean the normed Lebesgue measure. �


Set X+ = {(θ, x) : x ≤ ξ+(θ)}.

Proposition 2.4. The set X+ is invariant for F and the ω-limit set of every point ofS1 × [0, 1] is contained in X+. Moreover, if F n(θ, x) = (θn, xn) then xn ≤ (Sn1)(θn).

Proof. For n = 0, 1, 2, . . . , set Xn = {(θ, x) : x ≤ (Sn1)(θ)}. Since Sn1 is continuous,Xn is closed. For each θ we have

f([0, (Sn1)(θ)]) = [0, f(min{(Sn1)(θ), c})],and thus, in view of (2.2),

(2.4) F (Xn) = Xn+1.

By Lemma 2.2(b), we have

S1 × [0, 1] = X0 ⊃ X1 ⊃ X2 ⊃ · · ·and by the definitions of ξ+ and X+,

X+ =∞⋂

n=0

Xn.

From this and (2.4) we see that X+ is invariant for F . For every point of S1 × [0, 1],its ω-limit set is contained in Xn for every n, so it is contained in X+.

The inequality xn ≤ (Sn1)(θn) follows immediately from (2.4) by induction. �

Now we are in a position to define the third operator from P to itself. We call it aquartertransfer operator and define it as

(2.5) Qψ = min{Tψ, Tξ+}.The next lemmas study the properties of the operator Q in relation with the func-

tion ξ+.

Lemma 2.5. Assume that ϕ, ψ ∈ P and ϕ ≤ ψ ≤ ξ+. Then Qϕ ≤ Qψ.

Proof. Fix θ ∈ S1. If (Tϕ)(θ) ≤ (Tψ)(θ) then by the definition of the operator Q (for-mula (2.5)) we get (Qϕ)(θ) ≤ (Qψ)(θ). Now assume that (Tϕ)(θ) > (Tψ)(θ). Fromthe definition of the transfer operator (formula (2.1)) it follows that f(ϕ(R−1(θ))) >f(ψ(R−1(θ))). Hence, c < ψ(R−1(θ)) because f is increasing on [0, c]. Thus, since fis decreasing on [c, 1] and ψ(R−1(θ)) ≤ ξ+(R−1(θ)), we get

(Tξ+)(θ) ≤ (Tψ)(θ) < (Tϕ)(θ).

Therefore in this case (Qψ)(θ) = (Qϕ)(θ) = (Tξ+)(θ). This completes the proof. �

Given a function ψ ∈ P the essential supremum of ψ, denoted by ess supψ, is thesmallest supremum of ψ over any subset of S1 of full Lebesgue measure. Set β =f(sup ξ+) (by brevity we do not indicate arguments when writing on the supremumof a function in its domain) and β′ = f(ess sup ξ+). Note that ess sup ξ+ ≤ sup ξ+, soif ess sup ξ+ ≥ c then β′ ≥ β.

Lemma 2.6. For every ε ∈ (0, β] the following statements hold.

(a) εξ+ ≤ Q(εξ+).(b) If ψ ∈ P and εξ+ ≤ ψ ≤ ξ+ then εξ+ ≤ Tψ ≤ ξ+.


If ε ∈ (0, β′] then the above statements hold almost everywhere (in (b), “a.e.” can bealso added to the assumptions).

Proof. To prove (a) we have to show that for every θ ∈ S1

(2.6) εξ+(R(θ)) ≤ min{f(εξ+(θ)) · g(θ), f(ξ+(θ)) · g(θ)}.

Set z = min{ξ+(θ), c}. By (2.3), in order to prove (2.6), it is enough to show

(2.7) εf(z) ≤ f(εξ+(θ)),

and

(2.8) εf(z) ≤ f(ξ+(θ)).

If ξ+(θ) ≤ c then z = ξ+(θ) and, since ε ≤ 1, (2.8) holds. If ξ+(θ) > c then z = c.Thus, f(z) = 1, and, since c < ξ+(θ) ≤ sup ξ+ and f is decreasing on [c, 1], we obtain

(2.9) f(ξ+(θ)) ≥ f(sup ξ+) = β ≥ ε = εf(z).

Hence, (2.8) also holds in this case. This is the only place in the proof where theinequality ε ≤ β is used. By replacing the inequality ξ+(θ) ≤ sup ξ+ by ξ+(θ) ≤ess sup ξ+ a.e. and β by β′, (2.9) works for almost every θ.

To prove (2.7) we also consider two cases, but different than above. Set x = εξ+(θ).If x ≤ c then x ≤ z because ε ≤ 1. By the concavity of f and since f(0) = 0, we getf(x)/x ≥ f(z)/z. Thus, since by definition εz ≤ x, we get

f(εξ+(θ)) = f(x) ≥ xf(z)

z≥ εf(z),

which proves (2.7) in this case. If x > c then c < x ≤ ξ+(θ). Since f is deceasing on[c, 1], we get f(x) ≥ f(ξ+(θ)). Thus, in this case (2.7) follows from (2.8). This proves(a).

Now we prove (b). From (a) and Lemma 2.5 we get εξ+ ≤ Q(εξ+) ≤ Qψ. Bythe definition of Q we have Qψ ≤ Tψ. By the definition of S (formula (2.2)) andLemma 2.2(a) we have Tψ ≤ Sψ ≤ Sξ+. Putting all those inequalities together andusing the fact that ξ+ is a fixed point of the operator S we get εξ+ ≤ Tψ ≤ ξ+. Thisproves (b). �

Remark 2.7. Let F (θ, x) = (R(θ), x′). By the definitions of F and T , ψ(θ) = x isequivalent to (Tψ)(R(θ)) = x′. Thus, by Lemma 2.6 (b), if εξ+(θ) ≤ x ≤ ξ+(θ) thenεξ+(R(θ)) ≤ x′ ≤ ξ+(R(θ)). Hence, if F n(θ, x) = (θn, xn) and εξ+(θ) ≤ x ≤ ξ+(θ)then εξ+(θn) ≤ xn ≤ ξ+(θn) for each n ≥ 0. �

Lemma 2.8. If β > 0 then sup ξ+ < 1 and sup(Sn1) < 1 for all but finitely manyn ≥ 0.

Proof. If sup ξ+ = 1 then β = 0 by the definition of β.Assume that sup ξ+ < 1. By Lemma 2.2 (b), ({θ : (Sn1)(θ) = 1})∞n=0 is a descending

sequence of compact sets. Its intersection is empty, so only finitely many terms ofthis sequence are nonempty. If the n-th set is empty then, since Sn1 is continuous,we get sup(Sn1) < 1. �


Set

Y = {(θ, x) : βξ+(θ) ≤ x ≤ ξ+(θ)}.By Lemma 2.6 (b), Y is invariant for F . Therefore the set X =

⋂∞n=0 F

n(Y ) iscontained in Y and is also invariant for F . Since the intersection of Y with everyfiber is a closed interval or a point, the same is true for X. All interesting dynamicsof F takes place in X. The set X basically plays a role of the core of the unimodalmap (that is the interval [f 2(c), f(c)]).

3. Vertical Lyapunov exponent at x = 0

Since we do not assume that f is smooth, we cannot speak about vertical Lyapunovexponents almost everywhere on the cylinder (although we will consider them on theinvariant graph in Section 6 under an additional assumption of smoothness of f).However, since f is concave, there exists a one-sided derivative f ′+(0) of f at 0.Therefore, as we have done in the introduction with the vertical Lyapunov exponentof the Keller’s model (see Page 2), we can consider the vertical Lyapunov exponentat x = 0, or more precisely, on S1 × {0} in our case. It is equal to (see for instance[4]):

Λ = log f ′+(0) +

∫S1

log g(θ) dθ.

Here we assume that f ′+(0) is finite, but admit the possibility of∫

S1 log g(θ) dθ = −∞.The sign of Λ influences strongly the dynamics.

Theorem 3.1. If Λ < 0 then ξ+ ≡ 0, and for every θ ∈ S1 and every x ∈ [0, 1] thetrajectory of (θ, x) converges exponentially fast to S1 × {0}.

Proof. Assume that Λ < 0. Then we can find s > f ′+(0) and a > 0 such that ifg(θ) = max{g(θ), a} then

log s+

∫S1

log g(θ) dθ < 0.

Since f is concave, we have f(x) ≤ sx for every x. Thus, f(x)g(θ) ≤ sxg(θ). More-over, the function log g is continuous, so its ergodic averages (for R) converge uni-formly to its integral. Therefore there is a constant ε > 0 such that for all sufficientlylarge n,

log s+1

n

n−1∑k=0

log g < −ε.

Using the notation F n(θ0, x0) = (θn, xn), we get

xn ≤ x0sn

n−1∏k=0

g(θk) < x0e−ε.

This holds for all sufficiently large n, uniformly in (θ0, x0), so the theorem follows. �

The proof of the next theorem is a modification of a Keller’s proof, adapted to oursituation.

Theorem 3.2. If ξ+ = 0 a.e., then Λ ≤ 0.


Proof. If∫

S1 log g(θ) dθ is equal to −∞ then there is nothing to prove. Assume that

it is finite. Define a function f by f(x) = f(min{x, c}). This function is continuousand concave. Moreover, for any ψ ∈ P ,

Sψ = ((f ◦ ψ) · g) ◦R−1.

Therefore, for any n we have

(3.1)f ◦ (Sn1)

Sn1=

(Sn+11) ◦R(Sn1) · g

≤ (Sn1) ◦R(Sn1) · g

.

The value of the left-hand side of (3.1) at θ can be treated as a difference quotient of f

taken at points (Sn1)(θ) and 0. Since f ′+(0) = f ′+(0) is finite, this difference quotientis bounded and bounded away from 0 as a function of θ. Therefore its logarithm isintegrable. Hence,

logf ◦ (Sn1)

Sn1+ log g ≤ log

(Sn1) ◦R(Sn1)

and the left-hand side is integrable. Therefore, by Lemma 2 of [4], the integral ofthe right-hand side is 0 (we need this argument because a priori the right-hand sidecould be not integrable). Thus,

(3.2)

∫S1

logf ◦ (Sn1)

Sn1(θ) dθ +

∫S1

log g(θ) dθ ≤ 0.

Since (Sn1)∞n=0 is a monotone sequence of functions converging a.e. to ξ+ = 0, we canpass to the limit with n in (3.2) and we get Λ ≤ 0. �

By Remark 2.3, ξ+ is either positive a.e., or zero a.e. Therefore we get the followingcorollary.

Corollary 3.3. If Λ > 0 then ξ+ is positive a.e. (that is, ess sup ξ+ > 0).

4. An invariant graph

In this section we investigate mainly what happens if the essential supremum of ξ+

is positive. However, in order to have a complete picture, let us see what happens ifit is zero.

If ξ+ is zero a.e., then by its definition, the sequence (Sn1)∞n=0 converges a.e. tozero. By Remark 2.1, T ≤ S, and by Lemma 2.2, S is monotone. Therefore for everyfunction ψ ∈ P and every n we have T nψ ≤ Sn1. Thus, the sequence (T nψ))∞n=0

converges a.e. to zero. This is in a sharp contrast to the situation from Theorem 4.3,which describes the case when ξ+ is positive a.e.

Now we are going to impose some additional restrictions on our map F , in additionto the assumptions already made in Section 2. First of all we assume in this sectionthat the map f is strictly concave.

Recall that for a concave map f the one-sided derivatives are well defined. We willdenote the left (respectively right) one-sided derivative of f by f ′− (respectively f ′+).

Observe that, by the concavity of f and since f(0) = 0, for every x ∈ (0, c], we get∣∣f ′−(x)∣∣ < f(x)/x. This inequality can be clearly extended further to the right of c.


Indeed, we define

(4.1) b := sup

{x ∈ (c, 1] : − f ′−(x) <

f(x)

x

}.

Note that c < b < 1. In particular, f(b) > 0.We will have to deal with cross ratios related to the map f . To do this we introduce

necessary notation and prove two technical lemmas.Given points x, y ∈ [0, 1] such that x 6= y we set

κ(x, y) :=|y − x|

max{y, x}.

Also, to simplify the notation, when f(x) 6= f(y) (then automatically x 6= y) we set

τ(x, y) =κ(f(x), f(y))

κ(x, y).

Lemma 4.1. For every γ > 0 and η < b there exists µ < 1 such that τ(x, y) < µwhenever x, y ∈ [γ, η] and f(x) 6= f(y). In particular, if x, y ∈ (0, b) and f(x) 6= f(y)then τ(x, y) < 1. Moreover, there exists ν such that if min{x, y} ∈ (0, b) (but possiblymax{x, y} ≥ b) then τ(x, y) ≤ ν.

Proof. Assume for definiteness that γ ≤ x < y ≤ η. If f(y) > f(x) then

τ(x, y) =f(y)− f(x)

y − x

y

f(y)

is a decreasing function of x. Thus, it takes the largest value at x = γ:

(4.2)f(y)− f(x)

y − x

y

f(y)≤ f(y)− f(γ)

y − γ

y

f(y).

The latter expression is a continuous function of y on (γ, η], and

(4.3) limy↘γ

f(y)− f(γ)

y − γ

y

f(y)= f ′+(γ)

γ

f(γ).

By the strict concavity of f and since f(0) = 0 the right-hand sides of (4.2)and (4.3) are smaller than 1. Consequently, there exists a constant µ′ < 1 such thatτ(x, y) ≤ µ′ whenever x, y ∈ [γ, η] and f(y) > f(x).

If f(x) > f(y), then c < y ≤ η and

(4.4) τ(x, y) =f(x)− f(y)

y − x

y

f(x)<f(x)− f(y)

y − x

y

f(y),

which is an increasing function of x by the concavity of f . Thus,

f(x)− f(y)

y − x

y

f(y)≤ lim

x↗y

f(x)− f(y)

y − x

y

f(y)= −f ′−(y)

y

f(y)≤ −f ′−(η)

η

f(η)< 1.

Thus, if we set µ = max{µ′,−f ′−(η) ηf(η)

} then τ(x, y) ≤ µ for all x, y ∈ [γ, η]. This

proves the first statement of the lemma.To complete the proof of the lemma, let us see what happens in the preceding part

of the proof if instead of assuming that x, y ∈ (0, b) we assume only that x ∈ (0, b). Ify < b then the estimates from the first part of the proof hold and hence, τ(x, y) ≤ 1.Thus, in the rest of the proof we may assume that 0 < x < b ≤ y.


The case f(y) > f(x) is the same as in the first part of the proof, so we get againτ(x, y) ≤ 1.

To deal with the case f(x) > f(y) let b′ < c be the unique point such that f(b′) =f(b). Then, if x ≤ b′, we get

τ(x, y) =f(x)− f(y)

f(x)

y

y − x≤ y

y − x≤ 1

b− b′.

Otherwise, x > b′ and the first equality of (4.4) allows us to make estimates. We get

τ(x, y) ≤ 1

1− b

1

f(b).

Thus, in all cases we get τ(x, y) ≤ ν, where

ν =1

min{(1− b)f(b), b− b′}.

�

Lemma 4.2. Let γ, η and µ be as in the first part of Lemma 4.1. Fix n ∈ N,θ0 ∈ S1, x0, y0 ∈ [0, 1] and denote (θk, xk) = F k(θ0, x0), (θk, yk) = F k(θ0, y0) fork = 1, 2, . . . , n− 1. Assume that xk, yk ∈ (0, b) for each k ∈ {0, 1, . . . , n− 1}. Then,|xn − yn| ≤ µm(n), where m(n) is the number of indices k ∈ {0, 1, . . . , n−1} such thatxk, yk ∈ [γ, η].

Proof. If xn = yn then there is nothing to prove. Assume that xn 6= yn. Then alsoxk 6= yk for k = 0, 1, . . . , n− 1. We have

κ(xk+1, yk+1) =|xk+1 − yk+1|

max{xk+1, yk+1}=

|f(xk)g(θk)− f(yk)g(θk)|max{f(xk)g(θk), f(yk)g(θk)}

=|f(xk)− f(yk)|

max{f(xk), f(yk)}= κ(f(xk), f(yk)).

Therefore

|xn − yn| ≤ κ(xn, yn) = κ(x0, y0)n−1∏k=0

κ(xk+1, yk+1)

κ(xk, yk)

= κ(x0, y0)n−1∏k=0

τ(xk, yk) ≤n−1∏k=0

τ(xk, yk).

Thus, by Lemma 4.1, we get |xn − yn| ≤ µm(n). �

Now we will prove one of the two main theorems of this section.

Theorem 4.3. Assume that 0 < ess sup ξ+ < b. Then there exists a function ζ ∈ Psuch that

(a) 0 ≤ ζ ≤ ξ+ and ζ ≥ β′ξ+ almost everywhere;(b) Tζ = ζ;(c) if ψ ∈ P and εξ+ ≤ ψ ≤ ξ+ for some ε > 0 then T nψ converges to ζ almost

everywhere as n tends to infinity;(d) ζ is a measurable function;(e) ζ is positive almost everywhere.


Proof. Since 0 < ess sup ξ+ < b, there exist numbers δ, η such that 0 < δ < η < b andthe set

A = {θ ∈ S1 : δ ≤ ξ+(θ) ≤ η}has positive Lebesgue measure. By the ergodicity of R, for almost every θ ∈ S1 wehave R−k(θ) ∈ A for infinitely many positive integers k. On the other hand, underour assumptions we know that 0 < ξ+(θ) < b a.e. Consequently, for almost every θ itfollows that 0 < ξ+(R−k(θ)) < b for each k ≥ 0. Thus, there is a set of full Lebesguemeasure in S1 satisfying this property and such that R−k(θ) ∈ A for infinitely manypositive integers k. In the rest of the proof we fix such θ.

In view of the definition of the transfer operator it follows that if ψ ∈ P then(θ, (T nψ)(θ)

)= F n

(R−n(θ), ψ(R−n(θ))

). To compute limn→∞(T nψ)(θ) we fix n > 0,

denote θ0 = R−n(θ) and choose any x0, y0 ∈ [β′ξ+(θ0), ξ+(θ0)]. We also denote

(θk, yk) = F k(θ0, y0) for k = 1, 2, . . . . Then by an inductive use of Remark 2.7, we getxk, yk ∈ [β′ξ+(θk), ξ

+(θk)] for k = 1, 2, . . . , n. Set γ = δβ′. Since 0 < ess sup ξ+ < b,we have β′ > 0, so γ > 0. Whenever θk ∈ A, we have xk, yk ∈ [γ, η]. There-fore, by Lemma 4.2 we get |xn − yn| ≤ µm(n), where µ and m(n) are defined inthat lemma. This proves that the length of the segment In = F n

({R−n(θ)} ×

[β′ξ+(R−n(θ)), ξ+(R−n(θ))])

is at most µm(n). By Remark 2.7, the sequence of seg-ments In is descending, and by our choice of θ their lengths go to 0. Therefore theirintersection consists of one point in [β′ξ+(θ), ξ+(θ)]. We choose this point as ζ(θ). Atthe remaining values of θ (from a set B of measure 0) we set ζ(θ) = 0. Then make asmall modification (in order for (b) to hold everywhere, not only almost everywhere),by changing the value of ζ to 0 on the whole forward and backward trajectories ofthe points from B.

Now the statements (a–c) of the theorem follow immediately from the constructionof ζ. Statement (d) follows from (c). By (a), we have ζ ≥ β′ξ+ a.e. By the assumption,ess sup ξ+ > 0, so by Remark 2.3, ξ+ is positive a.e. This proves (e). �

Now we will investigate further properties of F under the assumptions of Theo-rem 4.3, related to the function ζ. When we speak about convergence of a trajectoryto a graph of a function, we measure the distance of a point from this graph in thecorresponding fiber.

Theorem 4.4. Assume that 0 < ess sup ξ+ < b. Then for almost every θ ∈ S1 andall x ∈ (0, 1) the trajectory of (θ, x) either converges exponentially fast to the graphof ζ or falls into S1 ×{1} and then stays in S1 ×{0}. In particular, for almost everyθ ∈ S1 and all but countable number of x ∈ (0, 1) the trajectory of (θ, x) convergesexponentially fast to the graph of ζ.

Proof. Fix η ∈ (ess sup ξ+, b). Since the sequence (Sn1)∞n=0 converges to ξ+ andess sup ξ+ < η, for every ρ ∈ (0, 1) there exists a set A1 ⊂ S1 of measure ρ and N ∈ Nsuch that (Sn1)(θ) < η for every θ ∈ A1 and n ≥ N .

Let A2 ⊂ S1 be the set of those θ ∈ S1 for which ζ(Rn(θ)) > 0 for n = 0, 1, 2, . . . .In particular, since ζ ≤ ξ+, if θ ∈ A2 then ξ+(Rn(θ)) > 0 for n = 0, 1, 2, . . . . ByTheorem 4.3 (e), ζ is positive a.e., so the set A2 has measure 1.

As in the proof of Theorem 4.3, there exists δ > 0 such that the set A3 = {θ ∈ S1 :ξ+(θ) ≥ δ} has measure λ > 0.


By the Birkhoff’s Ergodic Theorem, there is a set A ⊂ A2 of measure 1 such that ifθ ∈ A then the ergodic averages at θ of the characteristic functions of A1 and A1∩A3

converge respectively to the measures of those sets.Choose θ ∈ A, x ∈ (0, 1) and denote as always F n(θ, x) = (θn, xn). Let us make

an additional assumption that xn > 0 for n = 0, 1, 2, . . . . By Proposition 2.4, xn ≤(Sn1)(θn). Thus, if n ≥ N and θn ∈ A1 then by Lemma 4.1 we have τ(xn, yn) < 1,where yn = ζ(θn). We have x ≥ εξ+(θ) for some ε > 0 (we can choose ε < β′,since by the argument as in the proof of Theorem 4.3, β′ > 0), so by Remark 2.7,xn ≥ εξ+(θn) for all n. Thus, if θ ∈ A1 ∩ A3 and n ≥ N , by Lemma 4.2 (as in theproof of Theorem 4.3) we get τ(xn, yn) < µ, where µ is the constant from Lemma 4.1.By the estimates as in the proof of Lemma 4.2, for n ≥ N we get

|xn − yn| ≤ κ(xN , yN)n−1∏k=N

τ(xk, yk) ≤ κ(xN , yN)µm(n)νs(n),

where m(n) is the number of k’s between N and n − 1 for which θk ∈ A1 ∩ A3 ands(n) is the number of k’s between N and n − 1 for which θk /∈ A1. Thus, to getexponential convergence of |xn − yn| to 0 it suffices to show that

(4.5) lim supn→∞

1

n−Nlog(µm(n)νs(n)) < 0.

However, by the definition of A, m(n)/n converges to the measure of A1 ∩A3 (whichis at least ρ + λ − 1), and s(n)/n converges to 1 minus the measure of A1 (which is1− ρ). This means that the limit in (4.5) is equal to at most

(ρ+ λ− 1) log µ+ (1− ρ) log ν.

Since ρ can be taken arbitrarily close to 1, this expression can be made negative.This completes the proof under the additional assumption that xn > 0 for all n.

However, the only way we may have xn = 0 while xn−1 > 0, is that either g(θn−1) = 0or xn−1 = 1. The first possibility is ruled out because, by (2.3), we would haveξ+(θn) = 0, which contradicts the fact that θ ∈ A ⊂ A2. The second possibilitymeans that our trajectory falls into S1 × {1} and then stays in S1 × {0}.

Since f is strictly concave, in any fiber the preimage of a finite set is finite. Thereforein any fiber the set of points whose trajectories eventually fall into S1×{0} is at mostcountable. �

In most cases we can strengthen Theorem 4.4.

Theorem 4.5. Assume that 0 < ess sup ξ+ < b.

(a) If g 6= 1 almost everywhere then for almost every θ ∈ S1 and all x ∈ (0, 1) thetrajectory of (θ, x) converges exponentially fast to the graph of ζ.

(b) If β > 0 then for almost every θ ∈ S1 and all but finite number of x ∈ (0, 1) thetrajectory of (θ, x) converges exponentially fast to the graph of ζ.

Proof. To prove (a), just remove all values of θ for which g(θn) = 1 for some n. Bythe assumption, this is a set of measure zero.

To prove (b), note that xn−1 = 1 can happen only for finitely many n’s byLemma 2.8 since xn ≤ (Sn1)(θn). Thus, for a given θ, it can apply only to finitelymany initial x’s. �


5. Invariant measures

Here we discuss existence and properties of ergodic invariant probability measuresin the cases considered in preceding sections. One such measure is visible at once.Namely the circle S1 × {0} is invariant and F restricted to it is simply the rotationR. Thus, the normed Lebesgue measure on this circle is invariant and ergodic, andit is the only invariant probability measure on it. Let us denote it by m0.

Observe also that the rotation R is a factor of F , so the image of every F -invariantprobability measure under the projection of S1× [0, 1] to the first coordinate must bethe normed Lebesgue measure.

Thus, by Theorem 3.1, we get immediately the following result.

Theorem 5.1. If Λ < 0 then F is uniquely ergodic.

Proof. Let µ be an F -invariant ergodic probability measure. By Theorem 3.1 thetrajectory of almost every point converges to the closed set S1 × {0}. Thus, µ has tobe concentrated on S1 × {0} and hence µ = m0. �

In the above theorem we do not need an additional assumption that we made inSection 4, that f is strictly concave. However, we need it in the next theorem. Inthe situation described in Theorem 4.3, there is another invariant ergodic probabilitymeasure. This is the measure concentrated on the graph of ζ, whose projection tothe first coordinate is the normed Lebesgue measure. Let us denote this measureby mζ . It is invariant, since by Theorem 4.3 (b) the graph of ζ is invariant. Notethat the support of mζ may be larger than the graph of ζ, since this graph is notnecessarily closed. In particular, if g attains value 0, then the closure of the graph ofζ contains S1 × {0}, but mζ(S1 × {0}) = 0. The system (F,mζ) is isomorphic (in asense of measure preserving transformations) to the rotation R of the circle with thenormalized Lebesgue measure.

Theorem 5.2. Assume that f is strictly concave and 0 < ess sup ξ+ < b. Then F hasonly two invariant ergodic probability measures, namely m0 and mζ. In particular,the topological entropy of F is 0. The measure mζ is the (unique) Sinai-Ruelle-Bowenmeasure for F .

Proof. Denote by A the set of those θ ∈ S1 for which the point (θ, ζ(θ) is generic formζ and the conclusion of Theorem 4.4 holds. This set is of the full Lebesgue measure.

Let µ be an F -invariant ergodic probability measure. Take a generic point (θ, x) ofµ, such that θ ∈ A. The set of such points is of full measure µ. By Theorem 4.4, thedistance between F n(θ, x) and (Rn(θ), ζ(Rn(θ)) or between F n(θ, x) and (Rn(θ), 0)goes to 0 as n→∞. This means that the weak-* limit of the sequence of measures

(5.1)1

n

n−1∑k=0

δF n(θ,x)

(where δy is the probability measure concentrated at y) is either mζ or m0. However,since (θ, x) is a generic point of µ, this limit is equal to µ. This proves that eitherµ = mζ or µ = m0.

Both measures µ = mζ and µ = m0 have entropy 0, so by the Variational Principle,the topological entropy of F is 0.


By Theorem 4.4, for almost every θ ∈ S1 and x ∈ (0, 1) the trajectory of (θ, x)converges exponentially fast to the graph of ζ. Then the limit of (5.1) is mζ forLebesgue-a.e. (θ, x). Therefore mζ is the (unique) Sinai-Ruelle-Bowen measure forF . �

6. Lyapunov exponents of the invariant graph ζ

In addition to the assumptions made in the preceding sections, here we are going toassume that the map f is differentiable. If 0 < ess sup ξ+ < b, then by Theorem 5.2,there is only one candidate for a measure used for computing the Lyapunov exponentson the graph of ζ, namely the measure mζ . Then, as in [4], the vertical Lyapunovexponent on the graph of ζ is equal to

Λζ =

∫S1

log |f ′(ζ(θ))| dθ +

∫S1

log g(θ) dθ.

Observe that none of the integrals above can be +∞, so we can add them.The next result says that the “attractive character” of ζ described in Theorem 4.4

can also be derived from the value of Λζ . Recall that Λ is the vertical Lyapunovexponent on the circle x = 0.

Theorem 6.1. Assume that Λ > 0 and ess sup ξ+ < b. Then Λζ < 0.

Proof. We follow the proof of [4, Lemma 1(2)] with some changes due to the non-monotonicity of f .

By Corollary 3.3, the assumption Λ > 0 implies that ess sup ξ+ > 0, so we can useTheorem 5.2 and there is no ambiguity for which measure we should compute theLyapunov exponent on the graph of ζ. Moreover, the assumptions of Theorems 4.3and 4.4 are satisfied, so we can use those theorems.

Since∫

S1 log g(θ) dθ < +∞, the proposition holds when∫

S1 log |f ′(ζ(θ))| dθ = −∞.In the rest of the proof we assume that

∫S1 log |f ′(ζ(θ))| dθ > −∞. Moreover, we

have∫

S1 log g(θ) dθ = Λ− log f ′(0) > −∞.From Theorem 4.3(a) and (e) and the assumption ess sup ξ+ < b, it follows that

0 < ζ(θ) < b for almost every θ. Consequently, from the concavity of f and thedefinition of b we get for almost every θ

|f ′(ζ(θ))| < f(ζ(θ))

ζ(θ)=

ζ(R(θ))

ζ(θ) · g(θ)because ζ(R(θ)) = (Tζ)(θ) = f(ζ(θ)) · g(θ) by Theorem 4.3(b). Thus, the function

log |f ′(ζ(θ))| + log g(θ) is an integrable minorant of log ζ(R(θ))ζ(θ)

. Consequently, by [4,

Lemma 2], log ζ(R(θ))ζ(θ)

is integrable and∫

S1 log ζ(R(θ))ζ(θ)

dθ = 0. Thus,∫S1

log |f ′(ζ(θ))| dθ <∫

S1

logζ(R(θ))

ζ(θ)dθ −

∫S1

log g(θ) dθ = −∫

S1

log g(θ) dθ.

This completes the proof. �

Counting with multiplicities, our system has two Lyapunov exponents for mζ . Weknow already that one of them is negative. Now we will investigate the other one.

Theorem 6.2. Assume that Λ > 0 and ess sup ξ+ < b. Then all Lyapunov exponentsfor mζ are nonpositive.


Figure 3. Logistic map forced by a logistic map, a = 2.3. Theattractor seems to be below the level x = 1/2.

Proof. By Oseledec’s Theorem we know that either there is one double Lyapunovexponent Λζ for mζ , or there are two single ones, Λζ and %ζ . In the first case thestatement of the theorem follows from Theorem 6.1. In the second case for mζ-almostevery (θ, ζ(θ)) we have

Λζ + %ζ = limn→∞

1

nlog |detDF n(θ, ζ(θ))|

= limn→∞

1

nlog

∣∣∣∣det

(1 0

∂xn

∂θ∂xn

∂x

)∣∣∣∣ = limn→∞

1

nlog

∣∣∣∣∂xn

∂x

∣∣∣∣ = Λζ ,

where (θn, xn) denotes F n(θ, ζ(θ)) for every n ≥ 0. Therefore, %ζ = 0. �

The above theorem justifies the use of the word “nonchaotic” in the name “StrangeNonchaotic Attractor.”

7. Examples

When we specify a formula for g, we treat the circle as the interval [0, 1] withendpoints identified. Thus, g will be a function on [0, 1]. The rotation R can be anyirrational rotation, but for our numerical experiments we chose the golden numberrotation, so ω = (

√5− 1)/2.

Our first example is a logistic map forced by a logistic map. Thus, f(x) = 4x(1−x)and g(x) = ax(1 − x). The integral of log g is equal to log a − 2, and f ′(0) = 4.Thus, Λ = log(4a/e2), so the interesting case, Λ > 0, occurs when a > e2/4 ≈


Figure 4. Logistic map forced by a logistic map, a = 2.9. Theattractor seems to stick above the level x = b = 2/3.

1.84726402473266. However, even for slightly larger values of a (definitely for a < 2,because then S1 < 1/2) we have sup ξ+ < 1/2, so this is basically Keller’s case. Onthe other hand, if a is too large, then ess sup ξ+ > b = 2/3 and our main theoremsfrom Section 4 do not apply. Numerical estimates (which do not distinguish betweensup ξ+ and ess sup ξ+ - but perhaps those numbers are really equal in many cases?)suggest an interval of “good” values of a, in particular a = 2.6 (see Figures 3–5).

Since numerical evidence should be treated only as a suggestion, we will give nowan example of a family when we can prove that such “good” parameters exist.

Lemma 7.1. Let g be a logistic map, g(x) = ax(1− x), with a > e2/2, and let f besuch that its turning point c is 1/2. Then ess sup ξ+ > 1/2.

Proof. As we already computed, the integral of log g is equal to log a− 2. Thus, sincea > e2/2, we have

(7.1) log 2 +

∫S1

g(θ) dθ > 0.

Since f is concave, continuous, and f(c) = 1, we have f(x) ≥ 2x for x ∈ [0, 1/2]. Inparticular, f ′+(0) ≥ 2, so by (7.1), Λ > 0. Thus, by Corollary 3.3, ξ+ is positive a.e.Suppose that ξ+ ≤ 1/2 a.e. Then Tξ+ = Sξ+ = ξ+ a.e., so

log ξ+(θ) = log Tξ+(θ) = log f(ξ+(R−1(θ))) + log g(R−1(θ))

≥ log 2 + log ξ+(R−1(θ)) + log g(R−1(θ))


Figure 5. Logistic map forced by a logistic map, a = 2.6. Theattractor seems to be between the levels x = 1/2 and x = b = 2/3.

almost everywhere. By induction we get

log ξ+(θ)− log ξ+(R−n(θ)) ≥ n log 2 +n∑

k=1

log g(R−k(θ))

almost everywhere. This can be rewritten (by replacing θ by Rn(θ)) by

log ξ+(Rn(θ))− log ξ+(θ) ≥ n log 2 +n−1∑k=0

log g(Rk(θ))

almost everywhere. By the Ergodic Theorem and (7.1), the right-hand side abovegoes to infinity as n → ∞. This contradicts the facts that ξ+ is positive a.e. andbounded. Hence, ess sup ξ+ > 1/2. �

Another way of stating the above lemma is that if its assumptions are satisfied, thesystem cannot be reduced to the Keller’s case.

To find an example of a quasiperiodically forced system of the form (1.4) withg(x) = ax(1 − x) which cannot be reduced to the Keller’s case and such that ourresults apply, we only have to find an f which satisfies f(1/2) = 1 and e2/8 < b < 1.Then we will be able to choose a value of a such that e2/8 < a/4 < b. This impliesthat e2/2 < a and, since S1 ≤ a/4, we get 1/2 < ess sup ξ+ < b. Thus, Theorem 4.3applies, but the study of this system cannot be reduced to the Keller’s case. Now themap f can be taken of the form 1− (2x−1)2n with sufficiently large n (computationsshow that n ≥ 13 is sufficient to assure the existence of e2/8 < b < 1).


Figure 6. Tent map forced by a logistic map, a = 3.696, slightlymore than e2/2. The attractor sticks above the level x = 1/2.

The next example shows that if we do not assume that f is strictly concave, thesituation can be completely different. Namely, let us take again g(x) = ax(1 − x)with a > e2/2, but as f we take the tent map, f(x) = 1− |2x− 1|. Then the verticalLyapunov exponent is the same everywhere in S1 × [0, 1], and is positive. Thus, bythe results of Buzzi [2], there exists an invariant probability measure for F , absolutelycontinuous with respect to the Lebesgue measure. This implies that, in this case, theattractor is not a graph of a function (see Figures 6 and 7).

Note that for the above family, by Lemma 7.1 for all a > e2/2 we have ess sup ξ+ >1/2, while for all a < e2/2 we have Λ < 0, and thus by Theorem 3.1, ξ+ ≡ 0. This isin a sharp contrast to the family from the first example, where computer experimentssuggest continuous dependence of ess sup ξ+ on a.

8. Questions

There are some interesting questions connected with the dynamics and dependenceon parameters of the systems that we are investigating.

Question 8.1. Are Theorems 4.3 and 4.4 true without the assumption that theessential supremum of ξ+ is smaller than b? �

Question 8.2. When both f and g are logistic maps and g depends on the param-eter: g(x) = ax(1 − x) the computer experiments suggest some kind of continuousdependence of the attractor on the parameter. Is this dependence really continuous?


Figure 7. Tent map forced by a logistic map, a = 4.

If yes, in what sense (what topology)? If no, is at least the supremum (or the es-sential supremum) of ξ+ depending continuously on the parameter? Of course thesame question can be asked for other similar families. As we noted at the end of thepreceding section, the situation may be different if f is not strictly concave. �

Question 8.3. Are the supremum and the essential supremum of ξ+ always equal?If not, what natural assumptions imply this? �

Question 8.4. An attracting invariant graph is an analogue of an attracting fixedpoint for an interval map. However, for interval maps we see often periodic attractingpoints of periods n > 1. Can in our model an attracting periodic graph occur? Tobe more specific, we are asking about a possibility of an attracting invariant set thathas n points in almost every fiber and is in some sense irreducible. �

References

[1] Z. I. Bezhaeva and V. I. Oseledets. On an example of a “strange nonchaotic attractor”. Funkt-sional. Anal. i Prilozhen. 30 (1996), 1–9, 95.

[2] J. Buzzi. Absolutely continuous S.R.B. measures for random Lasota-Yorke maps. Trans. Amer.Math. Soc. 352 (2000), 3289–3303.

[3] C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke. Strange attractors that are not chaotic. Phys.D, 13 (1984), 261–268.

[4] G. Keller. A note on strange nonchaotic attractors. Fund. Math., 151 (1996), 139–148.


Departament de Matematiques, Edifici Cc, Universitat Autonoma de Barcelona,08913 Cerdanyola del Valles, Barcelona, Spain

E-mail address: [email protected]

Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indi-anapolis, IN 46202-3216

E-mail address: [email protected]

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