IN
DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS
,
STOCKHOLM SWEDEN 2018
Constructing Multidimensional
Dynamical Systems with Positive
Lyapunov Exponents
Constructing Multidimensional
Dynamical Systems with Positive
Lyapunov Exponents
MARTIN ANDERSSON
Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2018 Supervisor at KTH: Kristian Bjerklöv
TRITA-SCI-GRU 2018:255 MAT-E 2018:55
Royal Institute of Technology
School of Engineering Sciences KTH SCI
SE-100 44 Stockholm, Sweden URL: www.kth.se/sci
Abstract
In this thesis we adapt Marcelo Viana’s general construction of smooth transformations exhibiting non-uniform expansion in several dimensions. In our construction we, instead of coupling with a quadratic map, as Viana did, derive the expansion from a mapping with a cubic critical point. Further, we discuss how the argument can be extended and expansion derived from certain mappings with a critical point of arbitrarily high (odd) degree. We also discuss the issues in trying to use Viana’s method with the double standard map as our source of non-uniform expansion.
Konstruktion av Flerdimensionella
Dynamiska System med Positiva
Lyapunovexponenter
Sammanfattning
I denna uppsats anpassar vi Marcelo Vianas generella konstruk-tion av glatta transformakonstruk-tioner som uppvisar icke-likformig expansion i flera dimensioner. I vår konstruktion, istället för att koppla ihop vårt system med en kvadratisk avbildning, får vi expansionen från en av-bildning med en kubisk kritisk punkt. Dessutom så diskuterar vi hur argumentationen kan utvidgas och expansionen fås av vissa typer av avbildningar med en kritisk punkt av godtyckligt hög (udda) grad. Vi diskuterar också problemen i att försöka använda Vianas metod med den så kallade double standard-avbildningen som källan till den icke-likformiga expansionen.
Acknowledgements
I want to thank my supervisor Kristian Bjerklöv for all of his support, guid-ance and great enthusiasm for the subject.
Contents
1 Introduction 5 2 Uniform hyperbolicity 7 3 One-dimensional maps 11 4 Lyapunov exponents 12 4.1 Osedelec’s Theorem . . . 145 Results and comments on choice of system 17 6 Overview of proof 20 7 Admissible curves 21 8 Building expansion 25 9 Technical lemma 27 10 Putting it all together 35 11 The double standard map 40 11.1 Admissible curves . . . 41
11.2 Building expansion (again) . . . 41
11.3 Technical lemma (again) . . . 43
11.4 Sketch of attempted proof of the Technical lemma . . . 44
1
Introduction
Say we have a differentiable map f from a manifold M to itself. We may refer to this as a dynamical system, and throughout this thesis we will assume M is smooth. We could ask ourselves, what can we say about the orbits of this dynamical system? By this we mean the sets {x, f (x), f2(x), . . . }, sometimes
even {. . . , f−2(x), f−1(x), x, f (x), f2(x), . . . }, if f is a diffeomorphism. This turns out to often be a difficult question, since the orbits easily become very complicated. Perhaps we can relent a little, and disregard sets with zero measure, for some reasonable measure. This helps, and from a measure-theoretic viewpoint, is quite prudent. Perhaps we might even say it is enough only to know how the orbits behave asymptotically, say if they are attracted to or occupy only a small subset of M after enough iterations. Either that, or we could ask ourselves what measurements along the orbits are on average, for instance do the limits
lim n→∞ 1 n n−1 X j=0 g(fj(x)),
for some continuous function g, exist? If so, how do they behave? It is also of interest to know what kind of dynamical behavior will persist if we perturb the system. This is connected to the structural stability of a system, which says that other systems that are close to it have similar dynamics. To be able to talk about systems that are close we need a notion of closeness of two dynamical systems, and one that will be used throughout this thesis is the Cr-distance between two Cr functions f and g, written ||f − g||
Cr. This is
given by the supremum of the absolute value of the differences between the partial derivatives up to order r. For instance if f, g : R2 → R2 are two C1
functions, where f (x, y) = (f1(x, y), f2(x, y)) g(x, y) = (g1(x, y), g2(x, y)), then ||f − g||C1 = sup (x,y)∈R2 {|fi− gi|, |∂xfi− ∂xgi|, |∂yfi− ∂ygi|, for i = 1, 2}.
This is very important when we use dynamical systems to model real world systems, because more often than not, we do not have the exactly right model at hand. There are measurement errors and approximations involved, and it is problematic if small changes to the system could dramatically alter the
behavior of a dynamical system. These are important questions that we want to answer when examining dynamical systems.
One way of trying to find answers to these questions is by looking at the derivative of f at points in the orbits. We can have conditions of varying strength on the derivative, but in this thesis we focus on how the derivative behaves on average, by examining the limits
lim n→∞ 1 nlog ||D(f n (x))v|| , where x ∈ M and v ∈ TxM. (1)
By TxM we mean the tangent space at x. For a point x and non-zero vectors
v ∈ TxM , we refer to the limits as the Lyapunov exponents of f at x. Should
the limit above not exist, we can replace it with lim inf or lim sup, such that vectors for which lim inf > 0 imply a positive Lyapunov exponent, and those for which lim sup < 0 imply a negative Lyapunov exponent. Given U ⊂ M such that f (U ) ⊂ int(U ), we say f has non-uniformly expanding behavior on U if almost every x ∈ U have tangent vectors such that the limit (or lim inf if the limit does not exist) above is positive (in other words, at least one positive Lyapunov exponent). This also implies that if a point x ∈ M has more than one positive Lyapunov exponent, there is non-uniform expansion in more than one dimension. Later in this thesis we will discuss what these positive Lyapunov exponents can imply for a dynamical system.
Let us first note that it can be quite difficult to determine if we have non-uniform expansion if we have regions in U where ||D(f (x))v|| can get arbitrarily close to zero, for some non-zero v ∈ TxM . Because looking at the
limit lim n→∞ 1 nlog ||D(f n (x))v|| = lim n→∞ 1 nlog n−1 Y i=0 Df (fi(x)) ! v ,
we realize that we will need to know if a typical orbit is prone to enter these critical regions, where D(f (x))v can be close to zero. Another complicating factor is that there can be rotations of the tangent space, and expanding directions can be interchanged with contracting directions.
In 1997 Marcelo Viana in a novel way constructed smooth dynamical systems with several positive Lyapunov exponents at almost every point, a multidimensional non-uniform expansion. He also showed that this expansion persists in systems close in C3-distance. Viana’s strategy was to couple non-uniformly expanding quadratic maps x 7→ a0 − x2 with suitable systems
that have an everywhere large expansion. More precisely, he begins with the system ψα : S1× R → S1 × R, given by
where S1 = R/Z and d ≥ 2 is an integer. Then for certain values of a0 ∈ (1, 2)
and a compact interval I0 ⊂ (−2, 2) such that ψα(S1× I0) ⊂ int(S1× I0) for
small enough α, he proved the following:
Theorem 1.1 (Viana [18]). If d is large enough, in this case d ≥ 16, the following holds. For every α > 0 sufficiently small, the map ψα has two
positive Lyapunov exponents at Lebesgue almost every point (θ, x) ∈ S1× I 0.
Moreover, the same holds for every map ψ such that ||ψα−ψ||C3 is sufficiently
small.
In his argument, the fact that the critical point of a0− x2 is quadratic is
important. The main result of this thesis will be an extension of his argument to other systems, where we in the construction replace the quadratic map with a class of maps with instead a cubic critical point. We also have a few extra assumptions on our systems, similar to ones Viana has in the first part of his general proof. Otherwise we try to mirror the construction and method, to show our result. We will also describe how to extend our result if the critical point has higher order, for certain classes of systems. We will also discuss the problems in trying to replace the quadratic map above with the so called double standard map, x 7→ 2x + π1 sin 2πx mod 1, from S1 to
S1.
Dynamics that are characterized by expansion or contraction through the derivative are often called hyperbolic, and before coming to our main results we will go through some of the history and theory of hyperbolic dynamics.
2
Uniform hyperbolicity
As stated above, hyperbolic dynamics are characterized by the presence of expanding and contracting directions for the derivative. When this is the case the differential can give a lot of information about topological and measurable aspects of the dynamics. [8].
In particular uniformly hyperbolic dynamics has been a historically fruit-ful object of research. Uniform hyperbolicity is a stronger condition on these expanding and contracting directions of the derivative, than the kind of ex-pansion (or contraction) we can derive from looking at Lyapunov exponents. More precisely, it often means the following: [8]
Definition 2.1. Suppose f : M → M is a diffeomorphism. We say that f is uniformly hyperbolic or an Anosov diffeomorphism if for every x ∈ M there is a splitting of the tangent space TxM = Es(x)L Eu(x), where Es and Eu
are Df -invariant subspaces, and there are constants C > 0 and λ ∈ (0, 1) such that for every n ∈ N one has
||Dfn(v)|| ≤ Cλn||v|| for v ∈ Es(x) and
||Dfn(v)|| ≤ Cλ−n||v||
for v ∈ Eu(x)
We call Es(x) and Eu(x) the stable and unstable subspaces at x. Impor-tantly, the constant C is uniformly chosen for all x ∈ M .
Example 2.2. Famous examples of uniformly hyperbolic systems are toral automorphisms. These are dynamical systems where
M = Rk/Zk= S1
× · · · × S1
| {z }
k
, and the transformation is given by x 7→ Ax mod 1. Here A is a matrix with integer coefficients such that det A = 1 and A has no eigenvalues on the unit circle. For example, if k = 2 and
A =2 1 1 1
,
then Dfn = An. Also A has two eigenvectors v1, v2 with eigenvalues λ1 = 3−√5
2 ∈ (0, 1) and λ2 = 3+√5
2 . In the notation above then, E
s(x) is the
subspace generated by v1 and Eu is generated by v2.
In dynamical systems, the notion of Cr structural stability is important. This means that a system should be equivalent to other systems in a Cr
neighborhood, for some 0 < r ∈ N. Two dynamical systems f and g on a manifold M are equivalent if there exists a homeomorphism h : M → M such that the diagram below commutes.
M M
M M
f
h h
g
Example 2.3. The quadratic map F4 : [0, 1] → [0, 1] defined by x 7→ 4x(1−x)
, is topologically conjugated to the tent map T : [0, 1] → [0, 1], given by T (x) = 2x when 0 ≤ x ≤ 1/2 and T (x) = 2(1 − x) when 1/2 ≤ x ≤ 1. The homeomorphism h of the interval is given by h(x) = sin2(x).
Structural stability implies topological properties of the system are pre-served under perturbation. One reason why this is important is if we wish to model some phenomena with a dynamical system, our model is unlikely to
exactly mirror the phenomena, but maybe some perturbation of the system actually does. Then we would not want dynamically interesting properties of our model to disappear if we change the system a little bit. For instance: The system in Example 2.3 is part of the quadratic family Fa= ax(1−x),
and if a > 4 +√5, the map, this time from R to R, is C1 structurally stable
[11].
If we know f is a diffeomorphism, there is a very tight connection between uniformly hyperbolic systems and structural stability. Work done by Robbin, de Melo, Robinson and Mañé led to this important theorem: [5, Theorem 1.5]
Theorem 2.4. A C1 diffeomorphism on a compact manifold is C1
struc-turally stable if and only if it is uniformly hyperbolic and verifies the strong transversality condition.
We need not talk about what exactly strong transversality means, but the point is structural stability of a diffeomorphism is closely connected to uniform hyperbolicity.
A dynamical system can also have uniformly hyperbolic dynamics on a part of the state space M . A hyperbolic set is defined to be an invariant (under the action of f ) and compact set Λ ⊂ M such that every x ∈ Λ allows a splitting of the tangent space as above. Of particular interest are hyperbolic sets which are also attractors. A compact f -invariant set Λ is called an attractor if there is a neighborhood U of Λ such limk→∞∩k≥0fk(U ) ⊂ Λ. Such
a set Λ is called an Axiom A attractor if it is also a hyperbolic set. If we say an attractor is Λ is irreducible, then we mean it cannot be written as the union of two disjoint attractors. An f -invariant probability measure µ is a probability measure for which given any measurable set A, µ(f−1(A)) = µ(A). Sinai, Ruelle, Bowen showed [5, Theorem 1.7]
Theorem 2.5. Let f be a C2 diffeomorphism with an irreducible Axiom A attractor Λ. Then there is a unique f -invariant probability measure µ on Λ such that there exists a set V ⊂ U having full Lebesgue measure (with regards to U ) with the property that for every continuous function ϕ : U → R we have lim n→∞ 1 n n−1 X j=0 ϕ ◦ fj(x) = Z ϕ dµ, for every x ∈ V .
This can be interpreted to say that the behavior of orbits of typical points in the open set U are completely determined on the statistical level
[5], at least with regards to the measure µ. This can be contrasted with Birkhoff’s ergodic theorem (see section 4), which could only give us the same result for points on the attractor itself. This unique measure µ is called the Sinai-Ruelle-Bowen measure (SRB-measure). There are several special things about this measure. One is that it in some sense is observable, mean-ing that the measure of subsets by µ can be approximated by observmean-ing the fraction of the time the orbit of a typical point in the basin spends there. For this reason it is sometimes referred to as a physical measure [5]. By Df | Eu
we mean Df the restriction of Df as a function from Eu → Eu.
Theorem 2.6. The SRB-measure is also characterized by the fact that hµ(f ) =
Z
|det(Df | Eu)| dµ,
where hµ(f ) is the metric entropy of f .
The metric entropy of f , in a sense, is a measure of the randomness in a dynamical system [20]. Let us take a moment and introduce this measure, because it is also connected to Lyapunov exponents.
A measure preserving transformation, or map, f , of a probability space is one for which µ(f−1A) = µ(A). Given such a transformation f of a probability space (M, B, µ), we let α = {A1, A2, . . . , An} to be measurable partition of
M . Then for n ≤ m, set αm
n = f−nαW · · · W f−mα, where αW β is defined
as {A ∩ B, A ∈ α and B ∈ β}. Continuing, we define H(α) = −Xpilog pi where pi = µ(Ai),
and finally the entropy of a transformation f , hµ(f ), is defined as:
hµ(f ) = sup α h(f ; α), where h(f ; α) = lim n→∞ 1 nH(α n−1 0 ).
For proof that the above limit actually exists we refer to Walter’s chapter on entropy in [19].
3
One-dimensional maps
The field of one-dimensional dynamics have occupied a special place in the theory of non-linear dynamics [9]. The transformations are relatively simple, but the dynamical behaviour very rich. It has also served as a model for the study of dynamical systems in higher dimensions, and we will take a moment to talk about recent developments in the field. In particular, the quadratic family fa: [0, 1] → [0, 1] given by
fa: x 7→ ax(1 − x), a ∈ [1, 4],
has been a leading example in the area. It is an example of more general class of maps, called unimodal, which are maps from a closed interval I to itself, which has only one critical point in its interior. For technical reasons, we assume one of the endpoints of I is the preimage of the other one, which is also a repelling fixed point. If the number of critical points in the interior is ≥ 1 we say the map is multimodal. Let us say a uni- or multimodal map is hyperbolic, or regular, if there is a periodic attractor in I whose basin of attraction is I \K, and K an expanding Cantor set of zero length. Expanding meaning that there are C > 0 and σ > 1 such that |Dfn(x)| ≥ Cσn for
every x ∈ K. This implies that almost every orbit converges to the periodic attractor, and so has quite a simple behaviour. We say a map is of mixing type if:
There is a cycle of intervals fk(J ), k = 0, 1, . . . , p − 1, where
int(fn(J )) ∩ int(fm(J )) = ∅
for 0 ≤ m < n < p and fp(J ) ⊂ J , and such that the return map fp : J → J
is topologically mixing. A dynamical system f : M → M is topologically mixing if for every pair of non-empty and open subsets A, B ⊂ M , there exists an integer N > 0 such that for every n ≥ N , fn(A) ∩ B 6= ∅. To be of
mixing type, we also need of the maps that I \ ∪n≥0f−n(J ) is an expanding
Cantor set of zero length.
A big problem in one-dimensional dynamics have been whether or not the hyperbolic maps are dense among multimodal maps. It was first shown in 1997, by Graczyk together with Swiatek, and independently by Lyubich, that in the quadratic family, hyperbolic maps are dense. This result was then carried further to unimodal maps, and then finally Kozlovski, Shen and Van Strien showed in 2004 that [10]
Theorem 3.1. Hyperbolic maps are dense in the space of Ck maps of the compact interval or the circle, k = 1, 2, . . . , ∞.
But the picture is quite interesting and complicated, because for instance, among the unimodal maps we have the following result [5, Theorem 2.11]: Theorem 3.2. For an open class of families of C2 unimodal maps, includ-ing the quadratic family, the set of parameters for which the map has an absolutely continuous invariant probability measure υ has positive Lebesgue measure.
Recall that a measure υ is absolutely continuous (with respect to the or-dinary Lebesgue measure µ), if µ(A) = 0 implies υ(A) = 0. A map with an absolutely continous invariant probability measure as above is called stochas-tic. In this setting υ will be a SRB-measure, meaning as earlier that
lim n→∞ 1 n n−1 X k=0 ϕ(fkx) = Z ϕ dυ.
Then the dynamics are not necessarily very simple, but we have a very good probabilistic understanding of the dynamics. Lyubich proved that in the quadratic family, almost every real quadratic map is either hyperbolic or stochastic (meaning for almost every parameter a ∈ [1, 4] in the definition of Fa). After this, Lyubich and Avila extended this result [12]:
Theorem 3.3. In any nontrivial real analytic family of unimodal maps with quadratic critical point, almost every map is either hyperbolic or stochastic.
We see much important work began with the quadratic family and then extended to the general unimodal or even multimodal case. A further gen-eralisation is of course to families of maps of higher degree, for instance of form fc,d: z 7→ zd+ c. As late as 2011, Avila, Lyubich and Shen showed that
a typical polynomial fc,d of mixing type is stochastic (and from Theorem 3.1
we know hyperbolic maps are dense in these families).
4
Lyapunov exponents
In the history of dynamical systems, the study of uniform hyperbolicity has been very important for developing techniques and insights, but it is in many ways not a wide enough class of systems. For instance not every manifold admits an Anosov diffeomorphism [15].
Much work has been done to relax the conditions of uniform hyperbolicity, to allow a wider class of systems, but still potentially salvage some the insights and results from the study of uniform hyperbolicity. One way of doing this is instead of prescribing uniform bounds on the contraction and expansion by
the derivative, we instead look at the asymptotic behavior of orbits; can the tangent space be decomposed into subspaces that either contract or expand on average? One way of formulating this is through Lyapunov exponents.
Let the setting now be that f : M → M is a differentiable map, and M a smooth compact manifold. Given x ∈ M and v ∈ TM(x), let us consider
λ(x, v) defined as λ(x, v) = lim n→∞ 1 nlog ||D(f n (x))v||.
If there is no such limit one can can use instead λ(x, v) and λ(x, v), that is lim sup or lim inf of the above. As stated in the introduction, these limits are refered to as Lyapunov exponents. If λ(x, v) is positive, then ||Dfn(x)v||
grows exponentially and we can interpret it to mean an exponential diver-gence of nearby orbits.
An important theorem in the theory of Lyapunov exponents is Osedelec’s multiplicative theorem, which tells us when the limit above actually exists. It says that if µ is an f -invariant probability measure, then almost everywhere there exists for each x, numbers
λ1(x) > λ2(x) > · · · > λr(x)
with multiplicities m1(x), m2(x), . . . , mr(x)(x) such that
1. For every nonzero vector v ∈ TxM , λ(x, v) = λi(x) for some 1 ≤ i ≤
r(x).
2. The sumP
i≤r(x)mi(x) = dim(M ).
3. The sumP
i≤r(x)λi(x)mi(x) = limn→∞ n1 log || det(Dfn(x))||.
Should f also happen to be a diffeomorphism, the tangent space is decom-posed into Df -invariant subspaces E1(x), . . . , Er(x)(x). We define a+ :=
max{a, 0}, and then if f also is C2 we have the following fascinating
connec-tion with the entropy hµ (recall the definition from the section on Uniform
hyperbolicity) of a system:
Theorem 4.1. [22] Let f : M → M be a C2 diffeomorphism and µ an
f -invariant probability measure with compact support. Then the following holds: • In general hµ(f ) ≤ Z X i λ+i dim(Ei) dµ
• If µ is equivalent to the Lebesgue measure, then hµ(f ) = Z X i λ+i dim(Ei) dµ
• When λ1 > 1, the equality in (ii) holds if and only if µ is an SRB
measure.
Above then is an example how results on uniformly hyperbolic systems, Theorems 2.5 and 2.6, have analogues in a non-uniform setting.
4.1
Osedelec’s Theorem
Since Osedelec’s Theorem is central in the theory of Lyapunov exponents, let us state it properly and give a proof of it. We are working with Lyapunov exponents, and thus interested in how the differential operator D acts on fn(x), but the setting for the theorem is more general than that.
We assume first that (X, B, µ) is a probability space, and let T : X → X be a measure preserving transformation. We recall that a measure preserving transformation T : X → X is one for which µ(T−1A) = µ(A) for every A ∈ B. Then we let A : X → GL(m, R) be a measurable mapping. Let use define An := A(Tn−1x) · · · A(x). Let is also assume thatR log+||A|| dµ < ∞.
Theorem 4.2 (Osedelec’s Theorem [14]). Let (T, µ; A) be as above. Then at µ-a.e x, there exists a filtration of subspaces
{0} = V0(x) ( V1(x) ( · · · ( Vr(x) = Rm and numbers λ1(x) < · · · < λr(x) s.t. (1) ∀v ∈ Vi(x) − Vi−1(x), λ+(x, v) = λi(x) (2) lim n→∞ 1 n log|det A n
(x)| =Xλi(x) · [dim Vi(x) − dim Vi−1(x)].
The functions x → r(x), λi(x) and Vi(x) are measurable.
The proof for the full theorem is quite technical, and we will only prove it in the two dimensional case. To do this we will need Birkhoff’s ergodic theorem and also Kingmans’s subbadditive ergodic theorem. We will state these two without proofs. Recall that a measure is ergodic if for every A ∈ B, T−1(A) = A =⇒ µA = 0 or 1. Then
Theorem 4.3 (Birkhoffs Ergodic Theorem [4]). Let T : X → X be as above, and let ϕ ∈ L1(µ). Then there exists ϕ∗ ∈ L1(µ) such that
1 n n−1 X 0 ϕ ◦ Ti → ϕ∗a.e.
Morever, ϕ∗ ◦ T = ϕ∗a.e. and R ϕ∗ dµ = R ϕ dµ. It is also the case that if
(T, µ) is ergodic, then ϕ∗ =R ϕ dµ.
Theorem 4.4 (Kingman’s subadditive ergodic theorem [20]). Let T : (X, B, µ) be a measure preserving transformation. For n = 1, 2, . . . , let ϕn : X → R
be a sequence of measurable functions s.t. 1. R ϕ+1 < ∞
2. ϕm+n≤ ϕm+ ϕn◦ Tm a.e. for m, n ≥ 1.
Then there exists ϕ∗ : X → R ∪ {−∞} with ϕ∗◦ T = ϕ∗a.e. and R (ϕ∗)+ < ∞
such that
1
nϕn → ϕ
∗
a.e.
The following proof will be based on the exposition by Young in [20]: Proof. First we recall that any given matrix A ∈ GL(2, R), A has a singular value decomposition [16, Theorem 17.1], meaning that A = O2DO1, where
O1 and O2 are orthonormal matrices and D is diagonal matrix with
D =d1 0 0 d2
,
and d2 ≥ d1. Since the O1, O2 are orthogonal, there are unit vectors u1 and
u2 for which O1u1 = e1 and O1u2 = e2 and thus minimally and maximally
stretched by A. We also know that Au1 = O2(d1e1) and Au2 = O2(d2e2) are
orthogonal.
To prove the theorem we will apply the subadditivity theorem to the functions ϕ(k)n (x) = log sup W ⊂R2 |det(An(x) | W )| ,
where W is a k-dim subspace of R2, k ∈ {1, 2}, corresponding to subspaces
spanned by subsets of {u1, u2}, so that det(An(x) | W ) makes sense. The first
with the fact that X is a probability space. The second hypothesis is also easy since ϕ(k)m+n = log sup W ⊂R2 det(An+m(x) | W ) = log sup W ⊂R2 det(A(Tn+m−1(x) · · · A(x) | W ) ≤ log sup W ⊂R2 det(A(Tn+m−1(x) · · · A(Tm(x)) | W ) + log sup W ⊂R2 det(A(Tm−1(x) · · · A(x) | W ) = ϕ(k)n ◦ Tm+ ϕ(k) m .
Next we will examine how the singular values of An grows. We define
d(n)1 (x) ≤ d(n)2 (x) to be the singular values of An(x), u(n)1 (x) and u(n)2 (x) be unit vectors such that |An(x)u(n)1 | = |d(n)1 (x)| and |An(x)u(n)2 (x)| = |d(n)2 (x)|. We claim that the limits
lim n→∞ 1 n log d (n) 1 and lim n→∞ 1 n log d (n) 2
exists, almost everywhere. That the second one exists follows from the fact that n1 log d(n)2 = 1nlog ||An|| = 1
n|A nu(n) 2 | = 1 nϕ (1)
n (x), and using the
subad-ditivity. Then since 1nlog d(n)1 d(n)2 = n1log|det An| converges by the ergodic
theorem, n1d(n)1 = 1nlog | det An| − 1 nlog d
(n)
2 converges too. Let us denote the
limits of d(n)1 and d(n)2 by d1(x) and d2(x). Now let an x be typical such that
the limits d1 and d2 exists. If d1 = d2 then r(x) = 1, λ1 = d2 and V1 = R2.
Also (2) is true since lim n→∞ 1 nlog|det A n(x)| = lim n→∞ 1 nlog |d (n) 1 d (n) 2 | = d2· 2.
From now on then, assume d1 < d2. We will begin by showing now that u (n) 1
converges. Recall that each pair (u(n)1 , u(n)2 ) defines an orthonormal basis in R2. Let us then write u(n)1 = v1 + v2 with regards to the splitting u
(n+1) 1 ⊕
u(n+1)2 , and ψn be the angle between vectors u (n)
1 and u (n+1)
1 . Then v2 =
(u(n)1 · u(n+1)2 )u(n+1)2 , and |u(n)1 · u(n+1)2 | = | cos(π/2 − ψ)| = | sin ψn|. Now let
h(n) be a function going to 0 as n → ∞, dependent only on x. Let also c(n) denote a constant close to 1 dependent on n. The preceding gives us that
|An+1v 2| = | sin ψn||An+1u (n+1) 2 | = | sin ψn||d (n+1) 2 | = c|ψndn2| + h(n).
We also know |An+1v 2| = |An+1(u (n) 1 − v1)| ≤ |An+1u (n)
1 | = |A(Tnx)An(x)u (n) 1 |
≤ ||A(Tnx)|| · d(n)1 ≤ ||A(Tnx)|| · (dn1 + h(n)).
The first inequality follows since v1 is the direction of minimum stretch of
An+1. Putting the two inequalities above together, we have, for arbitrary ,
|ψndn2| ≤ ||A(T n x)||(·dn1 + h(n)) =⇒ |ψn| ≤ ||A(Tnx)||( d1 d2 n + h(n)) =⇒ |ψn| ≤ (1 + )n d1 d2 n ,
for large enough n. This is because ||A|| ∈ L1, and the ergodic theorem then gives 1nA◦Tn→ 0 a.e.. For small enough then, ψ
nis a converging gemoetric
sequence, and u(n)1 converges. We can also see that the angle between u1 and
u(n)1 is less than |ψ| ≤ const (1 + ) d1 d2 n .
Now we will first show that n1 log |Anu1| → log d1. Let c be a constant close
to 1. We write u1 = w1+ w2 with regards to the splitting u(n)1 ⊕ u (n)
2 . Then
Anw1 = cd(n)1 = cdn1 + h(n). Also, similar as before, Anw2 ≤ const (1 +
)n d1 d2 n d(n)2 ≤ const (1 + )n d1 d2 n
dn2+ h(n) = const (1 + )ndn+ h(n). Using also that Anw
1 is orthogonal to Anw2, we get that
log d1+ h(n) ≤
1 nlog |A
nu
1| ≤ log(1 + ) + log d1 + h(n),
and since → 0 as n → ∞, we are done.
We let V1(x) be the space spanned by un1. Now instead we look at v ∈
V2(x) = R2\ V1(x). Again we write v = v1+ v2 in the splitting u (n) 1 ⊕ u
(n) 2 .
Since we now know that u(n)1 converges, for sufficiently large n, there is a constant c > 0 such that |v2| ≥ d. Then Anv1 ≤ d1n+ h(n) and Anv2 ≥ cdn2,
which implies 1nlog |Anv| → log d
2, which was the last piece in proving the
theorem.
5
Results and comments on choice of system
Recall the setup behind and the conclusions in Theorem 1.1, given in the introduction. Viana begins his main argument by adding a few more as-sumptions on the systems ψ close to ψα. He assumes first that every ψ is
of skew-product form, meaning of the form ψ(θ, x) = (g(θ), f (θ, x)). He also assumes that ∂xf (θ, x) = 0 if and only if x = 0. In this first part, he further
only assumes that ψ is C2 and ||ψ − ψ
α||C2 ≤ α.
Before we come to our choice of system, some comments are in order. When starting work on this thesis, we tried at first to look for a similar set of C2 functions close to φ
α : S1× S1 → S1× S1, given by
φα(θ, x) = (dθ mod 1, 2x + α sin(2πθ) +
1
πsin(2πx) mod 1)
The reason being that the critical point x = 1/2 of x 7→ 2x + π1 sin(2πx) is cubic, and there was good reason to believe one could get similar results as in Viana’s section on building expansion [13]. However, the fact that the critical point was not quadratic gave some qualitatively different issues in the method. Without going into details now, Viana in [18, Lemma 2.5], seemingly, only needed that when orbits tend to be a certain distance away from the critical point, there is an average expansion greater than one, but in the ways we have tried to adapt the proof the actual magnitude of the expansion becomes important.
After some experimenting with different reasonable systems, we instead based the system to couple with on eh : S1 → S1, defined by
e h(x) = kx mod 1, x − 12 ≤ − k 32(x − 1 2) 3 mod 1, x − 12 < kx + D mod 1, x −12 ≥ ,
where k > 2 and D are chosen so that the system is continuous.1, and > 0.
See Figure 1 for a plot of eh with = 0.1, and note that k approaches 2 as → 0+. Although eh00(x) is not defined when |x − 1/2| = , eh0 is Lipschitz
continuous with Lipschitz constant 2k .
Then we can find a C3 map h : S1 → S1 such that the following holds: If
/2 ≤ |x − 1/2| ≤ 2: (i) |eh − h| ≤ α (ii) |eh0− h0| ≤ α (iii) |eh00− h00| ≤ 2k , where eh 00 is defined,
and eh = h otherwise. Both > 0 and α > 0 are small constants, and just how small will be determined through considerations in the main proof. The constant α can be much smaller than .
1We let k = 1 1 2− 2 3 and D = −k 12+23.
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Graph of h˜
Figure 1: Graph of eh with = 0.1.
Similar then to how Viana began his proof with the additional assump-tions on ψ above, we look at systems ϕ sufficiently close to ϕα : S1× S1 →
S1× S1, defined by:
ϕα(θ, x) = (dθ mod 1, fα(θ, x)),
where
fα(θ, x) = h(x) + α sin(2πθ) mod 1 (*)
and d an integer greater than 16. The assumptions on the systems ϕ close to ϕα we have to make are:
1. The systems are of skew-product form, so ϕ(θ, x) = (g(θ), f (θ, x)). 2. We assume ∂xf = ∂xxf = 0 if and only if x = 1/2.
The main result of this thesis is then the following:
Theorem 5.1. Let ϕα : S1× S1 → S1 × S1 be as defined above. Then if
and α are small enough and d ≥ 16, every ϕ such that ||ϕ − ϕα||C3 ≤ α,
with the added assumptions on ϕ as stated above, has two positive Lyapunov exponents Lebesgue almost everywhere.
Hopefully the extra assumptions can be removed in a similar way as Viana removed his assumptions above on ψ. We believe that the fact we need that
||ϕ − ϕα||C3 ≤ α and not only ||ϕα− ϕ||C2 ≤ α at this stage of the proof is
a consequence of having a cubic critical point.
We also do not see any issues with extending Theorem 5.1, making a very similar construction but replacing eh(x) with
b h(x) = bkx mod 1, x − 12 ≤ − b k nn−1(12 − x)n mod 1, x − 1 2 < bkx + bD mod 1, x −1 2 ≥ ,
where n = 2m + 1 > 3, and m is a positive integer. This is also reminiscent of work done on one-dimensional systems by Thunberg in [17].
6
Overview of proof
What we will essentially do is use and adapt Viana’s method where necessary. Exactly as in Viana’s case, it is clear that every point in S1×S1has at least one
positive Lyapunov exponent since (gn)0 grows exponentially fast as n → ∞.
lim n→∞ D(ϕn(θ0, x0) 1 0 = lim n→∞ n−1 Y j=0 A(θj, xj) ! 1 0 ≥ |g0|n ≥ (d − α)n, where (θj, xj) = ϕj(θ0, x0), (θ0, x0) ∈ S1× S1 and A(θj, xj) = g0(θj) 0 ∂θf (θj, xj) ∂xf (θj, xj) .
Then we know at least that the lim inf, λ
(θ, x),1 0
, is positive for every (θ, x) ∈ S1× S1, and we have one positive Lyapunov exponent. The task we
are set with, and the challenge, is to show that
lim n→∞ D(ϕ(θ0, x0)n) 0 1 = n−1 Y j=0 |∂xf (θj, xj)| ≥ ecn
for some c > 0 and almost every (θ, x) ∈ S1 × S1. Again, this is difficult
because we will need to have a good amount of control on how often and how close the xj:s get to the critical point 1/2, since there we can have an infinite
contraction. An important part of this strategy is the use of something we (and Viana) call admissible curves.
Definition 6.1. We say bX ⊂ S1×S1 is an admissible curve if bX = graph(X), where X : S1 → S1, and is such that X is C2 except maybe discontinuous on
the left at θ0, the fixed point of g. Also, |X0| ≤ α and |X00| ≤ α.
Given an admissible curve bX0, we let bXj(θ) = ϕj(θ, X0(θ)). In the main
proof we will show that, if n is large enough, then D(ϕn( bX1(θ)) 0 1 = n Y j=1 ∂xf ( bXj(θ)) ≥ e cn.
Or almost. It will be true everywhere except on a set En of values of θ, with
Lebesgue measure m(En) ≤ D1e−γ √
n, for some D
1, γ > 0 . Then we can see
that the set E = ∩n≥1∪k≥nEk will have measure 0, since
m(E) ≤X k≥n D1e−γ √ k = D 1e−γ √ nX k≥n e−γ( √ k−√n) = D1e−γ √ nX k≥0 e−γ √ k ≤ D 2e−γ √ n,
for some constant D2 > 0. This expression goes to 0 as n → ∞, meaning
m(E) = 0. If we can show all this, then all but a subset of S1× S1 of measure
0 will have two positive Lyapunov exponents, since the admissible curve bX0
was arbitrary.
Now the function f (θ, x) is such that ∂xf ≈ k > 2 for most values of x,
it is only close to the critical point x = 1/2 that we can lose expansion when we iterate f . What we will do is essentially two things then. We will show that orbits that get extremely close to S1 × {1/2} will be a very rare, and
orbits that only get very close will do so seldom enough so that the expansion will be enough to offset the contracting effects close to S1× {1/2}. What we mean by extremely and very close will be made precise later in the proof.
7
Admissible curves
In this section we will prove some very similar results about our admissible curves, as Viana’s (very similar since the curves are very similarly defined). First we will create a set of Markov partitions on S1. A Markov partition of S1
is simply a partition into intervals Ij, such that g
Ij(θ) is a homeomorphism
onto a union of other intervals in the partition. In our case we have a partition Pj for each j ≥ 0, and we build them up recursively. We let θ0 be the fixed
are ordered according to the orientation of S1 = R/Z inherited from R. Then we define
P1 := {[eθj−1, eθj), for j = 1, 2 . . . , d where eθd = eθ0}
and recursively
Pi+1= {All the connected components of g−1(ω), for each ω ∈ Pi}.
For example, if g(θ) = dθ mod 1, then {eθ0, eθ1, . . . eθd−1} = {0,1d,2d, . . . ,d−1d },
and P1 = {[0,1d), . . . , [d−1d , 0)}.
Given any ω ∈ Pk and admissible curve bX, we define bX | ω = graph(X |
ω). The following fact will be very useful
Lemma 7.1. If ω ∈ Pm, and bX is an admissible curve, then so is ϕm( bX | ω)
Proof. It is quite clear, by how we have defined the markov partitions Pm,
that the set ϕn( bX | ω) properly defines a C2 function over S1 with possibly
a discontinuity at eθ0. Let Y : S1 → S1 be defined by Y (g(θ)) = f (θ, X(θ)).
Then we just need to show that |Y0| ≤ α and |Y00| ≤ α, because then we
know this property is preserved on each iteration of ϕ. So Y0(g(θ))g0 = ∂θf + ∂xf (θ, X(θ))X0 and since g0 > 15, |∂θf | ≤ α2π| sin(2πθ)| + α ≤ 8α, |∂xf | ≤ 3, we know |Y0| ≤ 1 1512α ≤ α. Also Y00(g0)2+ Y0g00 = ∂θθ2 f + ∂θx2 f + ∂xx2 f (θ, X)X02+ ∂xf (θ, X)X00,
and similarly, since |g00| ≤ α, ∂θθ2 ≤ α(2π)2| sin(2πθ)| + α ≤ 50α, ∂θx2 f ≤ α, ∂xx2 f ≤ 2k + α, we know |Y 00| ≤ ( 1 15) 2100α ≤ α.
The above then proves that ϕn( bX | ω) too will be an admissible curve.
In the next lemmas we want to, given an admissible curve bX and an interval I ⊂ S1, find a bound on the measure of values of θ that upon iteration of ϕ on bX has x-values in I. This will be important to find a bound on the measure of values of θ such that ϕn( bX) are extremely close to S1× {1/2}.
First the following fact about the metric distortion of gn, where we define the metric distortion of a function h as sup |hinf |h00||.
Lemma 7.2. The metric distortion of gn is bounded on ωn∈ Pn for all n.
Proof. Since d − α ≤ g0 ≤ d + α and −α ≤ g00≤ α, the distortion of gn on ω
is bounded by sup(gn)0(θ) inf(gn)0(θ) ≤ n Y i=1 g0(xi) + m(ωi)α g0(x i) − m(ωi)α ≤ n Y i=1 (1 + 2m(ωi)α) ≤ n Y i=1 1 + α 2i ,
where xi ∈ ωi, and since m(ωi) ≤ (d−α1 )i ≤ 2·21i. But
log n Y i=1 1 + 2m(ωi)α = n X i=1 log1 + α 2i ≤ ∞ X i=1 Dα 2i < C∗
for some constants D and C∗, the latter then being a bound for the distortion.
Lemma 7.3. Given an admissible curve bX, let bX(θ) = (θ, X(θ)), and let b
Z(θ) = ϕ( bX(θ)) = ϕ(θ, X(θ)) = (g(θ), Z(θ)). Then given any interval I ∈ S1, m({θ ∈ S1 : bZ(θ) ∈ S1× I}) ≤ 2|I| α + 2 r |I| α
Proof. Here properties of sin(2πθ) and the particular bounds on |X0|, |X00|
will be important. First we partition S1 by setting A
1 = {θ ∈ S1 | sin(2πθ) ≤
1/3}, and A2 to be the complement of A1. Note that both A1 and A2 has
two connected components. Now by the trigonometric one, cos(2πθ) ≥ 11/12 for θ ∈ A1. Then if θ ∈ A1,
|Z0| = |∂θf + ∂xf (X(θ))X0| ≥ |α2π cos(2πθ) − α| − 3α ≥
11α
2 − 4α ≥ α By use of the mean value theorem it follows that, for a connected component [a, b] ∈ A1,
|Z(b) − Z(a)| ≥ α|b − a| which implies that |b − a| ≤ |Z(b) − Z(a)|
α .
Since there are two connected components in A1 and |Z(b) − Z(a)| ≤ |I|, we
get
m({θ ∈ A1 : Z(θ) ∈ I}) ≤
2|I| α .
Now instead assume that θ ∈ A2, so sin(2πθ) > 1/3. Then |Z00| = ∂θθf + 2∂θxf (θ, X)X0 + ∂xxf (θ, X)(X0)2+ ∂xf (θ, X)X00 ≥ α((2π) 2 3 − α) − 2α 2− 2k + 2α α2− 3α ≥ 6α. Then we claim that
m({θ ∈ A2 : Z(θ) ∈ I}) ≤ 2
√
2p|I|/(6α) ≤ 2p|I|/α
One way to see this is the following. Worst case is that |Z0(θ)| = 0 for some ξ in a connected component [a, b] of A2. Using taylor expansion around that
point we see that |Z(ξ + h) − Z(ξ)| ≥ Z00(η)h2 2 ≥ 6αh2 2 , for |h| ≥ (b − a)/2.
The result follows.
Corollary 7.4. There is an C1 > 0 such that given any admissible curve
b
X0, and an interval I ⊂ S1 with |I| ≤ α, the following holds:
m({θ ∈ S1 : bXj(θ) ∈ S1× I}) ≤ C1
r |I|
α for every j ≥ 1.
Proof. Again we will use properties of the Markov partitions. Let ω ∈ Pj−1,
and set bXω = ϕj−1( bX0) and bZω = ϕ( bXω). Then
m({θ ∈ S1 : bZω(θ) ∈ S1× I}) ≤ 2 |I| α + 2 r |I| α ≤ 4 r |I| α
since α ≤ |I|. Now by construction, bXj(θ) = bZω(gj−1(θ)). Then by Lemma
7.3,
m({θ ∈ S1 : bXj(θ) ∈ S1× I}) ≤ C∗4
r |I|
α m(ω).
The prove the last inequality, let us argue more generally. Let h(θ) := gn(θ)
and ω ∈ Pn, A be a subset of S1 and w0 the inverse image of A. Then
1 = m(S1) ≤ m(ω) sup h0 and
m(A) ≥ m(ω0) inf h0. This implies that
m(ω)sup h
0
inf h0 m(A) ≥ m(ω 0
), which is the same as
8
Building expansion
This section contains similar results as in the likewise named section in Viana’s paper [18], where he shows what kind of expanding behavior we can expect, depending on the distance of the x-variable to the critical point 1/2.
Given a point (θ0, x0), we let (θj, xj) = ϕj(θ0, x0). Throughout this
sec-tion, let di denote
xi− 12
. We introduce two constants. One will be C ∝ 2 and dependent on h, and C0 which is fixed with respect to , but otherwise dependent on h. We also introduce the small constant η = η(α) > 0. How small will be determined later in the proof.
Lemma 8.1. There exists an integer N = N (α) > 0 such that Qj=N −1 j=0 |∂xf (θj, xj)| ≥ x0−12 2 α−1+η whenever x0− 12 < 6α1/3.
Proof. Now |f (θ, x)| ≤ C(d30+ α), and given d0 < 6α1/3, f (θ, x) ≤ Cα. Then
since |xi| = (k + α)|xi−1| + C0α we see that
|xi| ≤ (1 + (k + α) + (k + α)2+ · · · + (k + α)i)Cα ≤
(k + α)i+1− 1
k − 1 Cα ≤ (k + α)
iCα.
Let eN be the smallest integer such that Cα(k + α)Ne > 1
C. This implies that
e N > 1 log(k + α) 2 log 1 C + log 1 α > log 1 C + log 1 α Setting N = eN + 1, we can estimate that
N −1 Y j=0 |∂xf (θ0, x0)| ≥ |∂xf (θ0, x0)|(k − α)Ne ≥ 1 Cd 2 0α −1 ≥ d2 0α −1+η ,
for α chosen small enough, and the smaller α is, the smaller we can make η(α). We also used that since for x = 1/2, ∂xf = ∂xxf = 0, taylor expansion
gives that |∂xf (θ0, x0)| ≥ const (x0− 1/2)2.
Lemma 8.2. There is a constant C2 > 0 such that for a given δ, less than
but close to 1, and all (θ0, x0) ∈ S1 × S1, with d0, d1, d2, . . . , da−1 ≥ α1/3,
Qa−1 j=0|∂xf (θj, xj)| ≥ C2α 2/3(δk)a. In addition, if d a < , then Qa−1 j=0|∂xf (θj, xj)| ≥ C2(δk)a.
Proof. Since |∂xf (θj, xj)| ≥ k − 2α ≥ δk whenever dj ≥ and α is small
enough, we need to check what happens if α1/3 < d
j < . Let t be the least
integer such that dt < . If xs > for all s > t, then for C2 small enough
(independent of α) a−1 Y j=0 |∂xf (θj, xj)| ≥ (C2d2t − C 0 α)(δk)a−1 ≥ (C2α2/3− C0α)(δk)a−1 ≥ C2α2/3(δk)a.
Suppose now there exists l > 0 such that dt+l ≤ . We assume l is the
smallest such integer. Similar as in the proof of lemma 8.1, xt+(i+1)
≤ (k + α)|xt+i| + C0α. Then |xt+i| ≤ (1 + (k + α) + (k + α)2+ · · · + (k + α)i−1)C0α + (k + α)iCd3t ≤ (k + α)i(Cα + Cd3 t) ≤ (k + α)iCd3t,
since dt ≥ α1/3. It is clear that l ≥ l0, where l0 is least integer such that
(k + α)l0Cd3 s ≥
1
C0. Then
l0log(k + α) − 2 log() + 3 log(dt) ≥ − log C0
l0 ≥ 1
log(k + α)(2 log() − 3 log(dt) − log C
0
) ≥ 2 log() − 3 log(ds) − log C0,
if k is close enough to 2. We can estimate
t+l−1
Y
j=t
|∂xf (θj, xj)| ≥ C2d2t(k − α)l−1.
Thus the question becomes is if l is big enough so that C2(k − α)l−1d2t ≥ (δk)
l
. But the above is equivalent to
C2d2t( k − α k ) l ≥ (δ)l which is implied if C2d2t ≥ (δ 0 )l, where 0 < δ0 < δ, by making α small enough. But
(δ0)l ≤ (δ0)l0 ≤ 1 C0
d3 s
Then if > ds is small enough, the inequality holds.
This means that anytime we know that the system returns to the neigh-borhood we gain enough expansion to prove the second part of the lemma. If we do not return, then we get the added factor C2α2/3, as seen above.
Remark 8.3. The above proof we believe could be adapted to show that for a more general type of system, meaning other choices of h(x) in (*) that do not need to be so close to a linear function in large parts of S1, we would get
expansion ≈ δ inf |h0(x)|.
Remark 8.4. Notice that Lemma 8.2 implies that if (θ, x) has an orbit that only intersects S1× I a finite number of times , where I = {x : |x − 1/2| ≥
α1/3} (and not in such that x = 1/2), (θ, x) will have two positive Lyapunov
exponents.
9
Technical lemma
This section is also based on Viana’s section Technical lemma, and now we come to the part of Viana’s proof the broke down when we tried to adapt it to our first choice of system. Let us comment on the issues as we come to them in the proof of the lemma.
Important sets in the proof will be J (r), which we define as J (r) := {x ∈ S1 : |x −1
2| < α
1/3e−r},
(2) where r ≥ 0. We also define M to be the largest integer such that (kp)Mα ≤ 1,
where 1 < p < 2. We emphasize that this p and its relation to δ will be important in a later discussion on generalising our main result, Theorem 5.1. What we want to bound is the recurrence of orbits into sets of the form S1× J(r), where we can have very high contraction.
Lemma 9.1. There are C3 > 0 and β > 0 such that, given any admissible
curve bY0 = graph(Y0) and any r ≥ (16 − η) log(α1),
m({θ ∈ S1 : bYM(θ) ∈ S1× J(r − 2)}) ≤ C3e−5βr
Remark 9.2. A comment on the condition r ≥ (1/6 − η) log α1. Note that Lemmas 8.1 and 8.2 imply that it is precisely when r ≥ (1/6 − η/2) logα1 that orbits which enter S1× J(r) might have non-positive Lyapunov exponents.
Before proving the above lemma, we need another preliminary result. Given an admissible curve bX and 1 ≤ j ≤ d, we set bZj = ϕ( bX | [eθj−1, eθj)) =
Lemma 9.3. There are H1, H2 ⊂ {1, . . . , d} with #H1, #H2 ≥ [d/16] such
that |Zj1(θ) − Zj2(θ)| ≥ α/100 for all θ ∈ S
1, j
1 ∈ H1 and j2 ∈ H2.
Proof. As in lemma 2.2, let bZ(θ) = (g(θ), Z(θ)) = ϕ( bX(θ), but also l = [d/16] and χ1 < χ2 be the fixed points of Z(θ). There are two and only two such
points since Z0(θ) ≥ α in one of the two component of A1 and Z0(θ) ≤ −α
in the other, and |Z00(θ)| ≥ 6α on A2. And of course χ1, χ2 ∈ A2. Let ki
for i = 1, 2 be such that χ ∈ [eθki−1, eθki) and suppose now that neither χ1
nor χ2 is in [1/4, 3/4] (1/4 and 3/4 are the fixed points of sin(2πθ), so χ1, χ2
should be close to these points). Then we can set H1 = {k1+ 1, . . . , k1 + l}
and H2 = {k2 − l, . . . , k2 − 1}. Now we observe that eθk1+l <
1 4 + l+1 d−α < 1 2− 1 2πarcsin 1 3, where l+1
d−α is the minimal length of l + 1 intervals in P1. The
last inequality is a calculation, and follows since 152 > d−αl+1. Similarly, we have that eθk2−l> 3 4− l d−α > 1 2+ 1 2πarcsin 1
3. What is this good for? Well, then we
know that for j = 1, . . . , l,
sin eθk1+j > sin( 1 2 − 1 2πarcsin 1 3) = 1 3, and sin eθk2−j < sin( 1 2+ 1 2πarcsin 1 3) = − 1 3, meaning they are all separated by at least 1πarcsin1
3. Since Z is monotone decreasing on [eθk1, eθk2−1) and Z 0 ≥ α on A 1, we get inf Z | [eθj1−1, eθj1) − sup Z | [eθj2−1, eθj2) ≥ α π arcsin 1 3 ≥ α 100 for every j1 ∈ H1 and j2 ∈ H2., which proves the lemma in this case.
Now suppose instead that χ1 ≥ 1/4 . Then we can choose H1 = {k1 −
l, . . . , k1 − 1} and H2 = {k2 − l, . . . , k2 − 1}. Then eθk1−l > 1/4 − l d−α > 1 2πarcsin 1
3 of course for every j = 1, . . . , l, eθk2−j < 0, so then the intervals
given by H1 and H2 are separated by at least 2π1 arcsin13, which together with
that Z is monotone increasing on [eθk2−1, eθk1−l) we get that
inf Z | [eθj1−1, eθj1) − sup Z | [eθj2−1, eθj2) ≥ α 2π arcsin 1 3 ≥ α 100. The case if χ2 ≤ 3/4 is symmetrical to this, and so the lemma is proved.
Remark 9.4. The above result essentially means that on admissible curves, since they have such a low derivative, the effects of α sin(2πθ) is on the same order as h(X(θ)). See Lemma 7.1. We do not expect the above result to be improved in any significant way to help us in other systems.
Now for the proof of Lemma 9.1:
Proof. Let Y0 be an admissible curve, and denote the distance
Yj(θ) − 12
by dj(θ). To prove the theorem, we can assume that for some τ ∈ S1,
dM(τ ) ≤ α1/3, otherwise the lemma is obviously true. What we will show
now is that this implies that dj(θ) ≥ α1/3 for all θ ∈ S1, j = 1, . . . , M − 1,
and that dM(θ) < , which will give us a high rate of expansion through
Lemma 8.2. First we define
osc(Yj) := sup θ,τ ∈S1
|Yj(θ) − Yj(τ )|
Then osc(Y0) = α and
osc(Yj) ≤ (sup ∂xf )osc(Yj−1)+α ≤ (k+C0α)osc(Yj−1)+α ≤ k osc(Yj−1)+C0α,
implying osc(Yj) ≤ C0kjα. Thus osc(YM) ≤ C0kMα ≤ C0α1−1/p < αq, for
some q > 0. This gives us that
dM(θ) ≤ C0αq < for all θ ∈ S1, (3)
if we make α much smaller than . We also know that for 0 ≤ j ≤ M − 1 and 1 ≤ i ≤ M − j
|Yi+j(θ)| ≤ C(k + α)i(dj(θ)3 + α), (4)
when dj(θ) ≤ , and
|Yi+j| ≤ C(k + α)i(|Yj(θ)| + α) (5)
when dj(θ) > . To show that dj(θ) ≥ α1/3 for j = 1, . . . , M − 1, let us argue
by contradiction and assume otherwise. Then dj(θ) ≤ α1/3, and by a similar
reasoning as in Lemma 8.1 and 8.2,
|YM(θ)| ≤ C(k + α)M −j(dj(θ)3+ α) ≤ (k + α)M −jCα ≤ C(k + α)Mα ≤ αq.
The last inequality is true, since we can make α small enough such that for for any fixed 1 < p < 2, k + α ≤ kp. But again, choosing α small enough,
this will contradict (3). This means, through Lemma 8.2 that given y ∈ bb Y0,
|∂xfM(by)| ≥ C2(δk)
M.
(6) The next step is to find a uniform bound for the metric distortion in the x-direction upon iteration of an admissible curve bY0. More precisely, we will
show that for 0 ≤ j ≤ M − 1 and 1 ≤ i ≤ M − j, we have that ∂xfi(θj, xj) ∂xfi(τj, yj) = j+i−1 Y m=j ∂xf (θm, xm) ∂xf (τj, yj) ≤ 2 (7)
where (θj, xj), (τj, yj) ∈ bYj. This is reasonable since we know points on the
curve bYj are at a distance greater than α1/3 from x = 1/2, and osc(Yj)
has a good bound. To begin with, we first note that (4) gives us that if α1/3≤ d j(θ) ≤ , then 1 C ≤ (k + α) M −jd j(θ)3.
Now if both xm and ym are at a distance greater than from 12, then
1 − ∂xf (θm, xm) ∂xf (τm, ym) ≤ 1 −k + Cα k − Cα = Cα k − Cα < Cα q .
If both xm and ym are within distance of 12 (this can also be handled in a
similar way as in [18, Technical Lemma], by observing again that ∂xf (θ, x) ≈
const(x − 1/2)2) when x is close to 1/2), then
1 −∂xf (θm, xm) ∂xf (τm, ym) ≤ k/ 2|((y m− 1/2)2− (xm− 1/2)2| + Cα) k/2((y m− 1/2)2− Cα) . Now |(ym− 1/2)2− (xm− 1/2)2| + Cα (ym− 1/2)2− Cα ≤ |(ym− 1/2) 2− (y m− 1/2 + 2(k + α)mα)2| + Cα 1 C2/3(k+α)(M −m)2/3 − Cα ,
the last inequality since we know |ym− xm| ≤ (k + α)mα, and using equation
(4). Continuing, the above is less than or equal to C(k + α)mα 1 C(k+α)(M −m) − Cα ≤ C(k + α) Mα 1 − C(k + α)Mα ≤ Cαq 1/2 ≤ Cα q
Now assume that (θm, xm) − 1/2 ≥ and (τm, ym) − 1/2 < , then the worst
case scenario is if: ∂xf (θm, xm) ∂xf (τm, ym) ≤ k + Cα k/2( − 2kmα)2− Cα = k + Cα k(1 − 4kmα/ − α2/2) − Cα ≤ k + Cα k − Cαq. Then 1 −∂xf (θm, xm) ∂xf (τm, ym) ≤ Cα q k − Cαq ≤ Cα q.
Which implies that ∂xfi(θm, xm) ∂xfi(τm, ym) ≤ (1 + Cαq)2i ≤ (1 + Cαq+(Cαq)2 2! + . . . ) 2M ≤ eCαqM ≤ 2, since M ≈ log(α1).
Now that we have a uniform bound for the metric distortion of fi(θj, xj),
we can fix an arbitrary by ∈ bY0, and derive a good bound by just looking at
∂xfi(y). What we want to show next is that, in a sense, the expansion isb somewhat evenly distributed through the iterations of fi(θ, x). We begin by
defining λj =
∂xfM −j(ϕj(y))b
. Now the result in (7) gives us that 1 2 λj λi+j ≤ ∂xfi(θj, xj) ≤ 2 λj λi+j . We set K = 400e2, and define t
1 < t2 < · · · ≤ M by t1 = 1 and ti+1 =
min{s : ti < s ≤ M and λti ≥ 2Kλs}. We assume r ≥ (1/6 − η) log(
1 α), and
we define k(r) = max{i : λti ≥
2Ke−r
α2/3 }.
Let us pause and ponder here for a second, try to get our bearings. We know that λj ≥ C2(δk)M −j. The choice of the constant K is to a large degree a
matter of convenience, the important part is the set {t1, t2, . . . } which gives
us a sense of how much expansion there is left of C2(δk)M, and k(r) tell
us how long we can expect to still have a lot of expansion. How much? Well more than 2Keα2/3−r, which, multiplied with
α
100 from Lemma 9.3 will be
≈ α1/3e−r, which is a motivation for the specific definition of K.
Let us continue with the proof. We set the average expansion σ = δk. Note that λti ≤ 2Kλti−1 ≤ (k + C
0α)2Kλ
ti+1 ≤ 6Kλti+1, for every i. If σ = δk,
this implies that C2σM −1 ≤ λ1 ≤ (6K)kλtk+1, and so λtk+1 ≥ C2σ
M −1(6K)−k.
By definition, λtk+1 ≤
2Ke−r
have that:
C2σM −1(6K)−k ≤
2Ke−r α2/3
⇐⇒
(M − 1) log σ − k log(6K) + C ≤ −r + 2/3 log 1 α ⇐⇒ M log(σ) + r − 2/3 log 1 α + C ≤ k log(6K) =⇒ log σ p log klog 1 α + r − 2/3 log 1 α ≤ k log(6K). But the LHS is greater than
r 1 − 2/3 − log δk p log k 1/6 − η ! ,
and if we can now show that 2/3−p log klog δk < 1/6−η we are done. Remembering how we defined the constant η(α), we know that by making α small we can make η arbitrarily small too. What we need then is that
log δk
p log k > 1/2 ⇐⇒ 2 log δ
log k + 2 > p.
Since we can use an arbitrary δ ∈ (0, 1) (by chosing small enough), we see that any 1 < p < 2 will work. Then we have shown that for η small enough, k(r) ≥ rγ, for a constant γ > 0.
Remark 9.5. In a situation where we have a similarly defined system, but the critical point has degree ≥ 2n + 1, for n ≥ 2, what would happen in the proof ? Can we generalise? Well, we would define J (r) := {x | |x − 1/2| ≤ α1/(2n+1)e−r}. An analogue to Lemma 8.1 would give us trouble when
1 2n(2n + 1) − η 2n log 1 α ≥ r.
The earlier reasoning in this proof should carry through the same, and we would end up needing to show that
1 −2n/(2n + 1) − log δk p log k 1/(2n + 1) − η ! > 0.
Which, since both p and δ can be made arbitrarily close to 1, the proof should go through. We also see no reason why we would not get similar building expansion lemmas.
Remark 9.6. For the double standard map, where σ > 1 is the expansion factor given by Lemma 8.2, we end up needing to show that
0 < 1 − 2/3 − log σ p log sup ∂xfα 1/6 − η ! = 1 − 2/3 − log σ p log 4 1/6 − η ! < 1 − 2/3 − log 2 p log 4 1/6 − η ! < 1 −2/3 − 1/2 1/6 − η ,
which will not work.
The remainder of the proof is very similar to Viana’s corresponding proof, but we write it out for completion.
Now for each ¯l = (l1, . . . , lm) ∈ {1, . . . , d}M we denote with ω(¯l), the ω ∈
Pm satisfying that gi−1(ω) ⊂ [eθli−1, eθli]. Also for 1 ≤ j ≤ M , bYj(¯l) =
graph(Yj(¯l)) = ϕj( bY0 | ω(¯l)). We call ¯l and ¯m incompatible if
YM(¯l, θ) − YM( ¯m, θ)
≥ 4e2−rα1/3.
Now by the previous lemma, there are H10, H100⊂ {1, . . . , d}, with #H10, #H100 ≥ [d/16], such that for every l0
1 ∈ H10 and l200∈ H100, we have that
|Y1(l10, . . . , lm, θ) − Y1(l001, . . . , lM, θ| ≥
α 100, for θ ∈ g(ω(l10, . . . , lM)) = g(ω(l001, . . . , lM)). These are equal since
g(ω(l10/l100, . . . , lM) = ω(l2, . . . , lM) ∈ PM −1. Then also |YM(l10, . . . , lm, θ) − YM(l001, . . . , lM, θ| ≥ λ1 2 α 100 ≥ 2Ke−r 2α2/3 α 100 ≥ e 2−rα1/3
So this means that all pairs (l10, . . . , lM), (l001, . . . , lM) are incompatible.
We can continue this line of argument for each of the succesive ti. For
instance, given Li = (l1, . . . , lti−1), and any l
0 ti ∈ H 0 i and lt00i ∈ H 00 i, Yti(Li, l 0 ti, lti+1, . . . , lM) − Yti(Li, l 00 ti, lti+1, . . . , lM) ≥ α 100
for θ ∈ gti(ω(L i, lt0i, lti+1, . . . , lM, θ)) = g ti(ω(L i, l00ti, lti+1, . . . , lM, θ)). Then it follows that YM(Li, lt0i, lti+1, . . . , lM, θ) − YM(Li, l 00 ti, lti+1, . . . , lM, θ) ≥ λti 2 α 100 ≥ 2Ke−r 2α2/3 α 100 ≥ e 2−r α1/3,
as long as k(r) ≥ i. This means that, looking at an arbitrary admissible curve b
YM(l1, . . . lt2, . . . , ltk, . . . , lM), at each position tj when j ≤ k, only elements
from either Hj0 or Hj00 are possible, and since #Hj0, #Hj00 ≥ [d/16], we can at most have dM −k(d − [d/16])k of the admissible curves intersecting any
segment {θ} × J (r − 2). Also (gM)0 ≥ (d − α)M, and since M ≤ const α1, we know (d−αd )M = (1 + α)M ≤ ((1 + α)log 1
α)const ≤ ((1 + α) 1
α)const ≤ const
for α small enough. Add that m(ω) ≤ (gM1)0, and it follows that
m{θ ∈ bYM(θ) ∈ S1 × J(r − 2)} ≤ #{¯l | bYM(¯l) ∩ S1× J(r − 2) 6= ∅} · 1 (d − α)M ≤ dM((d − [d/16])/d)k(r) (d − α)M ≤ const ( 99 100) γ1r.
Setting β = 15log10099 the lemma is proved. We do not need it for this lemma but there is some room for improvement, because the fact that
YM(Li, lt0i, lti+1, . . . , lM, θ) − YM(Li, l 00 ti, lti+1, . . . , lM, θ) ≥ λti 2 α 100 ≥ 2Ke −r 2α2/3 α 100 ≥ e 2−rα1/3, meaning (Li, l0ti, lti+1, . . . , lM, θ) and (Li, l 00 ti, lti+1, . . . , lM, θ) are incompatible
also implies that every pair of the form (l10, l2, . . . , lt2−1, l
0 t2, . . . , l 0 M), (l200, l2, . . . , lt2−1, l 00 t2, . . . , l 00
M) are incompatible, where for j ≥ k, the l 0 j, l
00 j are
arbitrary. To show this notice that Yti+1(Li, l 0 ti, . . . , lti+1, . . . , lM, θ) − Yti+1(Li, l 00 ti, . . . , lti+1, . . . , lM, θ) ≥ λti 2λti+1 α 100 ≥ 4e 2α. and that Yti+1(Li, l 0 ti, . . . , lti+1−1, l 0 ti+1, . . . , l 0 M, θ) − Yti+1(Li, l 0 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) ≤ osc(ϕ( bYti+1−1(Li, l 0 ti, . . . , lti+1−1))) ≤ 8α.
Since bYti+1−1(Li, l
0
ti, . . . , lti+1−1) is an admissible curve and |osc(Yj)| ≤ 2α4
j.
The same argument applies to show Yti+1(Li, l 00 ti, . . . , lti+1−1, l 00 ti+1, . . . , l 00 M, θ) − Yti+1(Li, l 00 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) ≤ osc(ϕ( bYti+1−1(Li, l 0 ti, . . . , lti+1−1))) ≤ 8α.
This means that, by use of the triangle inequality that YM(Li, l 0 ti, . . . , lti+1−1, l 0 ti+1, . . . , l 0 M, θ) − YM(Li, l00ti, . . . , lti+1−1, l 00 ti+1, . . . , l 00 M, θ) ≥ |Yti+1(Li, l 0 ti, . . . , lti+1−1, l 0 ti+1, . . . , l 0 M, θ) − Yti+1(Li, l 00 ti, . . . , lti+1−1, l 00 ti+1, . . . , l 00 M, θ)| · λti+1 2 , which is greater than or equal to
≥ |Yti+1(Li, l 0 ti, . . . , lti+1−1, l 0 ti+1, . . . , l 0 M, θ) − Yti+1(Li, l 0 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) + Yti+1(Li, l 0 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) − Yti+1(Li, l 00 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) + Yti+1(Li, l 00 ti, . . . , lti+1−1, lti+1, . . . , lM, θ) − Yti+1(Li, l 00 ti, . . . , lti+1−1, l 00 ti+1, . . . , l 00 M, θ)| · λti+1 2 ≥ (4e 2− 16)αλti+1 2 ≥ 4e2−rα1/3.
In this case as long as k(r) ≥ i + 1.
Remark 9.7. Some comments on how to get a more general result, where σ only need to be larger than one.
The problem is in large part due to the loss we incur through situations like Lemma 8.1, when |x − 1/2| is very small. We get in trouble as soon as r ≥ (1/6 − η) logα1. However, if we could handle the case (1/6 − η) logα1 ≤ r ≤ (2/3 − η) logα1 in some other way, we could easily adapt the proof and show that for all r ≥ 2/3 − η, the Technical lemma would work for any fixed σ > 1. This would give us much more freedom in choice of h(x) in (*), and we believe we would be able to get a similar result for a system where h(x) is instead the double standard map.
10
Putting it all together
Now finally we have all the lemmas we need to prove the main result, and given the earlier results no novel difficulties arise in regards to Vianas method.
This will center around proving that the recurrence of orbits into S1× J(r) = S1× {x ∈ S1 : |x − 12| < α1/3e−r} for large values of r is low enough, almost everywhere. To do this we first fix n log 1α to be fixed and large enough. We define m, a positive integer such that m2 ≤ n < (m+1)2, and l = m−M .
We fix an arbitrary admissible curve bX0 and introduce 1 ≤ υ ≤ n and the
graph γ = ϕυ( bX0 | ωυ+l) Let us first bound the measure of θ ∈ S1, that after
n iterations, have come extremely close to S × 1/2. More specifically we say, after Viana, that υ is a IIn-situation for θ ∈ ωυ+l if
γ ∩ (S1× J(m)) 6= ∅.
By Lemma 7.1, we know γ is also an admissible curve over some ωl, and
thus has a derivative bounded by α. This lets us bound the diameter in the x-direction of this graph with
α(d − α)−l= α(d − α)M(d − α)−m ≤ 1 α
const
(d − α)−m α1/3e−m
, since n α1. This in turn implies that γ ⊂ J (m − 1). Then we can use Corollary 7.4 and see that
m({θ ∈ S1 | some 1 ≤ υ ≤ n is a IIn-situation}) ≤ nC1 r J (m − 1) α = nC1 r α1/3em−1 α ≤ nC1α −1/3 em/2≤ e √ n/4 .
Let us call this set of θ:s B2(n). From now on will assume that any θ ∈ S1
do not come this extremely close to S1× {1/2}, but only perhaps very close.
More precisely, we say that υ is a In-situation for θ ∈ ωυ+l if
γ ∩ (S1× J(0)) 6= ∅ but γ ∩ (S1× J(m)) = ∅.
Note, by remark earlier, that In- and IIn-situations are the only ways in
which we can get contraction along the orbits.
Let then 1 ≤ υ1 < · · · < υs ≤ n be In-situations of θ. If we recall how N
was defined in Lemma (8.1), then we know that n ≥ (s − 1)N . And for each υi, let us set ri equal to the minimal integer such that γ ∩ S1 × J(ri) = ∅.
What we are interested then is how |∂xfn(X1(θ))| = n Y j=1 |∂xf (Xj(θ))| = υ1 Y j=1 |∂xf (Xj(θ))| s Y i=1 υi+N −1 Y j=υi |∂xf (Xj(θ))| υi+1 Y j=υi+N |∂xf (Xj(θ))| ! n Y υs |∂xf (Xj(θ))| (8)
Again we denote the expansion factor δk by σ to make the remaining proof easier to read. Then using Lemma 8.2 and 8.1, we get the following estimates:
υ1 Y j=1 |∂xf (Xj(θ))| ≥ C2συ1−1, (9) υi+N −1 Y j=υi |∂xf (Xj(θ))| ≥ (α1/3e−ri)2α−1+η = α−1/3+ηe−2ri (10) and υi+1−1 Y υi+N |∂xf (Xj(θ))| ≥ C2συi+1−υi−N for 1 ≤ i < s. (11) We also know n Y υs |∂xf (Xj(θ))| ≥ const |xυs − 1/2| 2 σn−υs ≥ const α2/3e−2rsσn−υ2. (12)
(The above since close to 1/2, ∂xf = const|xs − 1/2|2.) Above we see that
(9) and (11) give us expansion in our orbit, and (10) and (12) are causing us trouble. Taking the log of (8) we estimate
log n Y j=1 ∂xf ( bXj(θ)) ≥ log σ υ1−1+ s−1 X i=1 log e−2riα−1/3+η + s−1 X i=1 log συi+1−υi−N + log α2/3e−2rsσn−υs − (s − 1) log C 2 ≥ (n − (s − 1)N ) log σ + s−1 X i=1 (−2ri+ (−1/3 + η) log α) + 2/3 log α − 2rs − s · const ≥ (n − (s − 1)N ) log σ + s X i=1 (1/3 − η) log1 α − 2ri − (1/3 − η) log1 α − 2/3 log 1 α − s · const ≥ (n − (s − 1)N ) log σ + s X i=1 (1/3 − η) log 1 α − 2ri − log 1 α − s · const.
The worst cases in trying to find a lower bound for the above is if any of the ri ≥ (1/6 − η/2) logα1. If we let G = {i | ri ≥ (1/6 − η) log α1} we can see
that s X i=1 (1/3 − η) log 1 α − 2ri ≥ sη 2 log 1 α − X i∈G 2ri ≥ γ2N s − X i∈G 2ri,
for some λ2 > 0 and independent of α and n. We used that N ≈ logα1. This leads us to log n Y j=1 ∂xf ( bXj(θ)) ≥ (n − (s − 1)N ) log σ + γ2N s − X i∈G 2ri− s const − log 1 α If we assume log σ > γ2, then
(n − (s − 1)N ) log σ + γ2N s ≥ n log σ + N (s − 1)(γ2− log σ)
≥ n log σ + n(γ2− log σ) ≥ nγ2
and if log σ < γ2
(n − (s − 1)N ) log σ + γ2N s ≥ n log σ.
Then for c = 13min{γ2, log σ} we get a lower bound for the derivative of
3cn −X i∈G 2ri− s const − log 1 α ≥ 2cn − X i∈G 2ri,
using that s ≤ Nn + 1 and n N ≈ logα1. Here we deviate somewhat from Vianas proof, to be more explicit about some details in the proof. So observe that 2cn −X i∈G ri ≥ 2c − m − X i∈G0 ri ≥ 2c0n − X i∈G0 ri,
wbere c0 is a constant. This works since ri ≤ m (and m ≈
√
n) and G0 is the same as G; except if there possibly is a In-situation for 1 ≤ υ ≤ M , we
have assumed a maximal loss, and removed any such ri from G. This is a
technical detail which we feel is necessary, but is not explicitly stated in the original paper. If we now define B1(n) = {θ ∈ S1 | Pi∈G0 ≥ c0n}, and and
En = B1(n) ∪ B2(n), we know that for θ ∈ S1\ En,
log n Y j=1 ∂xf ( bXj(θ)) ≥ cn.
Thus if we can show B1(n) ≤ e−γ √
n, for some γ > 0, we will be done. We will
now use a large deviation argument, to show that B1(n) is indeed a very rare
occurence, asymptotically. We begin by fixing 0 ≤ q ≤ m − 1 and define the set Gq = {i ∈ G0 | υi ≡ q mod m, and we set mq = max{j | mj + q ≤ n}.
