Invariant graphs of a family of non-uniformly expanding skew products over Markov maps

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by  Charles Walkden

INVARIANT GRAPHS OF A FAMILY OF NON-UNIFORMLY EXPANDING SKEW
PRODUCTS OVER MARKOV MAPS
C.P.WALKDENANDT.WITHERS
Abstract. We consider a family of skew-products of the form (Tx,gx(t)):X×R→X×R where T is a
continuousexpandingMarkovmapandgx:R→RisafamilyofhomeomorphismsofR. Afunctionu:X→
Rissaidtobeaninvariantgraph ifgraph(u)={(x,u(x))|x∈X}isaninvariantsetfortheskew-product;
7
1 equivalently if u(T(x)) = gx(u(x)). A well-studied problem is to consider the existence, regularity and
dimension-theoretic properties of such functions, usually under strong contraction or expansion conditions
0
(in terms of Lyapunov exponents or partial hyperbolicity) in the fibre direction. Here we consider such
2
problemsinasettingwheretheLyapunovexponentinthefibredirectioniszeroonasetofperiodicorbits.

n Weprovethatueitherhasthestructureofa‘quasi-graph’(or‘bonygraph’)orisassmoothasthedynamics,
a andwegiveacriteriaforthistohappen.
J

3
2

] 1. Introduction and results
S
D 1.1. Introduction. Let T be a continuous, expanding Markov map of the circle, X. Consider the skew
product dynamical system T˜ :X×R→X×R defined by
. g
h
t T˜ (x,t)=(Tx,g(x,t)). (1)
a g
m where g(x,t):X×R→R. A function u:X →R is said to be an invariant graph if graph(u)={(x,u(x))}
[ is T˜ -invariant; equivalently
g

u(Tx)=g(x,u(x)). (2)
1
v We refer to X as the base and R as the fibre. We are interested in the case when g(x,·) : R → R is a
0 homeomorphism; we normally write g (t) = g(x,t), g : R → R and refer to g (·) as a skewing function.
2 x x x
When g is uniformly expanding, the invariant graph exists, is H¨older continuous and, under a partial
3 x
6 hyperbolicity assumption, generically has no higher regularity. As a particular example, let b≥2,b∈N and
0 let Tx=bxmod1. Let λ∈(0,1), λb>1, and let g (t)=cos(2πx)+λ−1t. In this case the invariant graph
x
1. u(x)=−(cid:80)∞n=0λncos2πbnx, the classical Weierstrass function.
0 More generally, if the Lyapunov exponent in the fibre direction is positive with respect to a given ref-
7 erence measure, then the invariant graph is measurable, (2) holds almost everywhere, and generically is
1
not continuous [Sta99, HNW02]. Note that [HNW02] requires a partial hyperbolicity assumption on the
:
v skew-product.
i In this note, we alter the non-uniform contraction condition in the fibre and assume that the Lyapunov
X
exponent in the fibre direction is zero for certain measures. In particular, we consider the case when the
r
a skewing function is the identity map on a given set of periodic orbits. We construct a family of measurable
invariant sets for the skew product and identify the set of measure zero on which (2) fails. Our invariant
sets are generically almost everywhere graphs of functions that are discontinuous on every open set and are
almost everywhere uniformly bounded.
Wedescribetheprecisestructureoftheinvariantsets. Thefollowingdichotomyholds: eithertheinvariant
set is of a discontinuous nature of the form described above or is as smooth as the dynamics. The former
case is generic; in this case (together with a partial hyperbolicity assumption) we also calculate the box
dimension of the invariant set in terms of thermodynamic formalism.
The invariant sets we obtain are an example of a family of so-called bony attractors—a bony attractor
is a closed set that intersects every almost every fibre at a single point and any other fibre at an interval.
2010 Mathematics Subject Classification. 37C70,37D25,37C45.
Key words and phrases. Invariantgraph,skewproduct,bonygraph.
T.WitherswaspartiallysupportedbyanEPSRCDTA.
1
2 C.P.WALKDENANDT.WITHERS
These sets were first described by [Kud10] and other examples occur in [KV14, GH16] as attractors of step
functions over shift maps.
AninvariantgraphdefinedonX\RP. Thequasi-graphinvariantset.
Figure1. Theinvariantgraphandaninvariantset(quasi-graph)oftheaffinereal-valuedskewproductT˜g(x,t)=
(Tx,sin(2πx)+γ−1(x)t), where γ(x) = 3+cos(2πx) and where T is the doubling map. The skew product is the
4
identityatthefixedpointx=0andtheinvariantgraphisdiscontinuousatallpre-imagesof0.
1.2. Results. Let T˜ be a skew product as defined in (1). We write g(x,t) = g (t) and assume that, for
g x
each x ∈ X, g : R → R is a homeomorphism. We also assume that, for each t ∈ R, x (cid:55)→ g (t) is α-
x x
H¨older continuous. We define gn(t) = g g ···g (t) so that T˜n(x,t) = (Tnx,gn(t)). We define
x Tn−1x Tn−2x x g x
h (t) = g (t)−1 so that gn(t)−1 = hn(t) = h ···h (t). Let P := (cid:83)ρ−1P ⊂ X be a collection of ρ
x x x x x Tn−1x r=0 r
distinct periodic orbits for T : X → X. For p ∈ P we denote by (cid:96)(p) the least period of p. Let R be the
P
set of pre-images of points in P; that is
R ={x∈X |Tnx∈P for some n≥0}.
P
Note that this is a countable set. We will often consider the set R and its complement X\R separately;
P P
both sets are T-invariant.
If h :R→R is a diffeomorphism, we define the derivative in the fibre direction to be
x
h (t+(cid:15))−h (t)
∂h (t)= lim x x .
x (cid:15)→0 (cid:15)
In §1.3 we will precisely define a set of skew products T˜ ∈ S(X,R,Cα) where T˜ is expanding in the fibre
g P g
direction except along the periodic orbits in P where g(cid:96)(p)(t) = t for all t ∈ R. We shall abuse notation
p
slightly and write g ∈Cα.
P
Remark 1.1. Examples of skew products that satisfy our hypotheses include affine maps g (t) = f(x)+
x
γ(x)−1t where f, γ are α-H¨older with γ(p) = 1, f(p) = 0 for p ∈ P and otherwise 0 < γ(x) < 1. Our
conditions remain satisfied for diffeomorphisms g : X → Diff(R,R) defined by small, sufficiently smooth
perturbations of affine maps preserving conditions (6) to (9) below. Figure 1 shows an explicit example.
First, we show that such invariant graphs exist. We prove that we have a unique invariant function u on
X\R ; hence the graph of this function is an invariant set of the restricted skew product T˜ | .
P g X\RP
Theorem 1. Let T˜ ∈ S(X,R,Cα). There exists a uniformly bounded function u : X \R → R such that
g P P
u(Tx) = g (u(x)) for x ∈ X \R . Any other uniformly bounded function v : X \R → R satisfying this
x P P
equation is equal to u; moreover, if µ is an ergodic measure for T not supported on R , then u is µ-a.e.
P
unique amongst the set of measurable functions.
We prove the following corollary to Theorem 1, showing that u is uniquely defined on X\R but can be
P
arbitrarily defined on R .
P
Corollary 2. Let T˜ ∈ S(X,R,Cα) and let u be as in Theorem 1. Suppose that P consists of ρ periodic
g P
orbits. For each r such that 0 ≤ r ≤ ρ − 1, choose one element of each periodic orbit p ∈ P ⊂ P.
r
There is a ρ-parameter family of invariant graphs for the skew product, u : X → R, where u (p) = s for
s s r
s=(s ,...,s )∈Rρ. Furthermore, u | =u.
0 ρ−1 s X\RP
INVARIANT GRAPHS OF SKEW PRODUCTS 3
Clearly, for x∈R , each fibre {x}×R is T˜ -invariant; by taking the union we have a dense T˜ -invariant
P g g
set. As X \ R is dense in X, we can consider a sequence x → x for any x ∈ R where each point
P 0 P
x ∈X\R .
0 P
Definition 1.2. LetT˜ ∈S(X,R,Cα). Definetheinvariant quasi-graph U ⊂X×RasU =(cid:83) U where
g P x∈X x
(cid:40)
(x,u(x)) if x∈X\R ,
U = P
x
{x}×[liminf u(y),limsup u(y)] if x∈R .
y∈X\RP,y→x y∈X\RP,y→x P
In other words, we connect the discontinuities between the values of the function u as we approach R ;
P
see Figure 1. Denote the length of the interval of the quasi-graph at x∈R by
P
(cid:12) (cid:12)
(cid:12) (cid:12)
|U |=(cid:12) limsup u(y)− liminf u(y)(cid:12);
x (cid:12) (cid:12)
(cid:12)y∈X\RP,y→x y∈X\RP,y→x (cid:12)
if x∈X\R then we set |U |=0. In Proposition 4.1 we show that U is a T˜ -invariant set.
P x g
We prove two main results about quasi-graphs, reminiscent of those found in the studies of Weierstrass
functions, [Bar15, HL93] and for dynamically-defined invariant graphs [Sta99, HNW02]. The first is a
dichotomy of the structure of the invariant graphs.
Theorem 3. Let T˜ ∈S(X,R,Cα). Then either:
g P
(1) the invariant quasi-graph U is the graph of a uniformly α-H¨older continuous function;
(2) for every x∈R , |U |>0.
P x
When we are in the second case, we will often say “the quasi-graph U is not the graph of an invariant
function”. As R is dense in X, we see that U is not the graph of a function on any open set and has a
P
‘vertical jump’ at each element of R . It is easy to construct examples of continuous invariant graphs, but
P
our second result proves that generically U is of the discontinuous type.
Theorem 4. Let T˜ ∈ S(X,R,Cα). There exists a Cα-open and C0-dense set of g ∈ Cα such that the
g P P
invariant quasi-graph U of T˜ is not the graph of a function.
g
For higher regularity of u, we need higher regularity on the base dynamics T and we consider Cr+α-
expanding endomorphisms of the circle, with r ∈N, r ≥1 and α>0. We also assume that that g :R→R
x
is Cr+α and, for each t, x(cid:55)→g (t) is Cr+α. Again, we abuse notation slightly and write g ∈Cr+α. Denote
x P
this set of skew products by T˜ ∈S (X,R,Cr+α). In this case we can strengthen the above dichotomy:
g Cr+α P
if U is the graph of a function then it is as smooth as the dynamics.
Theorem 5. Let α≥0, r ∈N and r ≥1. Let T˜ ∈S (X,R,Cr+α). Then either:
g Cr+α P
(1) the invariant quasi-graph U is the graph of a Cr+α function;
(2) for every x∈R , |U |>0.
P x
Our final result concerns the box dimension of the invariant quasi-graphs. Our result extends that in
[Bed89] to our setting, and we believe it is the first attempt to calculate the dimension of invariant graphs
of non-uniformly expanding skew products. The key difficulty is establishing a sufficiently strong form of
bounded distortion.
Letq :X →Rbeafunctionandletm(q)=inf q(x). Suppose∂g (t)>0; wewillshowinRemark3.2
x∈X x
that this causes no loss in generality. A skew product is partially hyperbolic if there exists κ>1 such that
1<κ≤m(∂h)m(|T(cid:48)|) (3)
where the infimum m(∂h) is taken over all (x,t)∈X×R and h =g−1.
x x
Suppose that g : X → C2(R,R) is a C2 diffeomorphism and we return to allowing T : X → X to be
a continuous expanding Markov map (with conditions specified in §1.3). Let x (cid:55)→ h t, x (cid:55)→ ∂h (t) and
x x
x(cid:55)→∂2h (t) be Lipschitz continuous. Denote this set of skew products by S(X,R,CLip,2)⊂S(X,R,Cα).
x P P
Define the function
(cid:40)
∂h (u(Tx)) for x∈X\R ,
Dh(x)= x P
∂h (t) for x∈R , where t=limsup u(y).
x P y∈X\RP,y→Tx
Let P(φ) be the topological pressure of a function φ:X →R, see Definition 7.12. We prove the following.
4 C.P.WALKDENANDT.WITHERS
Theorem 6. Let T˜ ∈S(X,R,CLip,2) be partially hyperbolic such that the invariant quasi-graph is not the
g P
graph of a continuous function on X. The box dimension of the quasi-graph U is the unique solution t to
the generalised Bowen equation
P((1−t)log|T(cid:48)|+log|Dh|)=0. (4)
As∂h (t),andsoDh(x),areuniformlyboundedfor(x,t)∈X×R,itiswell-knownthatauniquesolution
x
to such an equation for a partially-hyperbolic skew-products exists and often corresponds to the dimension
of graphs [Bed89, MW12].
1.3. The family of skew products. Here we list the technical hypotheses on the dynamics. We assume
that T is a C1 expanding Markov map. Specifically, there is a partition X = (cid:83)b−1X , X = [t ,t ]
j=0 j j j j+1
with 0 = t < t < ··· < t = 1, such that, for each j, T| : IntX → (cid:83) X is a C1 diffeomorphism.
0 1 b Xj j i∈Ij i
We assume that T is continuous. We assume there exists θ < 1 such that |T(cid:48)(x)| ≥ θ−1 > 1 for all
x ∈(cid:83)b−1Int(X ). We assume that T is locally eventually onto, namely that there exists N ∈N such that,
i=0 j
for all X , TNX = X (equivalently, TN is full branched for some N ∈ N). As T| is a diffeomorphism
j j (cid:83) Xj
onto its image, there exists a well-defined inverse branch ω : X → X and ω is a diffeomorphism.
j i∈Ij i j j
Note that d(ω x,ω y) ≤ θd(x,y). Define a cylinder of rank n by C = C = ω ω ...ω (X).
j j n j0j1...jn−1 j0 j1 jn−1
Note that C ⊂X is an interval.
n
We now state the hypotheses on the skewing function. Let P =(cid:83)ρ−1P be a finite set of periodic orbits.
r=0 r
If p∈P then we write (cid:96)(p) for the least period of p. Recall h (t)=g (t)−1.
r x x
Fix a constant C ≥1. Suppose the skewing function g is such that
S
g(cid:96)(p)(t)=t, (5)
p
for all t∈R and p∈P. Suppose that
|hn(t)−hn(t(cid:48))|≤C |t−t(cid:48)| (6)
x x S
for all x∈X. Let (cid:15)>0 be small. Define the collection of open balls (intervals) in X of radius (cid:15) centred on
p ∈ P by B (P) = (cid:83) B (p) and let G = X\B (P). For an orbit segment x,...,Tn−1x ∈ B (P), denote
(cid:15) p∈P (cid:15) (cid:15) (cid:15)
(cid:15) = inf d(Tjx,p). Suppose we can define a function λ : [0,(cid:15)] → R+ where logλ is α-H¨older for some
j p∈P
α>0, with λ((cid:15))>0 and λ(0)=1 such that
|hn(t)−hn(t(cid:48))|≥C−1λ((cid:15) )...λ((cid:15) )|t−t(cid:48)|. (7)
x x S 0 n−1
Let δ be such that 0 < δ ≤ (cid:15). Suppose that the orbit segment x,...,Tn−1x visits X\B (P) j-many times,
δ
let there exist 0<λ <1 such that
δ
|hn(t)−hn(t(cid:48))|≤C λj|t−t(cid:48)|, (8)
x x S δ
where λ is independent of x, t and t(cid:48). We also assume that both x (cid:55)→ g (t) and x (cid:55)→ h (t) are α-H¨older
δ x x
continuous,
|g (t)−g (t)|≤C (t)d(x,y)α and |h (t)−h (t)|≤C (t)d(x,y)α, (9)
x y g x y h
for some α>0. If t∈W and W ⊂R is compact, then define C =sup {C (t),C (t)}.
W t∈W g h
As x(cid:55)→g (t) is continuous and X is compact, there exists K >0 independent of x such that
x g
|gn(t)|≤ sup sup|gn(t)|=K (t,n). (10)
x x g
0≤j≤nx∈X
Finally, as g is invertible and continuous, there exists λ >0 such that
min
|hn(t)−hn(t(cid:48))|≥C−1λn |t−t(cid:48)|. (11)
x x S min
LetT :X →X beacontinuous, expanding, locallyeventuallyontoMarkovmap. Wedenotethesetofskew
products of the form (1) satisfying conditions (5) to (9) with fixed constant C by S(X,R,Cα).
S P
Remark 1.3. Suppose g (t) = f(x)+γ−1(x)t as in Remark 1.1, where (cid:80)(cid:96)(p)f(Tip) = 0, 0 < γ(x) ≤ 1
x i=0
and γ(p) = 1 if and only if p ∈ P. The function λ : [0,(cid:15)] → R+ in condition (7) can be defined as
λ((cid:15) )=|γ(x)| for x∈B (P) where (cid:15) =inf d(x,p). In this case, for δ ≤(cid:15), λ in condition (8) is chosen
x (cid:15) x p∈P δ
as sup |γ(x)|<1; also λ =m(|γ|).
x(cid:54)∈Bδ(P) min
For these skew products, the constant C =1. Allowing the constant to be larger than 1 allows |γ(x)|>
S
1 at some x ∈ X provided that |γn(x)| < C for all n, hence the Lyapunov exponent is non-positive.
S
INVARIANT GRAPHS OF SKEW PRODUCTS 5
Throughout, we fix the constant C as, when we make a small perturbation of the skew product, the
S
constant does not necessarily perturb independent of n. This is important in the proof of Theorem 4.
More generally, for g a diffeomorphism with 0 < λ ≤ |∂h (t)| < 1 for x (cid:54)∈ P and h(cid:96)(p)(t) = t if and
x min x p
only if p∈P, we define λ((cid:15) )=m(|∂h |) and λ =sup (cid:107)∂h (cid:107) .
x x δ x(cid:54)∈Bδ(P) x ∞
Remark 1.4. Notice, unlike [Bed89] and [HNW02] we have no partial hyperbolicity condition in general.
Remark 1.5. Suppose that the skewing function is not the identity over the periodic orbits of P but has
zero Lyapunov exponent. Say g (t) = f(x)+γ(x)−1t, where γ(cid:96)(p) = 1 but f(p) > 0 for p ∈ P. In this
x
setting, it is easy to see that the invariant graph is unbounded on pre-images of P. In fact, the two basins
(of points repelled to ±∞) appear intermingled in the sense of [AYYK92, AP11, Kel15].
Remark 1.6. For Theorem 1 and Corollary 2 we do not need T to be locally eventually onto, rather just
continuous and expanding.
2. Properties of Markov maps
To prove Theorem 1, we will need the following technical lemma. The content of the lemma is surely
well-known, but we include a proof for completeness.
Lemma 2.1. Let T :X →X be a continuous expanding Markov map of the circle. Let p be a periodic point
and let P denote the periodic orbit of p. There exists (cid:15) > 0 such that for all δ where 0 < δ ≤ (cid:15) if the orbit
sequence {x,Tx,...,Tnx}⊂B (P) with x∈B (p), then d(Tjx,Tjp)<δθn−j for all 0≤j ≤n.
δ δ
Proof. If x=p, then, as P is T-invariant, the result is trivial.
We first prove that if x ∈ B (p) and Tx ∈ B (P), then Tx ∈ B (Tp). Let t be the lower end-point of
δ δ δ j
the interval X . Let Q=P ∪{t }b−1. Let d=inf d(q ,q ). Choose (cid:15)<d/(cid:107)T(cid:48)(cid:107) 4 and let 0<δ ≤(cid:15).
j i i=0 qi(cid:54)=qj∈Q i j ∞
As (cid:107)T(cid:107) >1, δ <d/4 and so for p∈P the balls B (p) are pairwise disjoint.
∞ δ
For x(cid:54)=p, x is located in precisely one of (p−δ,p) or (p,p+δ) and on these sets T is differentiable (note,
p could be equal to t for some j, hence we split the interval B (p) at p). By the Mean Value Theorem,
j δ
d(Tx,Tp)≤(cid:107)T(cid:48)(cid:107) d(x,p)<d/4. Hence the set T(B (p)) has diameter at most d/2 and
∞ δ
T(B (p))⊂B (Tp). (12)
δ d/2
By assumption, Tx ∈ B (P). We show that Tx ∈ B (Tp). Suppose not, say Tx ∈ B (p(cid:48)) with p(cid:48) ∈ P,
δ δ δ
p(cid:48) (cid:54)= Tp. Then Tx ∈ B (p(cid:48))∩T(B (p)). By (12), Tx ∈ B (p(cid:48))∩B (Tp), contradicting that the points
δ δ d/2 d/2
of P are at least distance d apart. So Tx∈B (Tp).
δ
Suppose that p ∈ Int(X ) for some j; therefore ω (Tp) = p. By choice of δ, T : B (x) → X is a
j j δ
diffeomorphism on B (x) ⊂ X . Therefore, the inverse branch ω : T(B (x)) → X is a diffeomorphism on
δ j j δ j
T(B (x)). As T is expanding, T(B (p))⊃B (Tp). By the Mean Value Theorem, for x∈B (p),
δ δ δ δ
d(ω (Tx),ω (Tp)) d(x,p)
θ ≥ j j = . (13)
d(Tx,Tp) d(Tx,Tp)
As Tx∈B (Tp), we have d(Tx,Tp)<δ and so d(x,p)<θδ.
δ
Finally, consider the case where p is the unique point of intersection p∈X ∩X . Then ω (Tp)=p=
j−1 j j
ω (Tp). Either x ∈ (p−(cid:15),p) ⊂ X or x ∈ (p,p+(cid:15)) ⊂ X . In either case, by applying the appropriate
j−1 j−1 j
inverse branch, the result follows by the idea of (13) above, replacing the ball B (p) with intervals (p,p+(cid:15))
δ
or (p−(cid:15),p). By iterating this argument, if Tjx ∈ B (P) for all 0 ≤ j ≤ n, then d(Tjx,Tjp) < δθn−j, as
δ
required. (cid:3)
Thefollowingcorollaryisalsowell-known: theonlyorbitthatδ-shadowsaperiodicorbit(forδ sufficiently
small) is the periodic orbit itself.
Corollary 2.2. Let δ >0 be as in Lemma 2.1. Suppose for all n∈N we have Tnx∈B (P). Then x∈P.
δ
Proof. Suppose x (cid:54)∈ P. There exists η > 0 such that, for all p ∈ P, d(x,p) > η. As Tnx ∈ B (P) for all
δ
n ∈ N, for any m ∈ N the orbit segment {x,...,Tmx} ⊂ B (P). By Lemma 2.1, for some p ∈ P, we have
δ
d(x,p)<δθm. Choose m large such that δθm <η, contradicting our assumption that x(cid:54)∈P. (cid:3)
6 C.P.WALKDENANDT.WITHERS
In order to prove the existence of an invariant graph, we show that hn(t) → u(x). In our setting, when
x
x is very close to P, the rate of contraction of h (t) is not uniformly bounded below 1. Thus, supposing
x
that the orbit segment Tx,...,Tn−1x remains in B (P), the next result uses Lemma 2.1 to bound hn(t)
δ x
independently of x and n.
Lemma 2.3. Let T : X → X be a continuous expanding Markov Map of the interval. Let 0 < δ ≤ (cid:15) < 1
be as in Lemma 2.1. Let g ∈ Cα. Let n ≥ 2. If the orbit segment {Tx,...,Tn−1x} lies in B (P), then
P δ
|hn(t)−t|≤A (t), where A (t)>0 is independent of n and x, but not t.
x h h
Proof. Let (cid:96) be the least period of p∈P. Let Tx∈B (Tp). Let n be the smallest integer n ≥n divisible
δ (cid:96) (cid:96)
by (cid:96), so hnp(cid:96)(t)=(t) and n(cid:96)−n≤(cid:96)−1
As {Tx,...,Tn−1x} ⊂ B (P), by Lemma 2.1, for 0 ≤ i ≤ n−2, d(Ti(Tx),Ti(Tp)) ≤ θn−2−iδ. Let
δ
Tn(cid:96)−nq =p. As hn(cid:96) is the identity,
q
|hn(t)−t|=|hn(t)−hn(cid:96)(t)|≤|hn(t)−hn(t)|+|hn(t)−hn(hn(cid:96)−n(t))|. (14)
x x q x p p p q
We bound the first term of (14). By (6) and Lemma 2.1,
n−1
(cid:88)
|hn(t)−hn(t)|≤ |hih hn−1−i(t)−hih hn−1−i(t)|
x p x Tix Ti+1p x Tip Ti+1p
i=0
n−1
(cid:88)
≤C C (hn−1−i(t)) d(Tix,Tip)α
S g Ti+1p
i=0
(cid:32) n−2 (cid:33)
(cid:88)
≤C C (hn−1−i(t)) 1+ θα(n−2−i)δ ≤C((cid:96),t).
S g Ti+1p
i=0
as hn−1−i(t) ∈ [−K ((cid:96),t),K ((cid:96),t)] =: W, the H¨older constant C (hn−1−i(t)) < C . We bound the second
Ti+1p g g g Ti+1p W
term of (14) by
|hnp(t)−hnp(hnq(cid:96)−n(t))|≤CS|t−hnq(cid:96)−n(t)|≤CS(|t|+Kg(t,(cid:96)))
as n −n≤(cid:96). Hence, we have the bound as required. (cid:3)
(cid:96)
3. Existence of the invariant graph
We consider the existence and uniqueness of the invariant graph and prove Theorem 1 and Corollary 2.
Proof of Theorem 1. Let δ be sufficiently small so that Lemma 2.1 holds. Define u : X \R ×R → R by
n P
u (x,t) = hn(t). By Corollary 2.2, the set of points with orbit that visits X \B (P) infinitely often are
n x (cid:15)
precisely X\R . Denote the set X\B (P)=G.
P δ
Suppose that the orbit segment Tni−1+1x,...,Tni−1x ∈ Bδ(P) and Tnix ∈ G. If x (cid:54)∈ G, we let n−1 = 0
and n is the first visit to G, and if x∈G then n =0. In both cases n is the first return to G. Define
0 0 1
HTni−1x =hnTin−i−n1i−x1.
Let j be the number of times the orbit segment x,...,Tnx visits G. We have
hnx(t)=Hxj(t)=HxHTn1x...HTnj−2xHTnj−1x(t).
By Lemma 2.3, |HTni−1x(t)−t| ≤ Ah(t), where Ah is independent of n,i and x but depends on t. Fix
n,m∈N. Letnj ≤n<m≤nk whereTnjxisthelargestnj ≤nsuchthatTnjx∈Gandnk isthesmallest
nk ≥m such that Tnkx∈G. Fixing t∈R, we bound
|hn(t)−hm(t)|=|Hj(hn−nj(t))−Hj(hm−nj(t))|≤λj|hn−nj(t)−hm−nj(t)|.
x x x Tnjx x Tnjx δ Tnjx Tnjx
As n,m→∞, we have λj →0. By repeatedly applying Lemma 2.3,
δ
|hn−nj(t)−hm−nj(t)|=|hn−nj(t)−Hk−1−j(hm−nk−1(t))|
Tnjx Tnjx Tnjx Tnjx Tnk−1x
≤|hnT−njnxj(t)−t|+|t−HTnjx(t)|+|HTnjx(t)−HT2njx(t)|+
···+|Hk−j−1(t)−Hk−j−1(hm−nk−1(t))|
Tnjx Tnjx Tnk−1x
INVARIANT GRAPHS OF SKEW PRODUCTS 7
k−j−1
(cid:88)
≤A (t)+ λiA (t)≤C(t) (15)
h δ h
i=0
for some C(t)>0. Therefore, for each x,t, the sequence hn(t) is Cauchy. By (8), it follows that the limit is
x
independent of t. We can define a uniformly bounded (on X\R ) function u by u(x)=lim u (x,t)=
P n→∞ n
lim Hj(t). Then, u is an invariant graph on X \R as u(Tx)=lim hn (t)=g (lim hn(t))=
j→∞ x P n→∞ Tx x n→∞ x
g (u(x)). Toproveuniqueness,supposevisanotherinvariantgraphandsupposethatvisuniformlybounded
x
on X\R . For x∈X\R , and all n∈N, we have v(x)=hn(v(Tnx)). So,
P P x
hn(−(cid:107)v(cid:107) )≤v(x)≤hn((cid:107)v(cid:107) ).
x ∞ x ∞
As hn(t) converges to u(x) as n→∞ and the limit is independent of t, v(x)=u(x) for x∈X\R .
x P
Now, suppose that v is only measurable. Let B >0, Suppose W ={x||u(x)|<B}. Let µ be an ergodic
measure for T that is not supported on R . For B sufficiently large, µ(W)>0. By the ergodicity of T, for
P
µ-a.e. x there exists subsequence nm such that Tnmx∈W. Hence,
hnm(−B)≤u(x)≤hnm(B).
x x
As m→∞, we have hnm(t)→u(x) independently of t. Hence v =u µ-a.e. (cid:3)
x
While the bound on u is uniform in x, the rate of convergence is not uniform; this lack of uniform
convergence gives the invariant graph its interesting non-continuous structure.
3.1. Proof of Corollary 2. We now consider invariant graphs when the dynamics is restricted to the set
R and prove Corollary 2.
P
Proof of Corollary 2. Let P be a periodic orbit in P and, for each r, choose p ∈ P . Recall (cid:96)(p ) denotes
r r r r
the least period of p . As u is assumed to be T˜ -invariant on R , we have g (u(p )) = u(Tp ). Iterating
r g P pr r r
this we have gp(cid:96)r(u(pr)) = u(T(cid:96)(pr)pr) = u(pr). As gp(cid:96)(pr) is the identity, any value of u(pr) satisfies this
equation.
Choose (arbitrarily) s = (s ,...,s ) ∈ Rρ. For our choice of p , define u (p ) = s . As u (Tx) =
0 ρ−1 r s r r s
g (u(x)), this then defines u on P. Note that u (ω p))=h u (p); we can iterate this to define u on R .
x s s j ωjp s s P
Defineu =uonX\R . Itremainstoshowthatu hasabounddependingonlyonsandu:X\R →R.
s P s P
We only need to consider points in R as u (x)=u(x) for x∈X\R . Let x∈R , so for some N ∈N we
P s P P
have TNx ∈ P, say TNx = p. We have gN(u (x)) = u (p). As u (p) = s it suffices to bound hN(s ) for
x s s s r x r
s ∈ R. If x ∈ P, then x is a periodic point and, by (10), |gN(s )| ≤ K (N,s ) ≤ K ((cid:96),s ). Otherwise, let
r p r g r g r
J be the total number of visits of the orbit of x to G, denoting each visit TNix ∈ G for 0 ≤ i < J −1 and
N =NJ so that TNJx=p. Again, if x(cid:54)∈G, let N−1 =0 so that N0 is the first visit to G.
Using the notation of the proof of Theorem 1, we write hN(s ) = HJ−1hN−NJ−1(s ). By the same
x r x TNJ−1x r
arguments as (15), we can bound |HJ−1(hN−NJ−1(s ))−s |<C(s ) for some C(s )>0. Therefore hN(s )
x TNJ−1x r r r r x r
is bounded independently of N. (cid:3)
3.2. Reducingtofixedpoints. HavingshownthattheinvariantsetsdescribedinTheorem1andCorollary
2 exist for T˜ , it will be useful in what follows to replace T by a power and assume that P consists of fixed
g
points such that g (t) = t for all p ∈ P,t ∈ R. Moreover, we can also assume that T is full branched. The
p
following allows us to do this.
Proposition 3.1. Let P be a set of periodic orbits of X. Let the skew product T˜ ∈ S(X,R,Cα) and let
g P
N be the least integer such that TNX = X for all 0 ≤ j ≤ b−1. Let (cid:96) = k(cid:96) where k ∈ N and (cid:96) is the
j p p
lowest common multiple of the periods of orbits in P and N. Let R be the set of pre-images of P under
P
T˜ and R(cid:96) be the set of pre-images of P under T˜(cid:96). Then R =R(cid:96) , the skew product T˜(cid:96)| has unique,
g P g P P g X\R(cid:96)
P
uniformly bounded invariant graph u:X\R(cid:96) →R and the graph is equal to to the unique uniformly bounded
P
invariant graph of the skew product T˜ | .
g X\RP
Proof. By our choice of (cid:96), the set P consists of points that are fixed under T(cid:96) : X → X. Let x ∈ R . So,
P
there exists n ∈ N such that for n ≥ n we have Tnx ∈ P. Choose M > n the smallest integer such that
0 0 0
M =(cid:96)K for K ∈N. Let k ≥K. Then T(cid:96)kx∈P, so x∈R(cid:96) . For the other direction, if x∈R(cid:96) then there
P P
8 C.P.WALKDENANDT.WITHERS
exists K ∈ N such that, for all k ≥ K, T(cid:96)kx ∈ P. As P is T invariant, Tnx ∈ P for all n ≥ (cid:96)K, therefore
x∈R . Thus, R =R(cid:96) .
P P P
Let g(cid:96) = γ . Then, T˜(cid:96)(x,t) = (T(cid:96)x,γ (t)) is contained in S(X,R,Cα), hence satisfies the conditions of
x x g x P
Theorem 1 and so there exists a unique, bounded invariant graph of T˜ say u(cid:48) :X\R →R.
γ P
Let u : X \R → R be the unique invariant graph of T˜ | . Clearly, u is also an invariant graph of
P g X\RP
T˜ | . As R =R(cid:96) and the invariant graph is unique, u(cid:48) =u. (cid:3)
γ X\R(cid:96) P P
P
FromnowonwewillassumewithoutlossofgeneralitythattheskewproductissuchthatT isfullbranched
and P consists only of fixed points.
Remark 3.2. Proposition 3.1 also allows us to assume g is orientation preserving. As each g : R → R is
x
invertible, g2 is orientation preserving and by Proposition 3.1 the skew product T˜ has the same invariant
x g2
graph as T˜ . (Note that we cannot assume that T is orientation preserving: consider the tent map.)
g
4. The invariant quasi-graph
4.1. Structureoftheinvariantquasi-graph. RecallDefinition1.2ofthequasi-graphU. Astheinvariant
graph u is unique on X\R , the quasi-graph exists and is unique.
P
Proposition 4.1. The quasi-graph U is T˜ -invariant.
g
Proof. By definition U| = graph(u(x)) and we know that the graph of u : X \R → R is T˜ | -
X\RP P g X\RP
invariant, thus we only need to show that T˜ U =U for all x∈R . Let x∈R . By Remark 3.2, we can
g x Tx P P
assume each g is orientation preserving. As g is continuous and X\R is T-invariant,
x x P
(cid:32) (cid:33) (cid:32) (cid:33)
T˜ x, limsup u(y) = Tx, limsup g (u(y)) . (16)
g y
y∈X\RP,y→x y∈X\RP,y→x
For all y ∈X\R , we have g (u(y))=u(Ty), so
P y
(cid:32) (cid:33) (cid:32) (cid:33)
T˜ x, limsup u(y) = Tx, limsup u(Ty) .
g
y∈X\RP,y→x y∈X\RP,y→x
As X \R is T-invariant and T is continuous, as y → x we have Ty → Tx through a sequence in X \R .
P P
So, limsup u(Ty)=limsup u(Ty). By changing notation Ty (cid:55)→y, we have
y∈X\RP,y→x Ty∈X\RP,Ty→Tx
(cid:32) (cid:33) (cid:32) (cid:33)
T˜ x, limsup u(y) = Tx, limsup u(y) .
g
y∈X\RP,y→x y∈X\RP,y→Tx
An identical argument holds for the liminf u(y). By the Intermediate Value Theorem, for any
y∈X\RP,y→x
(x,t) ∈ U we have T˜ (x,t) ∈ U . Therefore, T˜ (U ) = U . Taking the union of all of these, as R is
x g Tx g x Tx P
T-invariant, T˜ (U )=U . We already know T˜ (U )=U , hence we have T˜ (U)=U. (cid:3)
g RP RP g X\RP X\RP g
We now prove that if the quasi-graph is discontinuous at any point p∈P, then it is discontinuous on the
dense set of pre-images of the fixed point p.
Proposition 4.2. Let R denote the pre-images of a point p ∈P. The length |U | =0 for some x ∈R if
p x p
and only if |U |=0 for every x∈R .
x p
Proof. Let x ∈ R and |U | = 0. We can define u(x) = lim u(y). Thus, we have extended u to
p x y∈X\RP,y→x
the point x ∈ R and u is continuous at x. As g is a continuous function and g (u(x)) = u(Tx), so u is
p x x
continuous at Tx. Iterating, as TNx=p for some N ∈N, we have that u is continuous at p.
Now, let z ∈R . There exists N such that TNz =p. By the invariance of u, hN(u(p))=u(z). As, hN is
p z z
continuous, u is continuous at z. So |U |=0. The proof of the other direction is trivial. (cid:3)
z
INVARIANT GRAPHS OF SKEW PRODUCTS 9
5. Proof of Theorems 3 and 4
5.1. Proof of Theorem 3.
Proof of Theorem 3. Suppose that there exists x ∈ R such that |U | = 0. Then, by Proposition 4.2,
P x
|U |=0 for some p∈P. We show that we can extend u:X\R →R to an α-H¨older function u :X →R,
p P s
for some choice of s ∈ Rρ, by iterating backwards towards the fixed point p. As U is the graph of u the
s
dichotomy is proved.
Suppose p∈X . As p is fixed under T and T is full branched, there exists an inverse branch ω :X →X
j j
such that ω(p)=p. As ω is contracting, for any x∈X, ωnx→p as n→∞. Hence, u(x)=gn (u(ωnx)).
ωnx
Let x,y ∈ X \R . Let m be such that θm < (cid:15). As d(ωx,ωy) ≤ θd(x,y), we have that ωm+ix ∈ B (P) for
P (cid:15)
all i∈N. Furthermore, by Lemma 2.1,
d(ωm+ix,P)≤θi(cid:15). (17)
Using (6), (7) and (11),
|t−t(cid:48)|=|hn (gn (t))−hn (gn (t(cid:48)))|
ωnx ωnx ωnx ωnx
=|hmhn−m(gn (t))−hmhn−m(gn (t(cid:48)))|
x ωnx ωnx x ωnx ωnx
≥C−1λm |hn−m(gn (t))−hn−m(gn (t(cid:48)))|
S min ωnx ωnx ωnx ωnx
≥C−1λm λ((cid:15))...λ(θn−m(cid:15))|gn (t)−gn (t(cid:48))|.
S min ωnx ωnx
As logλ is Cα as a function of (cid:15) and logλ(0)=0, using (17),
(cid:12) (cid:12)
(cid:12) n(cid:88)−m (cid:12) n(cid:88)−m (cid:88)∞
(cid:12)− logλ(θi(cid:15))(cid:12)≤ |logλ(0)−logλ(θi(cid:15))|≤ C |θαi(cid:15)|=:logΛ.
(cid:12) (cid:12) logλ
(cid:12) (cid:12)
i=0 i=0 i=0
Exponentiating, (λ((cid:15))...λ(θn−m(cid:15)))−1 ≤Λ for some Λ>0 independent of n. As m is fixed by (cid:15) and T, the
λm term is absorbed into the constant. So, there exists C >0 independent of x, n and t such that
min 0
|gn (t)−gn (t(cid:48))|≤C Λ−1|t−t(cid:48)|. (18)
ωnx ωnx 0
Consider
|u(x)−u(y)|=|gn (u(ωnx))−gn (u(ωny))|
ωnx ωny
n−1
(cid:88)
≤ |gi g gn−1−i(u(ωny))−gi g gn−1−i(u(ωny))|
ωix wri+1x ωny ωix wri+1y ωny
i=0
+|gn (u(ωnx))−gn (u(ωny))|. (19)
ωnx ωnx
Using (18), we know that
|gn (u(ωnx))−gn (u(ωny))|≤C Λ−1|u(ωnx)−u(ωny)|.
ωnx ωnx 0
As ωnx,ωny → p and we assume that u is continuous at p, this term tends to 0 as n → ∞. Letting
s=g (t) and s(cid:48) =g (t) where t=gn−1−i(u(ωny)), we have
wri+1x wri+1y ωny
|gi (s)−gi (s(cid:48))|≤C Λ−1|s−s(cid:48)|.
ωix ωix 0
and
|s−s(cid:48)|=|g (t)−g (t)|≤C (t)d(wi+1x,wi+1y)α ≤C (t)θα(i+1)d(x,y)α.
wri+1x wri+1y g r r g
AsuisuniformlyboundedandT˜ -invariant,wehavet=u(ωn−1−iy)andso|t|≤(cid:107)u(cid:107) . Hencetiscontained
g ∞
in a compact set W ⊂ R. Thus C (t) ≤ C . As θ < 1, we can find a constant C > 0 such that (19) is
g W u
bounded by
|u(x)−u(y)|<C d(x,y)α.
u
Therefore, u is a uniformly α-H¨older continuous function on X \R . As X \R is dense in X, we can
P P
uniquely extend u to a uniformly α-H¨older continuous function u : X → R for some s ∈ Rρ. As g,T and
s
u are continuous and u is an invariant graph on X \R , u must be an invariant graph on X. Therefore,
s P s
the quasi-graph U =graph(u) is an α-H¨older continuous invariant graph of the skew product. (cid:3)
10 C.P.WALKDENANDT.WITHERS
Remark 5.1. By (18), we have
|gn+1 (t)−gn (t)|=|gn (g (t))−gn (t)|≤C Λ−1|g (t)−t|.
ωn+1x ωnx ωnx ωn+1x ωnx 0 ωn+1x
Furthermore, for n sufficiently large, by (17), d(ωn+1x,p)≤Cθn−m(cid:15) for fixed m independent of x. So,
|g (t)−t|=|g (t)−g (t)|≤C (t)λ(θn(cid:15))≤C (t)Cθα(n−m)(cid:15)α.
ωn+1x ωn+1x p g g
As θ < 1, the sequence gn (t) is Cauchy (in n) and converges pointwise with gn (t) → D (t) for some
ωnx ωnx x
function D :R→R. By (18) we have D is Lipschitz continuous as a function of t.
x x
5.2. Proof of Theorem 4. We prove this result in two parts, first the density of such skew products and
second the openness.
Lemma 5.2. Let T˜ ∈S(X,R,Cα). There exists a C0-dense set of g ∈Cα such that U is not the graph of
g P P
a function.
Proof. By Theorem 3, it suffices to assume that there exists s ∈ Rρ such that the invariant graph u of
s
T˜ ∈ S(X,R,Cα) is continuous at p ∈ P and find a small perturbation of g (cid:55)→ gˆ within Cα such that, for
g P P
all s ∈ Rρ, the invariant graph uˆ of T˜ is not continuous at p. Equivalently, the quasi-graph of uˆ is not
s gˆ
the graph of a function. To do this, we assume that both u and uˆ are continuous for some s,sˆ∈ Rρ and
s s
obtain a contradiction. As u ,uˆ are the unique, continuous extensions of u,uˆ:X\R →R, we denote the
s s P
invariant graphs u =u, uˆ =uˆ
s s
Let q (cid:54)∈R be a periodic point. By Proposition 3.1, we can assume that T is full branched with at least
P
twoinversebranches. Therefore,thereexistdisjointsequencesofpre-imagesofq,sayωnq →pandνnq →p.
As the orbit of q ∈/ P ⊂R , for n>N sufficiently large ωnq is sufficiently close to p that ωnq cannot be in
P
the orbit of q. Let V be a small neighbourhood of ωjq for some j >N, disjoint from: νnq for all n∈N, ωnq
for n (cid:54)= j, the orbit of q, and the set P. By making a C0 perturbation g (cid:55)→ gˆ on V with gˆ ∈ Cα, we can
P
ensure that
|gˆj(u(p))−gj(u(p))|≥(cid:15) >0
q q 1
for some (cid:15) >0. We have not perturbed g at ωnq for n(cid:54)=j, so g =gˆ . By inverting condition (6), for
1 ωnq ωnq
all n>j
|gˆn(u(p))−gn(u(p))|=|gn−jgˆj(u(p))−gn−jgj(u(p))|≥C−1(cid:15) .
q q Tjq q Tjq q S 1
By Remark 5.1, as n→∞, we have
|D (u(p))−Dˆ (u(p))|≥C−1(cid:15) . (20)
q q S 1
AsV isdisjointfromtheorbitofq,wehaveh =hˆ foralli≥0. Asq ∈/ R ,byTheorem1,u(q)=uˆ(q).
Tiq Tiq P
AsV is,also,disjointfromνiq foralli≥0,wealsohaveg =gˆ foralli≥0. So,gn (u(νnq))=u(q)
νiq νiq νnq
and gˆn (uˆ(νnq)) = gn (uˆ(νnq)) = uˆ(q). Thus, u(νnq) = uˆ(νnq) for all n ∈ N. In particular, as u,uˆ are
νnq νnq
continuous,
uˆ(p)=u(p). (21)
As both gn (u(ωnq))=u(q) and gˆn (uˆ(ωnq))=uˆ(q), we obtain
ωnq ωnq
0=u(q)−uˆ(q)=gn (u(ωnq))−gˆn (uˆ(ωnq))
ωnq ωnq
=gn (u(ωnq))−gˆn (u(ωnq))+gˆn (u(ωnq))−gˆn (uˆ(ωnq)). (22)
ωnq ωnq ωnq ωnq
Asuanduˆarecontinuous,byassumption,andD isLipschitzcontinuous,wehavegn (u(ωnq))→D (u(p))
q ωnq q
and similarly for Dˆ (u(p)). By (20), for all n>N with N sufficiently large, there exists δ >0 such that
q 0
|gn (u(ωnq))−gˆn (u(ωnq))|>δ . (23)
ωnq ωnq 0
Substituting (23) into (22),
|gˆn (u(ωnq))−gˆn (uˆ(ωnq))|>δ ,
ωnq ωnq 0
for all n>N. Therefore, Dˆ (u(p))(cid:54)=Dˆ (uˆ(p)), hence u(p)(cid:54)=uˆ(p), contradicting (21). As the perturbation
q q
g (cid:55)→gˆis arbitrarily small in the uniform norm, we have a C0-dense set of g ∈Cα such that for all s∈R the
P
invariant graph u is not continuous. (cid:3)
s
We now show that if U is not the graph of an invariant function u, then for any small Cα perturbation of
g (cid:55)→gˆ within Cα, the quasi-graph Uˆ is, also, not the graph of an invariant function uˆ.
P