Infinite cascade case

Here we deal with the case where we have some I0 such that Fi are central for

i= 0,1, . . .. In this case we will find that |Ii+1|

|Ii| gets very close to 1. See Figure 1.5 to see what a typical such map will look. In particular, Ii will not shrink down to a point (the critical point c) as i increases so we can’t use the method above to bound sums of intervals which get very close toc.

Whenf is only finitely renormalisable, we don’t encounter this phenomenon. The principal tool is an extension given to us by a result of [K2].

We start by letting I0 be any nice interval about c. We assume that we have

some infinite cascade. This means thatFi is central for alli, where Fi is defined in the usual way. The main idea here is that we can still find good bounds on some intervalI0,0 and then apply the methods of Section 1.5 to it. Then we need

to find another interval I1,0 aroundc which is smaller than all I0,i, also has good bounds and is uniformly smaller than I0,0. In such a way, we obtain a sequence

of intervalsIi,0 which can each be treated as in the cascade case above and which

I

I

I

I

Figure 1.5: An infinite cascade.

central for all i, j ≥ 0. Otherwise we can simply choose I0 such that we never

have an infinite cascade.

For all i the central branch of Fi has two fixed points, q0 and p0 to the left and right of crespectively (as usual, we assume that Fi(c) is a maximum for Fi|Ii+1).

We letq0

0 be the point in Ii+1 not equal toq0 which maps byFi to q0. We define p0

0 similarly. We define I0,0 to be (p00, p0). Let F0,0 :

S

jI j

0,0 → I0,0 be the first

return map toI0,0(whereI00,0 =I0,1is the central domain). We have the following

lemma.

Lemma 1.7.3. There exists some χ >ˆ 0 depending only on f such that I0,0 is a

ˆ

χ-scaled neighbourhood of every domain I₀j_,0.

Proof: Clearly, Ii tends to (q0, q00). So we denote (q0, q00) by I∞. We will first

show that I∞ is uniformly larger than I0,0 and then show that all non-central

domains of the first entry map to I0,0 have an extension to I∞ and show what

this means for I₀j_,0.

In a similar manner to the exceptional case, we will find an upper bound for |DFi|Ii+1. This will allow us to get good bounds for the first return map to I0,0

For large i, the ratio Ii has |I_|i_I+1_i|| close to 1. The following lemma, an adaptation of Lemma 7.2 of [K2], allows us to bound |DFi|Ii+1.

Lemma 1.7.4. If f ∈ NF2 _{then there exist constants 0 < τ}

2 < 1 and τ3 > 0

with the following property. If T is any sufficiently small nice interval around the critical point, RT is the first entry map to T and its central domain J is

sufficiently big, i.e. _||J_T|_| > τ2, then there is an interval W which is a τ3-scaled

neighbourhood of the interval T such that if c ∈ RT(J) then the range of any

branch of RT :V →T can be extended to W provided that V is not J.

This lemma is only given as a C3 _{result in [K2], but it easily extends to ourC}2

case.

In the infinite cascade case we always have this c∈RT(J) condition.

It is straightforward to see that the above lemma is sufficient to prove a version of Lemma 1.6.8 in our case. That is, for large i, there exists some ˆC0 _{such that}

|DFi|Ii+1 < Cˆ0. This implies that there exists some 0 < θ < 1 depending only

on f such that |I0,0| < θ|I∞| and, equivalently, some ˜χ > 0 such that I∞ is a

˜

χ-scaled neighbourhood ofI0,0.

Now, for the moment we let F0,0 also denote the first entry map and

S

jI j

0,0 also

denote the first entry domains. Suppose that there exists a domain I₀j_,0 disjoint fromI0,0 which does not have an extension to I∞. That is, supposing F0,0|_Ij

0,0 =

fn(j)_|

I₀j_,0 there is no interval V such that fn(j) :V →I∞ is a diffeomorphism. Let 0≤k < n(j)−1 be maximal such that fn(j)−k _:fk_(Ij

0,0)→I0,0 has no extension

to I∞ (clearly if I0,0 is small f : fn(j)−1(I0j,0) → I0,0 always has an extension

so k < n(j) −1). Then there exists some interval V ⊃ fk+1_(Ij

0,0) such that fn(j)−k−1 _{: V → I}

∞ is a diffeomorphism and the element of f−1(V) containing

fk_(Ij

0,0) containsc.

Since I∞ is a nice interval, V ⊂ I∞. We also know that fk(I0j,0) ⊂ I∞ \I0,0.

ThereforeV contains eitherp0orp0

0. But then eitherfn(j)−k−1(p0) orfn(j)−k−1(p00)

is contained inI∞\I0,0 which is not possible.

Now considerf(I₀j_,0) for somej = 0 where6 I₀j_,0 ⊂I0,0is a domain of the first return

map. There exists some V ⊃f(I₀j_,0), where fn(j) _{:V → I}

∞ is a diffeomorphism

and V is a ˜χ-scaled neighbourhood of f(I₀j_,0). Let V(f(c)) denote the maximal interval aroundcwhich pulls back byf−1 _toI

0,0. IfV is not contained inV(f(c))

then eitherp0orp00 is contained inV0 the respective pullback byf−1 ofV (the one

which containsI₀j_,0). Thus, fn(j)_(p

So V0 _{⊂ I}

0,0 and I0,0 is a ˆχ-scaled neighbourhood of I0j,0 where ˆχ = min

¡ ˜ χ0_,1 2 ¢ . The case of the central branch follows in the usual manner.

2 So we are in a type of well bounded case for F0,0. Furthermore, we may assume

that F0,0 has an infinite cascade too. We sum for F0,0, F1,0, . . . as in the cascade

case. We let q1, q01, p1, p01 be defined as above for the fixed points of F0,0|I0,1. We

letI0,∞ denote (q1, q10). We may apply the same ideas as above to find some new

interval I1,0 := (p1, p01) which has |I1,0| < θ|I0,∞|. We may define Ii,j for i ≥ 2, and 0≤j ≤ ∞ in a similar way.

Let fNi(T) be the last iterate of T which lies inside I

0,i. Let Ni0 > Ni+1 be the

last time thatfN0

i(T) containsp_i (if no such integer exists, set N0

i =Ni+1). Then

we can prove the following proposition.

Proposition 1.7.5. For all ξ >0 there exists some Cinf >0 such that NXi−Ni0

k=1

|fk+N0

i(T)|1+ξ< C infσˆi

where σˆi is defined as follows. Let σi := supV∈domFi,0

P_n(V)

j=1 |fj(V)| (and n(V) is

defined as k where Fi,0|V =fk). LetVˆ ⊂Ii,0\Ii,1 be an interval such thatfnˆ( ˆV)

is one of the connected components of Ii,0 \Ii,1 and fj( ˆV) is disjoint from both Ii,0 \Ii,1 and Ii+1,0 for 0< j <nˆ( ˆV). Then σˆi is the supremum of all such sums

P_{ˆn( ˆV)}

j=1 |fj( ˆV)| and σi.

It is clear that we can prove the proposition when we don’t enter Ii,∞. When

we enterIi,∞\Ii+1,0 we simply use the Minimum Principle as usual to show that

|DFi|Ii,∞\Ii+1,0 is uniformly greater then 1. Then this gives us decay of the size of

these pullbacks.

In order to bound the whole sum, we must boundPNi0−Ni+1

k=1 |fk+Ni+1(T)|too. To

do this we split fN0

i(T) into the intervals fNi0(T−

i ) and fN

0

i(T+

i ) to the left and right of pi respectively. If we denote Mi ≥ 0 to be the last time that an iterate of T−

i contains pi+1. Then using Proposition 1.7.5, we have

N0 i−Mi X k=1 |fk+Mi_{(T)|< C} infσˆi+1.

Such logic then leads us to conclude that Ni X k=0 |fk+Ni+1_{(T)|< C} inf ∞ X k=i (k−i)ˆσk.

While this sum is not obviously bounded, this is not the sum we need to bound whenf ∈NF2+ξ_{. As in the cascade case, we bound}PNi−Ni0

k=1 |fk+N

0

i(T)|1+ξ. This is bounded by a sum of the form Cinf|I0,0|sup0≤iσi

P_∞

i=0iθξi. Clearly this is

bounded.