Adaptive trains for attracting sequences of holomorphic automorphisms
© Peters and Smit; licensee Springer. 2015
Received: 21 August 2014
Accepted: 7 December 2014
Published: 12 August 2015
Consider a holomorphic automorphism acting hyperbolically on an invariant compact set. It has been conjectured that the arising stable manifolds are all biholomorphic to Euclidean space. Such a stable manifold is always equivalent to the basin of a uniformly attracting sequence of maps. The equivalence of such a basin to Euclidean space has been shown under various additional assumptions. Recently, Majer and Abbondandolo achieved new results by non-autonomously conjugating to normal forms on larger and larger time intervals. We show here that their results can be improved by adapting these time intervals to the sequence of maps. Under the additional assumption that all maps have linear diagonal part, the adaptation is quite natural and quickly leads to significant improvements. We show how this construction can be emulated in the non-diagonal setting.
Let X be a complex manifold equipped with a Riemannian metric, and let f:X→X be an automorphism that acts hyperbolically on some invariant compact subset K⊂X. Let p∈K and write for the stable manifold of f through p. is a complex manifold, say of complex dimension m. In the special case where p is a fixed point, it is known that is biholomorphically equivalent to . It was conjectured by Bedford  that this equivalence holds for any p∈K.
Conjecture 1 (Bedford)
It was shown by Fornæss and Stensønes that a positive answer to the following conjecture implies a positive answer to Conjecture 1.
Here we present the following new results, both giving positive answers to Conjecture 2 under additional hypotheses.
In the next section, we will place these two results in a historical perspective. While Theorems 3 and 4 are significant improvements over previously known results, our main contribution lies in a new way of thinking about trains, as introduced by Abbondandolo and Majer in .
The authors would like to thank Alberto Abbondandolo, Leandro Arosio, John Erik Fornæss, Jasmin Raissy, Pietro Majer, and Liz Vivas for many stimulating discussions. The first author was supported by a SP3-People Marie Curie Actions grant in the project Complex Dynamics (FP7-PEOPLE-2009-RG, 248443).
2 A short history
Theorem 5 (Sternberg, Rosay-Rudin).
If p is not attracting but hyperbolic, one considers the restriction of f to the stable manifold. This restriction is an automorphism of with an attracting fixed point at p. Moreover, its basin of attraction is equal to the entire stable manifold, which is therefore equivalent to . This naturally raised Conjecture 1. Equivalence to of generic stable manifolds was proved by Jonsson and Varolin in .
Theorem 6 (Jonsson-Varolin).
Here, generic refers to a subset of K which has full measure with respect to any invariant probability measure on K. In fact, Jonsson and Varolin showed that Conjecture 1 holds for the so-called Oseledec points. The results of Jonsson and Varolin were extended by Berteloot et al. in . More recently, the following was shown in :
Theorem 7 (Abate-Abbondandolo-Majer).
where |a 0|<1 and |a n+1|≤|a n |2.
Theorem 8 (Fornæss).
on any uniform neighborhood of the origin. In Theorem 8, the rate of contraction is not uniformly bounded from below.
Let (f n ) be a sequence of automorphisms which satisfies the conditions in Conjecture 2, and suppose that D k <C for some . Suppose further that the maps f n all have order of contact k. Then, the basin of attraction is biholomorphic to .
Here, a map f has order of contact k if f(z)=l(z)+O(∥z∥ k ), where l(z) is linear.
The following was recently shown in :
Theorem 10 (Abbondandolo-Majer).
It was noted in  that for maps with order of contact k, one should be able to obtain the same result if D k+ε <C, where ε=ε(m,k).
While this result only seems marginally stronger (indeed, ε-stronger), it is in fact a much deeper result. If D k <C one can ignore all but the linear terms, see Lemma 16 in the next section. But if D k ≥C, one has to deal with the terms of order k, which is a major difficulty. In fact, looking at more classical results in local complex dynamics, the major difficulty in describing the behavior near a fixed point usually lies in controlling the lowest order terms which are not trivial. Theorems 10 and 3 may therefore be important steps towards a complete understanding of Conjecture 2.
Now that techniques have been found to deal with these terms of degree k, it is natural to ask whether the condition D k+ε <C can be pushed to the condition D k+1<C, since as long as D k+1<C is satisfied one can ignore terms of degree strictly greater than k. In Theorem 3, we show that we can indeed weaken the requirement to D k+1<C, at least in two complex dimensions and under the additional assumption that the linear parts of all the maps f n are diagonal.
Whether diagonality is a serious extra assumption or merely simplifies the computations remains to be seen, but we will see that some of the techniques we introduce to prove Theorem 3 can be applied without the diagonality assumption as well, leading to Theorem 4.
Since the computations in the general case are much more intensive than in the diagonal case, we have chosen not to prove Theorem 4 for maps with higher order of contact. We expect that the interested reader will be able to generalize Theorem 4 to maps with higher order of contact.
We conclude our historical overview with the following two results. The first was proved in .
Theorem 11 (Peters-Wold).
Let (f j ) be a sequence of automorphism of , each with an attracting fixed point at the origin. Then there exists a sequence of integers (n j ) such that the attracting basin of the sequence is equivalent to .
The next result, from , will be important to us not so much for the statement itself but for the ideas used in the proof. See also the more technical Lemma 17, which will be discussed in the next section.
Theorem 12 (Peters).
The combination of Theorems 11 and 12 provides a good idea for the techniques used to prove Theorems 3 and 4: If, after applying suitable changes of coordinates, we can find sufficiently long stretches in the sequence (f n ) that behave similarly, then we can show that the basin will be biholomorphic to . What is meant by behave similarly will be made more precise in later sections.
The theorems described above have been used to prove a number of results which are at first sight unrelated. The first example of such an application is of course the classical result of Fatou and Bieberbach which states that there exists a proper subdomain of which is biholomorphic to . Indeed, it is not hard to find an automorphism of with an attracting fixed point but whose basin of attraction is not equal to the entire . Proper subdomains of that are equivalent to are now called Fatou-Bieberbach domains.
It should be no surprise that for the construction of Fatou-Bieberbach domains with specific properties, it is useful to work with non-autonomous basins. Working with sequences of maps gives much more freedom than working with a single automorphism. Using Theorem 9, it is fairly easy to construct a sequence of automorphisms satisfying various global properties, while making sure that the attracting basin is equivalent to . We give two examples of results that have been obtained in this way.
Theorem 13 (Wold).
The question of whether such Fatou-Bieberbach domains exist was raised by Rosay and Rudin in .
Theorem 14 (Peters-Wold).
Here, we note that Fatou-Bieberbach domains with Hausdorff dimension equal to any h∈(3,4) were constructed by Wolf  using autonomous attracting basins. Hausdorff dimension 3 (and in fact C ∞ boundary) was obtained earlier by Stensønes in , who also used an iterative procedure involving a sequence of automorphisms of . See  for another application of non-autonomous basins to the theory of Fatou Bieberbach domains.
The last application we would like to mention is the Loewner partial differential equation. The link with non-autonomous attracting basins was made in , where Arosio used a construction due to Fornæss and Stensønes (see Theorem 20 below) to prove the existence of solutions to the Loewner PDE. See also  for the relationship between the Loewner PDE and non-autonomous attracting basins.
3.1 The autonomous case
Essentially, the only available method for proving that a domain Ω is equivalent to is by explicitly constructing a biholomorphic map from Ω to . Let us first review how this is done for autonomous basins, before we look at how this proof can be adapted to the non-autonomous setting. We follow the proof in the appendix of , but see also the survey  written by Berteloot.
Important here is that the lower triangular polynomial maps behave very similarly to linear maps. For example, it follows immediately by induction on m that the degrees of the iterates G n are uniformly bounded. One can also easily see that the basin of attraction of a lower triangular map with an attracting fixed point at the origin is always equal to the whole set .
Now let D>0 be such that ∥F(z)∥<D∥z∥ for . Then there exists a such that D k+1·β<1. It follows from Equation 3 that, with this choice of k, the maps Φ n converge, uniformly on compact subsets of Ω F , to a biholomorphic map from Ω F to .
which is summable over n. Hence, the sequence (Φ n ) forms a Cauchy sequence and converges to a limit map Φ. Since D Φ n (0)=Id for all n, it follows that Φ is biholomorphic onto its image. To prove the surjectivity of Φ, one shows that for each there exists a compact K⊂Ω F such that for all . The fact that follows since we are dealing with open maps.
3.2 Non-autonomous conjugation
In the non-autonomous setting, it is very rare that a single change of coordinates simplifies the sequence of maps. Instead, we use a sequence of coordinate changes.
If the maps (g n ) are all lower triangular polynomials then one still has that the basin of the sequence (g n ) is equal to . This simple fact was used in  to prove the following lemma, which for simplicity we state in the case m=2.
with |b n |2<ξ|a n | for some uniform constant ξ<1. Then, we can find bounded sequences (g n ) and (h n )as in Lemma 16. Moreover, the maps g n can be chosen to be lower triangular polynomials, and hence .
with . Then by defining , we obtain a new sequence (g n ) whose basin is equivalent (by the biholomorphic map U 0) to the basin of the sequence (f n ). Notice that the linear parts of all the maps (g n ) are lower triangular. In this construction, we are free to choose the initial tangent vector v 0.
where the U j are unitary matrices and the maps g n are all lower triangular. In the spirit of Theorem 11, one would expect that the basin of this new sequence is equal to as long as the sequence (p j ) is sparse enough. Indeed this is the case, as follows from the following lemma, proved by Abbondandolo and Majer in .
Then, the basin of the sequence in Equation (5) is equal to
The proof of Theorem 10 from  can now be sketched as follows. Define p j+1=k j +p j , and on each interval find a tangent vector v j which is contracted most rapidly by the maps . Next, find the non-autonomous change of coordinates by unitary matrices such that the maps g n as defined above all have lower triangular linear part. Then on each interval I j =[p j ,p j+1], the maps g n satisfy the conditions of Lemma 17 'on average.’ This is enough to find another non-autonomous change of coordinates after which the maps g n are lower triangular polynomial maps on each interval I j . Then it follows from Lemma 18 that the basin of the sequence (f n ) is equivalent to .
Now, we arrive at one of the main points presented in this article. In the argument of Abbondandolo and Majer, the intervals [p j ,p j+1] are chosen without taking the maps (f n ) into consideration; it is sufficient to make a simple choice such that Eq. 6 is satisfied. In the last section of this article, we will show that we can obtain stronger results if we instead let the intervals [p j ,p j+1] depend on the maps (f n ) or, to be more precise, on the linear parts of the maps (f n ).
3.3 Abstract basins
We refer to as the abstract basin of attraction, sometimes also called the tail space. We have now used the notation for both abstract and non-autonomous basins, but thanks to the following lemma this will not cause any problems.
Theorem 20 (Fornæss-Stensønes).
Working with abstract basins can be very convenient. For example, Lemma 16 also holds for abstract basins which, in conjunction with Lemma 19, implies that in Diagram 4 we do not need to worry about whether the maps h n and g n are globally defined automorphisms. From the fact that the sequences (g n ) and (h n ) are uniformly bounded, it follows that their restrictions to some uniform neighborhood of the origin are biholomorphisms, which is all that is needed.
4 Definition of the trains
Let us go back to the proof of 10 by Abbondandolo and Majer. Instead of conjugating to lower triangular maps on the entire sequence (f n ), they introduced a partition of into intervals of rapidly increasing size and on each interval changed coordinates to either lower or upper triangular maps. These intervals were referred to as trains, and we will build on this terminology.
Each train [p j ,p j+1) will be headed by an interval [p j ,q j ) which we will call the engine. On the engine, we will have very good estimates of the linear maps. Our estimates are not nearly as strong on the interval [q j ,p j+1), but the good estimates on the engine will be used as a buffer to deal with estimates on the rest of the train. In fact, as soon as the buffer from the engine fails to be sufficient, we start a new train.
We proceed to define the trains explicitly, both in the diagonal and the general case.
4.1 The diagonal case
is maximal. We will write I j =[p j ,p j+1) and refer to this interval as the j th train. The interval [p j ,q j ) we will call the engine of the train I j .
which will later allow us to apply Lemma 18.
All three properties follow immediately from the definitions.
4.2 The general case
We say that g is lower triangular with respect to v if U g U -1 is a lower triangular polynomial. Note that this definition is independent of the choice of U.
Looking back, we see that in the diagonal case, our maps g n are all lower triangular with respect to either [0,1] or [1,0]. In the general case, we allow the vector v to be any unit vector in , but v must remain constant on each train. This motivates the following definition.
For k>x>1, recursively define an increasing sequence of integers p 1,q 1,p 2,q 2,… and a sequence of unit-vectors v 1,v 2,… as follows:
Our strategy to prove inequalities like those in Proposition 22 on an engine [p j ,q j ) is the following. We will work not only with the diagonal entries a n and b n in the current coordinates but first with the entries α n and β n , which are the diagonal entries in the coordinates that were used for the previous train. By the recursive definition of the train, we first obtain estimates for α n and β n , and then translate these to estimates on a n and b n in the new coordinates. In the translation from old to new coordinates, the off-diagonal entries, denoted by c n and γ n , will play an important role.
which completes (i).
by minimality of q j+1. The second half of statement (ii) follows from our choice of p j+1.
Since is another eigenvector of , we now introduce notation that emphasizes the role of this vector. Pick a unitary matrix S j which maps to [0,1]. Then, pick unitary matrices T i for i=p j +1,…q j such that [0,1] is an eigenvector of . Note that we can take .
for all i.
This new notation allows us to derive additional inequalities for the engines of the trains:
since m<q j , which proves the right-hand side of inequality (iii).
which implies the left-hand side of inequality (iii).
We can now give an upper estimate for |γ n,m | on the engine as well:
where we used that |γ i |<D and |β i |>C.
We can now translate Lemma 25 to the following generalization of Proposition 22.
When we continue the proof of the general case, we will work with the sequence the sequence instead of (f n ). The basins of these two sequences must be equivalent since at any time the composed functions differ only by multiplication with one last unitary matrix.
5 Completion of the proof in the diagonal case
Let us recall the statement of Theorem 3
Let (f n ) be a sequence of automorphisms of which satisfies the conditions in Conjecture 2, and suppose that the maps f n all have order of contact k. Assume further that the linear part of each map f n is diagonal. Then if D k+1<C, the basin of attraction is biholomorphic to .
The proof will be completed in two steps. In the first step, we will find a non-autonomous conjugation on each train by linear maps such that in each train the new maps all contract the same coordinate most rapidly. This means that we can apply Lemma 30 below on each train separately. In the second step, we worry about what happens at times p j where we switch from one train to the other. After these two steps, we construct the maps (g n ) and (h n ) on each of the trains using the following lemma, a special case of Lemma 17.
Suppose that each map of the sequence (f n ) has linear part of the form (z,w)↦(a n z,b n w), with |b n |≤|a n |. Then for any k≥2, we can find bounded sequences (g n ) and (h n ), with the maps g n lower triangular, such that diagram (4) commutes up to jets of order k.
Step 1: Directing the trains.
Follows easily by induction on n.
for all q≤m≤n for which σ n,m ≤0. The statement in the lemma now follows from (11).
Finally, we need to check that θ r and τ r are large enough to satisfy the starting hypothesis for the next interval I j+1.
Our conclusion is the following:
for all n∈I j .
The condition on the coefficients of the linear parts of the maps follows from the discussion earlier in this section. The fact that the sequence is bounded follows from the facts that the linear parts stay bounded, and that the higher order terms grow by at most a uniform constant.
To see that the two basins of attraction are biholomorphically equivalent (with biholomorphism l 0), note that . Since the entries of the diagonal linear maps l n+1 are always strictly greater than 1, it follows that the basin of the sequence is contained in the l 0 image of the basin of the sequence (f n ). The other direction follows from the fact that the coefficients of the maps l n grow at a strictly lower rate than the rate at which orbits are contracted to the origin by the sequence (f n ).
Step 2: Connecting the trains.
The following is a direct consequence of Lemma 17:
Let (f n ) be a bounded sequence of automorphisms of whose linear parts are of the form (z,w)↦(a n z,b n w) and whose order of contact is k. Suppose that |a n |≥|b n | for all n=0,1,…. Then there exists bounded sequences (h n ) and (g n ) such that g n ∘h n =h n+1∘f n +O(k+1) for all n. Here, the maps g n can be chosen to be lower triangular maps, and the maps h n can be chosen to be of the form (z,w)↦Id+h.o.t..
Note that both coefficients of these affine maps are uniformly bounded from above in norm, and that . It follows that there exist a unique bounded orbit for this sequence of affine maps. In other words, we can choose a sequence (h n ) whose k-th degree terms are uniformly bounded. It follows directly that the maps g n are also uniformly bounded, which completes the proof.
Moreover, we note that we have much more flexibility if we have a sequence of intervals (I j ) (with j odd) and we want to find sequences (h n ) and (g n ) on each I j that are uniformly bounded over all j. In fact for each interval I j , we can start with any value for and inductively define the maps h n backwards. We merely need to choose uniformly bounded starting values .
We note that while the map does depend on the coefficient , the coefficient (and equivalently also the map ) only depends on the coefficients of and not on those of . Hence we can find uniformly bounded maps h n on each of the intervals, and therefore also uniformly bounded maps g n , which are lower triangular for n in each interval [p j ,r j ) with j odd, and upper triangular when j is even. This completes the proof of Theorem 3.
6 Proof in the general case
In what follows, we drop our assumption of diagonality of the linear parts of the maps f n . Our approach will be different this time. We will not able to use linear maps l n as we did in the diagonal setting. The problem is that trying to keep bounded off-diagonal terms in the linear parts of the maps places strong restrictions on the maps l n , even if we use lower-triangular matrices instead of diagonal matrices.
Our plan for proving Theorem 4 is the following: We use the trains defined in Definition 23, where we pick k such that and D k <C, and set . Then, we will analyze the conditions on the coefficients of the maps (g n ) and (h n ) that arise from the commutative diagram. We prove that it is possible to chose a specific sequence of maps for which the coefficients remain bounded at the start and end of each train. In the rest of the paper, we assume that we have made such a choice of maps.
In the second step, we find an estimate for the growth of the coefficients of (g n ) and (h n ) in the middle of trains and show that the basin of the sequence (g n ) is equal to . In the third step, we use the usual construction of the maps Φ n . Most work goes into proving that these maps converge on compact subsets to a holomorphic map Φ from the basin of the sequence (f n ) to the basin of the sequence . We end by showing that Φ is the required biholomorphism.
The choice of x we made seems rather random, but it turns out that the exact value of x does not influence our result. The bottleneck of our estimates on the quadratic coefficients of (g n ) and (h n ) depends on estimates on the wagons and the length of the trains. When x increases, the estimates on the wagons (Lemma 24) get weaker while the trains get longer. These effects cancel out.
Step 1: Exploring new coordinates
The values of up to can be studied using the relations above. We know that |a n | and |b n | are at most D<1. The value of is small on average, by the choice of our trains. Therefore, when we start with and work backwards, the quadratic constants up to should not get too big. This is made more precise in the lemma below.
where Y=Y(C,D,ε)is a constant independent of both j and δ.
We obtain the following similar estimate on the coefficients of g.
which plays an important role in the above estimates. This allows us to estimate the quadratic terms at the beginning of train j.
where K 3=K 3(C,D,k,x) is a constant independent of j.
Case I: t≥q j .
Case II: t≤q j .
Suppose we are now given and set . Recall that the M j are unitary matrices arising from the definition of the trains. A consequence of the fact that each map has at most six quadratic terms is the following.
The lemma follows.
The following is now an immediate consequence of Lemmas 36, 38, and 39.
Therefore, there exist an inverse orbit of the sequence ψ 0,ψ 1,…. That is, a sequence h 0,h 1,… in S with ψ i (h i+1)=h i . Thus, we have proved the following.
We can find sequences (h n ) and (g n ) whose coefficients are bounded by X at the start and end of every train j, where j≥j 0.
Step 2: Properties of these new coordinates.
What remains to show is that with the construction from the previous step gives a basin which is on the one hand equal to all of and on the other hand equivalent to the basin . In order to draw these conclusions, we will need finer estimates on the size of the coefficients of the maps h n and g n .
where K 3=K 3(C,D,k,x) as in Lemma 38.
q j ≤s≤t≤p j+1
p j ≤s≤q j ≤t≤p j+1
p j ≤s≤t≤q j
Step 3: The biholomorphism.
The proof follows immediately from our estimates on the coefficients of the maps h n .
Recall that the linear and quadratic parts of the map are exactly equal to . Therefore, we need to find estimates on the higher order terms of . Since the matrices M j are unitary, we can work with the f’s, g’s, and h’s instead.
Plugging in our bound on ∥z∥, we obtain the required estimate.
to give ourselves some wiggle room.
We prove both statements simultaneously by induction on n. For n=0 the first statement is trivial, and the second statement follows immediately from Lemma 46.
The sequence (Φ n ) converges uniformly on each V u .
For m,n≥N and z∈V u , we therefore have ∥Φ n (z)-Φ m (z)∥<η, which proves the uniform convergence on V u .
Since , Theorem 49 shows that the maps Φ n converge uniformly on compact subsets to a map . We will now prove that this limit Φ maps biholomorphically onto . The maps Φ n are compositions of a number of global biholomorphisms, plus a holomorphic map which is injective on a small ball whose radius decreases with n. We first estimate the size of these radii
The map Φ is a biholomorphism.
Let and n≥v(u). Then for z∈V u , we have . By Lemma 50(ii) and the fact that all and are biholomorphisms, the map Φ n must be injective on V u . Therefore, is a uniform limit of biholomorphisms, and we know that D Φ n (0)=I for all n. By Hurwitz’ theorem, we can conclude that is also a biholomorphism. The statement follows since is the increasing union of the sets V u .
With this lemma, we have proved Theorem 4.
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