Wandering domains for diffeomorphisms of the k-torus: a remark on a theorem by Norton and Sullivan
WWandering domains for diffeomorphisms of the k -torus:a remark on a theorem by Norton and Sullivan Andr´es Navas
Abstract:
We show that there is no C k +1 diffeomorphism of the k -torus which is semi-conjugate to a minimal translation and has a wandering domain all of whose iterates areEuclidean balls. Keywords:
Denjoy’s theorem, quasi-conformal map, distortion.
Mathematical Subject Classification:
Introduction
Answering (in the negative) a question raised by Poincar´e [16], Denjoy proved one ofhis most famous theorems [4], namely the action of every C (orientation preserving) dif-feomorphism of the circle with irrational rotation number is minimal. This result can beconsidered as one of the the starting points of the theory of differentiable dynamics, andmany generalizations have been proposed in the context of one-dimensional dynamics (see,for example, [3, 5, 6, 18, 19, 24]). The search for higher dimensional analogues of the DenjoyTheorem is a natural problem that has attracted some interest in recent years. Althoughthere is some partial evidence in the positive direction, no definitive result is known.Let us be more precise. As is well known, the C (or C Lip ) hypothesis is necessary inthe statement of the Denjoy Theorem. Indeed, Denjoy himself gave C counter-examplesfor his result. (It should be noticed that the first C counter-examples were constructedby Bohl in [2].) These examples were improved by Herman in [7] up to the class C − ε forevery ε > k ≥
2, there existsa dense family ( θ , . . . , θ k ) of Q / Z -independent numbers in R / Z such that, for any ε > k +1 − ε diffeomorphism of the k -torus that has a wandering (topological) disk andis semiconjugate to the translation by ( θ , . . . , θ k ). Unfortunately, the family of rotationvectors which appear in Mac Swiggen’s constructions is countable (they are all algebraic,and therefore Diophantine), and it is unclear whether similar examples do exist for anytranslation vector with the above properties.Mac Swiggen’s examples show that the natural differentiability where we should look for1 a r X i v : . [ m a t h . D S ] F e b n analogue of the Denjoy Theorem on the k -torus is C k +1 . This is confirmed by the factthat, by a straightforward application of the KAM Theory, if f is a small C k +1+ ε perturbationof an irrational Diophantine translation of the k -torus, then f is C k conjugate to it.In the one-dimensional case, the blowing-up procedure for the construction of counter-examples is necessarily “conformal”: one replaces points along orbits by intervals. In higherdimensions, one should replace points by continua, and the case where these continua are nottopological disks is interesting by itself: see, for instance, [1, 17]. (We should point out that,in this case, perhaps a Denjoy type theorem holds in regularity smaller than C k +1 .) In thecase of wandering topological disks, several partial results are known in dimension 2 (see forexample [13, 14, 15]). For instance, in [14], Norton and Sullivan show that it is impossiblefor a C diffeomorphism of the 2-torus to be semiconjugate and non-conjugate to a minimaltranslation, provided that the preimages of points have some “uniform” conformal geometryalong the orbits.The aim of this Note is to show how a simple modification of some of the ideas of[14] allows proving similar results for C k +1 diffeomorphisms of the k -torus. It should beemphasized that these do not follow from Norton-Sullivan’s arguments, as these stronglyrely on the Morrey-Bojarsky-Ahlfors-Bers integration theorem, which is no longer available inhigher dimensions. In particular, one of the key arguments of [14] uses Sullivan’s integrabilitytheorem, which states that every uniformly quasi-conformal group of homeomorphisms of asurface is quasiconformally conjugate to a group of conformal maps [20], and which is knownto be false in dimension greater than two [22].For the sake of concreteness, we only prove the following theorem, which is somewhatthe core of [14]. Theorem.
Let k ≥ , and let f be a diffeomorphism of the k -torus that is semiconjugate toa minimal translation without being conjugate to it. If the preimage by the semiconjugacy ofeach point is either a point or an Euclidean ball, then f cannot be of class C k +1 . Note that this result is still true in dimension one (where it follows from the classicalDenjoy Theorem), but our arguments only work in higher dimensions (see, however, [12,Exercise 3.1.4], which is somewhat related to our arguments here).We should stress that we do not know whether the C k +1 regularity hypothesis is actuallyneeded for our Theorem (assuming that the wandering domains are Euclidean balls). Indeed,in Mac Swiggen’s examples, the wandering domains have a very irregular geometry, and thereis even no uniform bound for the diameter of their lifts. See also [9] for a recent interestingresult concerning topological entropy of this kind of maps.Combining the methods of this Note with those of [8], one can show that if f i , i ∈{ , . . . , d } , are respectively C k i diffeomorphisms of the k -torus that are semiconjugate tominimal translations and whose translation vectors are independent over Q / Z , then one has k + . . . + k d ≤ k provided the f i ’s commute and the preimages of points by the (simultane-ous) conjugacy to translations are either points or Euclidean balls. (The constants k i ’s aresupposed to be positive but not necessarily integer numbers). Actually, the same statement2olds without the hypothesis that the f i commute but asking for the commutativity of theirpermutation action along an orbit of balls arising from the blowing-up procedure. Acknowledgments.
This Note circulated as a manuscript more than ten years ago. I’mindebted to J. Kiwi, V. Kleptsyn and M. Ponce for their comments at that time, to all mycolleges who asked me to make this available (despite no progress has been made since then),and to J. Bochi for his insight concerning Lemma 1.The preparation of this text was funded by the CONICYT Project 1415 “Geometr´ıa enLa Frontera” and the Fondecyt grant 11060541. I also whish to thank the Institute of PureMathematics of Teheran (Iran) for the hospitality during this task, and M. Nassiri for hisinvitation.
Proof of the Theorem
Let Conf( k ) ∼ GL( k, R ) / (SO( k, R ) × R ) denote the space of conformal structures on R k . This is a simply-connected space that carries a nonpositively curved metric which isinvariant under the GL( k, R )-action given by A · [ B ] := [ AB ]. In particular, the distancefunction dist k on it is smooth. For simplicity, we denote σ := [ Id ]. Lemma 1. If f is a C k diffeomorphism satisfying the hypothesis of the Theorem, then thereexists a constant M such that dist k ([ Df n ( x )] , σ ) ≤ M holds for all x ∈ T k and all n ≥ .Proof. Let ϕ denote the semiconjugacy of f to the corresponding translation. ThenΓ := T k \ (cid:91) interior( { ϕ − ( x ) : x ∈ T k } )is a connected, nonwhere dense, minimal invariant set for f (see [15]). Moreover, sinceall the wandering topological balls are Euclidean, [ Df ( x )] is identically equal to σ on Γ.Furthermore, as dist k is smooth, the derivatives of the function x (cid:55)→ dist k ([ Df ( x )] , σ )vanish up to order k . By a successive application of the Mean Value Theorem, this impliesthat there exists a constant C such that, if x belongs to the interior of a ball B x that collapsesto a single point under ϕ , then dist k ([ Df ( x )] , σ ) ≤ C (cid:96) ( x ) k , where (cid:96) ( x ) is the half of the length of the shortest chord of B x through x . Since (cid:96) ( x ) ≤ radius( B x ), this shows that dist k ([ Df ( x )] , σ ) ≤ M vol( B x )for a certain constant M . This yields, for each n ≥ x / ∈ Γ,dist k ([ Df n ( x )] , σ ) = dist k ( Df n ( x ) · σ , σ )3 n − (cid:88) i =0 dist k ( Df i +1 ( f n − i − ( x )) · σ , Df i ( f n − i ( x )) · σ )= n − (cid:88) i =0 dist k ( Df i ( f n − i ( x ))) · [ Df ( f n − i − ( x )] , Df i ( f n − i ( x )) · σ )= n − (cid:88) i =0 dist k ([ Df ( f n − i − ( x ))] , σ ) ≤ n − (cid:88) i =0 M vol( B f n − i − ( x ) ) ≤ M, where the last inequality holds because the balls B f j ( x ) are two-by-two disjoint. Since Γ isnonwhere dense in T k , the estimate above holds for every x ∈ T k . Remark.
The dilatation of an invertible linear map A : R k → R k is defined as dil ( A ) := max (cid:107) v (cid:107) =1 (cid:107) A ( v ) (cid:107) min (cid:107) w (cid:107) =1 (cid:107) A ( w ) (cid:107) . This induces a function on Conf( n ) that measures the degree of non-conformality of matrices.In the 2-dimensional case, this is a smooth function, as is shown by the well-known formula dil ( A ) = 1 + (cid:107) µ (cid:107) − (cid:107) µ (cid:107) = exp(dist hyp (0 , µ )) , where µ denotes the Beltrami differential and dist hyp stands for the hyperbolic distance onthe Poincar´e disk. However, in higher dimension, this function is no longer smooth (yet itis locally Lipschitz). This is why, unlike [14], we do not deal with the function dil , and wedirectly consider the function dist k . Lemma 2.
For each
M > there exists λ > and < λ (cid:48) with the following property: if g is a diffeomorphism of R k that commutes with the translations by vectors in Z k and suchthat dist k ([ Dg ( x )] , σ ) ≤ M for all x ∈ R k and g ( B ( x , α )) = B ( y , β ) for some x , y in R k and some positive numbers α, β , then g ( B ( x , λα )) ⊂ B ( y , λ (cid:48) β ) . Proof.
This follows directly from the equicontinuity of the family of restrictions to B ( x , α )of the maps g satisfying the properties above; see [23, § B = B ( x , α ), and4enote by x n (resp. α n ) the center (resp. the radius) of f n ( B ). Clearly, α n goes to zero as n goes to infinite, and for each ε > f n k ( B ) ⊂ B ( x , α + ε ) for an increasing sequence( n k ) of positive numbers. Fix such an n = n k so that α n < ( λ − α λ (cid:48) and dist( x , x n ) < α + (cid:18) λ − (cid:19) α . We can apply Lemma 2 to any covering map ˜ f of f , thus obtaining˜ f n k (cid:16) B (˜ x , λα ) (cid:17) ⊂ B (˜ x n , λ (cid:48) α n ) ⊂ B (cid:18) ˜ x , α + (cid:16) λ − (cid:17) α + λ (cid:48) α n (cid:19) ⊂ B (˜ x , λα ) . Therefore, f n ( B ( x , λα )) ⊂ B ( x , λα ), and by Brouwer’s fixed point theorem, f n hasa fixed point in B ( x , λα ) . However, this is absurd, since f is semiconjugate to a minimal(and therefore periodic-point free) torus translation. This contradiction completes the proof. References [1]
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Comptes Rendus Acad. Sci.Paris (1984), 141-144.Andr´es NavasDpto de Matem´atica y C.C., Fac. de Ciencia, Univ. de Santiago de ChileAlameda 3363, Estaci´on Central, Santiago, ChileEmail address: [email protected](1984), 141-144.Andr´es NavasDpto de Matem´atica y C.C., Fac. de Ciencia, Univ. de Santiago de ChileAlameda 3363, Estaci´on Central, Santiago, ChileEmail address: [email protected]