This is a preprint of: “Regularization of Brouwer Translation Theorem”, Armengol Gasull, Jorge Groisman, Francesc Mañosas, Qual. Theory Dyn. Syst., vol. 21, 135, 2022. DOI: [10.1007/s12346-022-00666-y] REGULARIZATION OF BROUWER TRANSLATION THEOREM. ARMENGOL GASULL†,‡ , JORGE GROISMAN[ AND FRANCESC MAÑOSAS† Abstract. The celebrated Brouwer translation theorem asserts that for a preserving orientation fixed point free homeomorphism of the plane, each point belongs to an invariant region where the dynamics is continuously conjugate to a translation. In this work we prove that if we start with a C m , m ∈ N ∪ {∞}, diffeomorphism then the referred conjugacy has the same kind of regularity. 2000 MSC: Primary: 37C15. Secondary: 54H20. Keywords: Brouwer translation theorem, smooth conjugation, homeo- morphism, diffeomorphism Introduction and Main result Let F be an orientation-preserving homeomorphism of R2 which is fixed point free. The Brouwer’s plane translation theorem asserts that every z ∈ R2 is contained in a domain of translation U for F, that is an open connected subset of R2 , whose boundary is L∪F (L) where L is the image of a proper embedding of R in R2 , such that L separates S F (L) −1 n and F (L) (L is called a Brouwer line). Moreover, if E = n∈Z F (U), where U = U ∪ L ∪ F (L), there exists a homeomorphism Φ : E → R2 such that (Φ ◦ F ◦ Φ−1 )(z) = z + (1, 0). This result was proved in 1912 by Brouwer ([1]) although many other researchers, like for instance Scherrer, Kerékjártó, Terasaka or Sperner, during the 20’s and the 30’s gave their own proofs trying to fix and clarify his approach. For more details, it is interesting to read Section 2 of Guillou’s paper [5] dedicated to some historical remarks about this Brouwer’s result. Nowadays the interested reader can also take a look for instance to any of the proofs given in the more recent works of Franks ([4]), Guillou ([5]), or Le Calvez and Sauzet ([7]). From now on if L is a Brower line for a planar S preserving orientation fixed point free map we denote by EL = n∈Z F n (U) the F -invariant space associated to L. 1 2 A. GASULL, J. GROISMAN AND F. MAÑOSAS In a few words, in this note we will prove that if F is a C m diffeo- morphism, m ∈ N ∪ {∞}, then the above conjugacy Φ can be choosed of class C m . Our result is: Theorem A. Let F : R2 → R2 be a preserving orientation fixed point free C m diffeomorphism, m ∈ N ∪ {∞}. Let L be a Brouwer line. Then, there exists a C m diffeomorphism Ψ : EL → R2 such that (Ψ ◦ F ◦ Ψ−1 )(z) = z + (1, 0). Note that there is no loss of derivatives in this result, that is we find a map Ψ which is as smooth as the original map F . Throughout the paper m takes any value in N ∪ {∞} and all diffeomorphisms are of class C m , m ∈ N ∪ {∞}. The study of the regularity of conjugacies between some dynamical systems is a classical subject of interest and goes back to Poincaré. For instance, nowadays this question is well understood for smooth diffeo- morphisms of the circle, see [3] and its references. It is interesting to observe that for some dynamical systems results an agreement between the regularity of the dynamical system and the one of the conjugation happens while for others do not. Like in Theorem A, another example of the first situation we have the Kerékjártó’s theorem of local lin- earization of periodic maps in a neighborhood of a fixed point ([2]). A well-known result of the second situation is given by the Grobman- Hartman theorem which states that a C m diffeomorphism, with m a positive integer, can be locally C 0 linearized near a hyperbolic fixed point but in general it can not be C m linearized for m > 0 ([8]). To show that if F is of class C m then it is possible to C m conjugate F |E , with a translation we proceed in several steps. A first simple reduction comes by using the celebrated Riemann mapping theorem. Recall that it affirms that any non-empty simply connected open sub- set W of the complex number plane C, different of the whole plane, can be biholomorphycally mapped to the open unit disk. Then, in case that W is not the whole plane, we can finally transform C ∞ diffeomor- phically the unit disc into R2 . In short it is not restrictive to assume that F has a Brouwer line L such that EL = R2 . Therefore we will prove Theorem A only in the case that EL = R2 . The general case follows from the previous observation. Next, we will prove that the C m diffeomorphism F, on R2 , is succes- sively C m conjugated, via some C m conjugations Ψ1 , Ψ2 , Ψ3 , with some C m diffeomorphisms of R2 , say G1 , G2 and G3 , respectively, obtaining finally [ G3 (0, y) = (1, y) and Gn3 (U) = R2 , n∈Z REGULARIZATION OF BROUWER TRANSLATION THEOREM 3 where here U = {(x, y) : 0 ≤ x ≤ 1}. More concretely, we prove: • The diffeomorphism Ψ1 can be taken such that Ψ1 (L) is the straight line {0} × R and it is a Brouwer line for G1 . • The diffeomorphism Ψ2 can be constructed such that the straight line {0} × R is a Brouwer line and G2 (0, y) = (1, p(y)) for some increasing diffeomorphism p of R. • The diffeomorphism Ψ3 can be taken such that the straight line {0} × R is a Brouwer line and G3 (0, y) = (1, y). The next three sections of the paper are devoted to prove each of the steps of the proof we described above. The final section 4 contains the proof of the main theorem. Next, we say some words about how we will prove it. By the above reductions, we can start with a diffeomorphism G3 = FSthat has L = {0}×R as a Brouwer line, satisfying F (0, y) = (1, y) and n∈Z F n (U) = R2 , where U = {(x, y) : 0 ≤ x ≤ 1}. We want to prove the existence of a C m diffeomorphism Ψ, defined on the whole plane, such that (1) Ψ(F (x)) = T (Ψ(x)), where T (x, y) = (x + 1, y). From this equality it is apparent that once Ψ is defined on U, then Ψ restricted to F (U) is determined by Ψ on U because of condition (1). Hence, the main point is to be able to construct a suitable seed on U such that its extension to U ∪ F (U) is of class C m on the common boundary {1} × R. As we will see, one of the main tools in our con- struction, and also in other parts of this paper, will be the sometimes called Smoothing theorem, due to Hirsch ([6]). We will recall it in Theorem 2.2. In a few words, it allows to modify piecewise smooth homeomorphism in order to produce diffeomorphisms. The question that we tackle in Theorem A, but for analytical diffeo- morhisms, seems also natural and interesting. Unfortunately, it is not covered at all by our approach. 1. First step: construction of G1 . To prove this step first we show that it is possible to choose a C m Brouwer line, and then, by conjugacy, we will transform it into the straight line {0} × R. Next lemma shows how to do this first choice. One of the key points to prove next result is that a proper embedding of R in R2 gives rise to a Jordan curve when the plane is compactified to S2 , by adding one point. Lemma 1.1. Let F : R2 → R2 be a preserving orientation fixed point free C m diffeomorphism. Let L be a Brouwer line. Then there exists a C ∞ Brouwer line S ⊂ EL satisfying that EL = ES . 4 A. GASULL, J. GROISMAN AND F. MAÑOSAS Proof. As before we denote by U the connected component of R2 \ ({L} ∪ {F (L)}) whose boundary is L ∪ F (L). Our first proposal is to construct a C ∞ curve that is a proper embedding of R in R2 that separates L and F (L). To do this, let σ be a simple arc that joints a point in L with a point in F (L) through U. That is, σ is the image of a continuous map and injective map ϕ : [0, 1] −→ R2 such that ϕ(0) ∈ L, ϕ(1) ∈ F (L) and ϕ(t) ∈ U for all t ∈ (0, 1). Clearly σ separates U in two unbounded, open and simply connected components both having only one boundary component. We denote by U + and U − these open components. Now we choose z ∈ σ not in L∪F (L) and we will construct a C ∞ sim- ple curve ϕ1 : [0, ∞) :−→ U + satisfying ϕ1 (0) = z and limt→∞ kϕ1 (t)k = ∞. To do this we consider Ψ : U + −→ A a holomorphic homeomor- phism, given by the Riemann mapping theorem, between U + and the unit ball A and we choose a sequence xn ∈ U + satisfying that the se- quences kxn k and kΨ(xn )k are strictly increasing and lim kxn k = ∞. Note that in this case we get lim kΨ(xn )k = 1. Now we consider a C ∞ injective map α : [1, ∞) −→ A such that α(i) = Ψ(xi ) for all i ∈ N and kα(t)k ∈ [kΨ(xi )k, kΨ(xi+1 )k] if t ∈ (i, i + 1). Then Ψ−1 ◦ α gives a C ∞ arc in U + beginning at x1 such that limt→∞ k(Ψ−1 ◦ α)(t)k = ∞. Now joining this arc with a C ∞ arc in U + that begins at z and ends at x1 we obtain the desired C ∞ simple curve. Using a symmetric construction we can obtain a C ∞ simple curve ϕ2 : (−∞, 0) :−→ U − satisfying ϕ2 (0) = z and limt→−∞ kϕ2 (t)k = ∞. Lastly joining ϕ1 and ϕ2 and modifying both in a little neighborhood of 0 we obtain ϕ : R −→ U a proper C ∞ embedding of R in R2 . Set S = ϕ(R). Now we consider S2 the compactification of R2 by adding one point and we obtain that S, e and f] e L (L), the corresponding com- pactifications of L, F (L), S, are Jordan curves in S2 . Furthermore it is clear that Le and F] (L) belongs to a different connected components e of S \ S. Thus S separates L and F (L) and this fact implies that S 2 separates F (S) and F −1 (S). So S is a C ∞ Brouwer line. Let V be the open set that has boundary S S ∪ F (S). S Clearly V ⊂ U ∪ F (U) and U ⊂ V ∪ F −1 (V). Therefore n∈Z F n (V) = n∈Z F n (U). This ends the proof of the Lemma. We remark that, although inSthe introduction we had reduced the problem to the situation where n∈Z F n (U) = R2 , in this proof we have not used at all this reduction.  To end the construction of G1 started in lemma 1.1 we need next result, that corresponds to Lemma 3.6 of [2]: REGULARIZATION OF BROUWER TRANSLATION THEOREM 5 Proposition 1.1. Let C be a closed, connected and non-compact one dimensional C m submanifold of R2 . Then there exists a C m diffeomor- phism ϕ : R2 → R2 such that ϕ(C) = {0} × R. Lemma 1.2. Let F be a preserving orientation fixed point free C m diffeomorphism. Let L a C ∞ Brouwer line L and assume that EL = R2 . Then F is C m conjugated via Ψ1 to a C m diffeomorphism G1 of R2 and Ψ1 (L) = {0} × R is a Brouwer line for G1 . Moreover E{0}×R = R2 . Proof. By lemma 1.1 we can choose a C ∞ Brouwer line L Then, by Proposition 1.1 there exists a C ∞ diffeomorphism Ψ1 : R2 → R2 such that Ψ1 (L) = {0} × R. Set G1 = Ψ1 ◦ f ◦ Ψ−1 1 . Then G1 is a C m diffeomorphism, Ψ1 conjugates F and G1 and {0} × R is a Brouwer line for G1 . Moreover E{0}×R = Ψ1 (EL ) = Ψ1 (R2 ) = R2 .  2. Second step: construction of G2 . We start this section with a couple of preliminary results. The- orem 3.11 in [6] asserts that two connected compact surfaces are C∞ diffeomorphic if and only if they have the same Euler characteristic and the same number of boundary components. We will use the following specific non-compact version: Theorem 2.1. Two open connected subsets of R2 whose boundary con- sists of the same number of boundary components, where each one of them is a C m proper embedding of R in R2 , are C m diffeomorphic. Next we recall the Smoothing theorem, that allows to modify a piece- wise smooth homeomorphism in order to obtain a diffeomorphism. This result is proved in Hirsch book [9] in the C ∞ context. We will also use a C m version with m ∈ N given in item (a) of [2, Thm. 3.2]. Theorem 2.2. Let W0 and W1 be two manifolds without boundary of dimension n and assume that they can be decomposed in the form Wi = Mi ∪ Ni , i = 0, 1, where Mi and Ni are closed n-dimensional sub-manifolds satisfying Mi ∩ Ni = ∂Mi = ∂Ni = Vi . In addition assume that Φ : W0 → W1 is a homeomorphism mapping C m diffeomorphically M0 onto M1 and N0 onto N1 . Then there exists a C m diffeomorphism Ψ : W0 → W1 such that Ψ(M0 ) = M1 , Ψ(N0 ) = N1 and Ψ = Φ on V0 . Moreover Ψ can be chosen in such a way that it coincides with Φ outside a given neighborhood of V0 . 6 A. GASULL, J. GROISMAN AND F. MAÑOSAS Now we are ready to prove the goal of this section. Lemma 2.1. Let G1 be the C m diffeomorphism given in Lemma 1.2 that has the Browuer line 0 × R. There exists a C m diffeomorphism G2 conjugated to G1 , such that {0} × R is a Brouwer line and G2 (0, y) = (1, p(y)), for some C m increasing diffeomorphism p of R.. Proof. Consider the manifolds (with boundary) A, B, A0 and B 0 where A is the adherence of the region between L = {0} × R and G1 (L), the set B is the adherence of the connected component of R2 \(L∪{G1 (L)}) whose boundary is exactly {G1 (L)}, and A0 = {(x, y) : 0 ≤ x ≤ 1}, B 0 = {(x, y) : x ≥ 1}, see Figure 1 for more details. By Theorem 2.1 there exists a C m diffeo- morphism Θ between A and A0 and a C m diffeomorphism ∆ between B and B 0 . ∆ L G1 (L) A B A′ B′ {0} × R {1} × R Θ Figure 1. Diffeomorphisms Θ and ∆ Let r be the C m diffeomorphism such that Θ ◦ ∆−1 (1, y) = (1, r(y)). Let us consider now R : B 0 → B 0 defined by R(x, y) = (x, r(y)). Notice also that Θ(0, y) = (0, s(y)) where s is another C m diffeomorphism. Let Φ : R2 → R2 be defined as   (x, s(y)), if x ≤ 0; Φ(x, y) = Θ(x, y), if (x, y) ∈ A;  R ◦ ∆(x, y), if (x, y) ∈ B. Then Φ is a plane homeomorphism which is a C m diffeomorphism in R2 \ D where D = L ∪ {G1 (L)}. Applying Theorem 2.2 to Φ we find a REGULARIZATION OF BROUWER TRANSLATION THEOREM 7 C m diffeomorphism Ψ2 in R2 such that Ψ2 |D = Φ|D . Finally, G2 = Ψ2 ◦ G1 ◦ Ψ−12 is the desired C m diffeomorphism. Note that by construction G2 (0, y) = (1, p(y)) for some diffeomorphism p that must be increasing because G2 preserves orientation and sends points (x, y) with x ∈ (0, 1) to points with first coordinate greater than one.  3. Third step: construction of G3 . Lemma 3.1. Let G2 be the C m diffeomorphism given in lemma 2.1. There exists a C m diffeomorphism G3 , conjugated to G2 , for which {0}× R is a Brouwer line and G3 (0, y) = (1, y). Proof. We know that G2 (0, y) = (1, p(y)), for some C m increasing dif- feomorphism p. Let Φ : R2 → R2 be defined as   (x, y) if x ≤ 0; Φ(x, y) = (x, xp−1 (y) + (1 − x)y) if x ∈ [0, 1];  (x, p−1 (y)) if x ≥ 1. Clearly Φ is a homeomophism that restictet to R2 \ {{0} × R ∪ {1} × R} is a C m diffeomorphism. Applying Theorem 2.2 to the function Φ, we obtain a C m diffeomorphism Ψ3 which restriction to ({0}×R)∪({1}×R) agrees with Φ. Thus, conjugating G2 by Ψ3 we obtain G3 with the property stated in this lemma.  4. Proof of Theorem A. Recall that from all the previous reductions we can start S with a Cm n plane diffeomorphism F satisfying F (0, y) = (1, y) and n∈Z F (U) = R2 , where U = [0, 1] × R. Recall also, that as we have already explained in the introduction, if a conjugation Ψ satisfying Ψ◦F ◦Ψ−1 = T, where T (x, y) = (x+1, y), is defined on U, automatically it is also defined on the whole plane. This is so, because the conjugation condition writes as Ψ(F (x)) = T (Ψ(x)). In short, Ψ restricted to F (U) is determined by the definition of Ψ in U. Hence, we start with a C m diffeomorphism Ψ only defined on U, and we firstly need to find conditions on it, that imply that its extension to U ∪ F (U) is also of class C m on the common boundary {1} × R. With this aim, we introduce G(x, y) = F (x, y) − (1, 0). We claim that if we define a C m diffeomorphism Ψ : U → U satisfying: • The map Ψ identically coincides with G in a neighborhood of {0} × R, • The map Ψ is the identity in a neighborhood {1} × R, 8 A. GASULL, J. GROISMAN AND F. MAÑOSAS then its extension to U ∪ F (U) also is a C m diffeomorphism. From it, clearly, this extension procedure S can be continued until Ψ is defined on the whole plane because n∈Z F n (U) = R2 . Let us prove the claim. The conjugacy condition can be written as Ψ = T ◦ Ψ ◦ F −1 . We know that in U and near {1} × R the map Ψ is simply the identity. Therefore, to prove that its extension to U ∪ F (U) gives rise to a C m map it suffices to prove that it is also the identity near {1}×R but in F (U). If (x, y) ∈ F (U) and close enough to {1}×R, F −1 (x, y) is in U and near {0} × R. Hence, for (x, y) ∈ F (U) and close enough to {1} × R, Ψ(x, y) =T ◦ Ψ ◦ F −1 (x, y) =Ψ(F −1 (x, y)) + (1, 0) = G(F −1 (x, y)) + (1, 0) =F (F −1 ((x, y)) − (1, 0) + (1, 0) = (x, y), as we wanted to prove. Notice that we have used that Ψ coincides with G in a neighborhood of {0} × R. To construct the seed Ψ on U, satisfying the above two properties, we will proceed by defining it on several pieces of U and then C m joining them by applying several times Theorem 2.2. Let us start our construction. Let V be a neighborhood of {0} × R of the form V = {(x, y) : 0 ≤ x ≤ σ(y)}, where σ is a C ∞ function, and such that G(V ) ⊂ [0, 1/2) × R, see Figure 2.(a). This can be done, simply by the continuity of G, because G(0, y) = (0, y). In that figure Σ = {(y, σ(y)) , y ∈ R}. Let k ∈ R be such that 1/2 < k < 1. Set A = {(x, y) : σ(y) ≤ x ≤ k} which is C m diffeomorphic to [0, 1]×R (through a C m diffeomorphism ∆) which moreover we can assume that satisfies ∆(σ(y), y) = (0, y). Let B be the subset of U whose boundary is the curve G(Σ) and the straight line x = k which as above is also C m diffeomorphic to [0, 1] × R (through a C m diffeomorphism Θ) that satisfies Θ(G(σ(y), y)) = (0, y), see Figure 2.(b). Then Θ−1 ◦ ∆ : A → B is a C m diffeomorphism that agrees with G restricted to the curve Σ. Thus, we are able to define S : [0, k] × R → [0, k] × R as ( G(x, y), if (x, y) ∈ V ; S(x, y) = Θ−1 ◦ ∆(x, y), if (x, y) ∈ A, which is a diffeomorphism in [0, k] × R \ Σ, although is a global homeo- morphism. Applying Theorem 2.2 we obtain Φ1 : [0, k] × R → [0, k] × R which agrees with G in some neighborhood of {0} × R and such that REGULARIZATION OF BROUWER TRANSLATION THEOREM 9 G(V ) V A B 0 1/2 k 1 Σ G(Σ) x=0 G(Σ) x=k x=1 (a) (b) Figure 2. Subsets of the strip [0, 1] × R introduced to define the diffeomorphisms Θ and ∆ in the proof of The- orem A Φ1 (k, y) = (k, `(y)) for some C m diffeomorphism `. Let δ > 0 be such that k < δ < 1 and consider R : [k, 1] × R → [k, 1] × R defined as     x, x − k y + δ − x `(y) , if k ≤ x ≤ δ; R(x, y) = δ−k δ−k  (x, y), if δ ≤ x ≤ 1. Notice that R is a C m diffeomorphism on each of the above two pieces and a global homemorphism on the strip. Applying once more Theo- rem 2.2 we obtain a C m diffeomorphism defined on [k, 1] × R, that we call Φ2 . Notice that Φ2 agrees with Φ1 on x = k and is the identity in [δ, 1]×R. Thus, applying again Theorem 2.2 to Φ : [0, 1]×R → [0, 1]×R defined as  Φ1 (x, y), if 0 ≤ x ≤ k; Φ(x, y) = Φ2 (x, y), if k ≤ x ≤ 1, we obtain the desired C m diffeomorphism Ψ defined on U = [0, 1] × R that is the searched seed. This is so, because near x = 0 the diffeo- morphism Φ coincides with Φ1 which, in turn, coincides with G, and near x = 1 it coincides with Φ2 that in that region is the identity. As a consequence, the theorem follows. 10 A. GASULL, J. GROISMAN AND F. MAÑOSAS Acknowledgements The first author is partially supported by the project PID2019- 104658GB-I00 grant) and Severo Ochoa and Marı́a de Maeztu Pro- gram for Centers and Units of Excellence in R&D (CEX2020-001084- M), and the third author by the project MTM2017-86795-C3-1-P, all of the Spanish Ministerio de Ciencia e Innovación. The first and third authors are also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. References [1] L. E. J. Brouwer. Proof of the plane translation theorem (in German). Math. Ann. 72 (1912), 37–54. [2] A. Cima, A. Gasull, F. Manosas and R. Ortega. Smooth linearization of planar periodic maps. Math. Proc. Cambridge Philos. Soc. 167 (2019), 295– 320. [3] R. de la Llave and N. P. Petrov. Regularity of conjugacies between critical circle maps: an experimental study. Experiment. Math. 11 (2002), 219– 241. [4] J. Franks. A new proof of the Brouwer plane translation theorem. Ergod. Th. and Dynamic. Sys. 12 (1991), 217–226. [5] L. Guillou. Brouwer’s plane translation theorem and generalizations of the Poincaré-Birkhoff theorem (in French). Topology 33 (1994), 331–351. [6] M. W. Hirsch. Differential Topology. Springer-Verlag, New York, 1976. [7] P. Le Calvez and A. Sauzet. A dynamical proof of Brouwer’s translation theorem (in French). Expo. Math. 14 (1996), 277–287. [8] W. Zhang, K. Lu and W. Zhang Differentiability of the conjugacy in the Hartman-Grobman theorem. Trans. Amer. Math. Soc. 369 (2017), 4995– 5030. † Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Ed- ifici C, 08193 Cerdanyola del Vallès (Barcelona), Spain. E-mail address: gasull@mat.uab.cat, manyosas@mat.uab.cat ‡ Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès (Barcelona), Spain [ IMERL, Facultad de Ingenierı́a, Universidad de la República, Mon- tevideo Uruguay E-mail address: jorgeg@fing.edu.uy