Injectivity of the composition operators of \'etale mappings

We consider the semigroup of \'etale polynomial mappings $\mathbb{C}^2\rightarrow\mathbb{C}^2$ where the binary operation is composition. We prove that both the right and the left composition operators on this semigroup are injective. This is in contrast to the situation in the semigroup of the entire functions $\mathbb{C}\rightarrow\mathbb{C}$ which are locally injective, where the left composition operator is not injective. Our interest in the injectivity (of the left composition operator) results from a new approach to deal with the two dimensional Jacobian Conjecture. In this approach we construct a fractal like structure on the above (first) semigroup in order to use it to settle the conjecture (in preparation). Injectivity is crucial for that construction.


Introduction
Let be a topological space. A mapping : → is called a local homeomorphism of or anétale mapping of if for any point ∈ there exists a neighborhood of such that the restriction of to , denoted by | , is an homeomorphism. The set of all theétale mappings of , denoted by et( ), is a semigroup with a unit with the composition of mappings taken to be the binary operation. If ∈ et( ), then the -right composition operator on et( ) is defined by The -left composition operator on et( ) is defined by We were interested in the injectivity of these two composition operators in two particular cases. The first is the case of entire functions C → C that areétale (and normalized). The second case is that of the polynomial mappings C 2 → C 2 with determinant of their Jacobian matrix equal (identically) to 1 and whose -degrees equal their total degrees. For the first case we use the following.
Thus we use in this case the symbol elh(C) instead of et( ).
Then we have the following.
Theorem 3 (see [1]). Let ∈ elh (C). Then is not injective if and only if This settled the first case. It should be noted (see [1]) that the proof for the left composition operator is much more involved than the proof for the right composition operator (which follows directly from the Picard little theorem). It is in fact the second case that initiated our interest in the injectivity of the composition operators. It results from a new approach 2 Algebra to studyétale polynomial mappings C 2 → C 2 and in particular the two-dimensional Jacobian conjecture [2][3][4]. This approach constructs a fractal structure on the semigroup of the (

normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation).
The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators.
The set of all such mappings will be denoted by et (C 2 ). This semigroup (with respect to composition of mappings) is the parallel of the semigroup elh(C) for entire functions. The 2dimensional Jacobian conjecture can be rephrased in each of the following forms: For the next survey of results we refer to the following paper: [5]. We denote by ( ) the asymptotic variety of , that is, the curve of all the asymptotic values of the mapping . The canonical geometric basis of will be denoted by 0 ( ). This basis consists of finitely many rational mappings of the following form: ( , ) = ( − , + − Φ( )), where ∈ Z + , ∈ Z + ∪ {0}, Φ( ) ∈ C[ ], and deg Φ < + . Also the effective powers in + + Φ( ) have a gcd which equals 1. The cardinality of the geometric basis, | 0 ( )|, equals the number of components of the affine algebraic curve ( ). ∀ ∈ 0 ( ) we have the double asymptotic identity ∘ = ∈ C[ , ] 2 , where the polynomial mapping is called the -dual of . Each ∈ 0 ( ) generates exactly one component of ( ). This component is normally parametrized by { (0, ) | ∈ C}. We will denote by ( , ) = 0 the implicit representation of this component in terms of the irreducible polynomial ∈ C[ , ]. There exists a natural number ( ) ≥ 2 and a polynomial ( , ) ∈ C[ , ]. The affine curve ( , ) = 0 is called thephantom curve of . The -component of ( ), ( , ) = 0, is a polynomial curve, which is not isomorphic to A 1 , and hence in particular must be a singular irreducible curve. We have the relation ( ( , )) = ( ) ( , ). The exponent ( ) satisfies the double inequality 2 ≤ ( ) ≤ − .
In our case of the canonical rational mappings ∈ 0 ( ), we have sing( ) = { = 0}. The following is true: Thus the -preimage of the -component of ( ) (which is the -image of sing( )) is the union of two curves: the first is sing( ) and the second is the so-called -phantom curve of . Even if for a single ( , ) the -phantom curve is empty, then JC(2) follows. Also if ∀ ∈ 0 ( ) sing( ) ∩ { ( , ) = 0} = 0, then is a surjective mapping. Proof.
The proposition tells us that compositions ofétale mappings do not decrease the geometric basis of the right factor and consequently do not decrease the left image of its asymptotic variety. We naturally ask under what conditions the geometric basis of ∘ is actually larger than that of ? In other words, we would like to know when is it true that 0 ( ) ⊂ 0 ( ∘ )? This happens exactly when ∃ ∈ 0 ( ∘ ) − 0 ( ). This means that ( ∘ ) ∘ ∈ C[ , ] 2 , ∘ ∉ C[ , ] 2 . Let ( , ) = ( − , + − Φ( )), ( , ) = ( ( , ), ( , )). Then We clearly have sing( ∘ ) ⊆ sing( ) and so sing( . This is not necessarily a member of the geometric basis of . The canonical geometric basis of , 0 ( ) contains finitely many rational mappings of the form ( , ) = ( − , + − Ψ( )). Since ∈ et(C 2 ), it follows that |C 2 − (C 2 )| < ∞ (a similar phenomenon as the Picard little theorem). If is an asymptotic tract of , then −1 ( ) cannot be a bounded subset of C 2 . The reason is that if −1 ( ) is compact, then ( −1 ( )) is compact and since ⊆ ( −1 ( )) ⊆ ( −1 ( )) this would imply the contradiction that is bounded (and hence cannot be an asymptotic tract). Hence −1 ( ) has at least one component, say 1 , that goes to infinity. This is because the number of components of −1 ( ) is finite and −1 ( ) is not bounded. So ∘ has a limit along 1 which equals the above asymptotic value of . This proves the following generalization of the second part of Proposition 4, namely, the following proposition.

The Composition
Proof. Consider the following: Proposition 8. is injective. Proof.
Based on our experience with entire functions we tend to prove that the answer to the question is negative. Indeed this is the case and the proof is almost identical to the entire case; see [1]. Namely, if the answer is affirmative, then we have two types of points in C 2 : ( , ) ∈ C 2 for which ( , ) ̸ = ( , ) and the complimentary set, where both sets are nonempty. Let us denote by the first subset of C 2 ; that is, The subset of . This implies that in any strong neighborhood of ( , ) = ( , ) there are different points, say ( , ) ̸ = ( , ) for ∈ Z + large enough, so that ( ( , )) = ( ( , )). Hence is not injective in any strong neighborhood of ( , ) = ( , ). Thus ∉ et(C 2 ). This contradiction proves the following.

The Size, , of the Generic Fiber of a Keller
Mapping ∈ et(C 2 ) We will need the generic size of a fiber of a mapping = ( , ) ∈ et(C 2 ). Moreover, there is a number that we will denote by such that generically in ( , ) we have | −1 ( , )| = . This means that {( , ) ∈ C 2 | | −1 ( , )| ̸ = } is a closed and proper Zariski subset of C 2 . In fact ∀( ,
This is a well-known result. We include one of its proofs for convenience.
of all the prime mappings will be denoted by et (C 2 ). Thus the set of all the compositeétale mappings is et( is not a prime number. Equivalently, , > 1 (by Proposition 12 and the fact that = 1 ⇔ ∈ Aut(C 2 )) ⇒ is a composite integer.
Theorem 15. The following hold true.

The Metric Spaces (et(C 2 ), )
We will need a special kind of four (real) dimensional subsets of R 4 . These will serve us to construct suitable metric structures on et(C 2 ).

Definition 16.
Let be an open subset of C 2 with respect to the strong topology that satisfies the following conditions.
(2) is a compact subset of C 2 (in the strong topology).
We define the following real valued function: Here we use the standard set-theoretic notation of the symmetric difference between two sets and ; that is, Remark 17. It is not clear how to construct an open subset of C 2 that will satisfy the three properties that are required in Definition 16. We will postpone for a while the demonstration that such open sets exist.

Proposition 18.
is a metric on et (C 2 ).
where the last equivalence follows by the fact that 1 and 2 are local homeomorphisms in the strong topology and because of condition (1) in Definition 16) (3) Here we use a little technical set-theoretic containment. Namely, for any three sets , , and we have ≤ (the volume of 1 ( ) Δ 2 ( )) + (the volume of 2 ( ) Δ 3 ( )) .
Hence the triangle inequality So far we thought of the volume of 1 ( )Δ 2 ( ) as the volume of the open set which is the symmetric difference between the 1 image and the 2 image of the open set . However, the mappings 1 and 2 areétale and in particular need not be injective. We will take into the volume computation the multiplicities of 1 and of 2 . By Theorem 3 on page 39 of [6] we have the following: given that : Thus the Jacobian condition det ≡ 1 implies that det̃≡ 1. So the real mapping preserves the usual volume form. In order to take into account the multiplicities of theétale mappings 1 and 2 when computing the volume of the symmetric difference 1 ( )Δ 2 ( ) we had to do the following. For any ∈ et(C 2 ) instead of computing, we compute For every = 1, 2, . . . , we denote by the subset of such that for each point of there are exactly points of that are mapped by to the same image of that point. In other words, = { ∈ | |̃− 1 (̃( )) ∩ | = }. We assume that is large enough so that ∀ = 1, . . . , we have ̸ = 0. For ourétale mappings it is well known that if < , then dim < dim , so the volume of these 's contribution equals 0. The dimension claim follows by the well-known fact that the size of a generic fiber | −1 ( )| equals and that is also the maximal size of any of the fibers of . However, for the sake of treating more general families of mappings, we denote by vol( ) the volume of the set . Then has a partition into exactly subsets of equal volume. The volume of each such a set is vol( )/ and each such a set has exactly one of the points iñ− 1 (̃( )) ∩ for each ∈ . We note that vol(̃( )) = vol( )/ by the Jacobian Condition. Thus the volume with the multiplicity of̃taken into account is given by We note that̃( ) = ⋃ =1̃( ) is a partition, so vol(̃( )) = ∑ =1 vol(̃( )). Hence we can express the desired volume by We note that this equals ∑ =1 vol( ) and since = ⋃ =1 is a partition we have vol( ) = ∑ =1 vol( ). As expected, the volume computation that takes into account the multiplicity of is in general larger than the geometric volume vol(̃( )).
The access can be expressed in several forms: Coming back to the computation of the metric distance ( 1 , 2 ) = the volume of 1 ( )Δ 2 ( ) we compute the volume of 1 ( ) − 2 ( ) with the multiplicity of 1 while the volume of 2 ( ) − 1 ( ) is computed with the multiplicity of 2 .

Characteristic Sets of Families of Holomorphic Local Homeomorphisms
In this section we prove the existence of sets that satisfy the three properties that are required in Definition 16. The third property will turn out to be the tricky one.
Definition 19. Let Γ be a family of holomorphic local homeomorphisms : C 2 → C 2 . A subset ⊆ C 2 is called a characteristic set of Γ if it satisfies the following condition: We start by recalling the well-known rigidity property of holomorphic functions in one complex variable. Also known as the permanence principle, or the identity theorem. The identity theorem for analytic functions of one complex variable says that if ⊆ C is a domain (an open and a connected set) and if is a subset of that has a nonisolated point and if ( ) is an analytic function defined on and vanishing on , then ( ) = 0 for all ∈ .
There is an identity theorem for analytic functions in several complex variables, but for more than one variable the above statement is false. One correct statement is as This type of elementary arguments that was used to construct a thin set for identity purpose is not new. For example, see the following.
Theorem 22 (see [8]). Let ⊆ C be a domain, and let be a subset of that has a nonisolated point. Let ( , ) be a function defined for , ∈ such that ( , ) is analytic in for each fixed ∈ and analytic in for each fixed ∈ . If ( , ) = 0 whenever and both belong to , then ( , ) = 0 for all , ∈ .
Advancing along the lines of the construction of the thin set in Proposition 20 we note that if { } ∞ =1 is a sequence of different numbers that converges to lim = , and if for each = 1, 2, 3, . . . there is a straight line segment [ , ] of 's such that two entire functions ( , ) and ( , ) agree on the union (a countable union) of the segments { }× [ , ], that is, ( , ) = ( , ), ∀ ∈ [ , ], then ( , ) ≡ ( , ), ∀( , ) ∈ C 2 . We now will construct characteristic sets of families Γ of holomorphic local homeomorphisms : Definition 23. Let be a natural number and ∈ C 2 . An -star at is the union of line segments, so that any pair intersects in .
Definition 24. Let be a line segment and let { } be a countable dense subset of . Let { } be a sequence of different natural numbers and ∀ , let be an -star at such that one of the star's segments lies on , and such that ∀ 1 ̸ = 2 , 1 ∩̃2 = 0. Here we denoted̃= − . Moreover, we group the stars in bundles of, say 5, thus getting the sequence of star bundles: and for each bundle of five we take the maximal length of its rays to be at most 1/10 the length of the maximal length of the previous bundle. We define Let { } ∞ =1 be a sequence of different complex numbers that converges to lim = . Let {{ ( ) } ∞ =1 } ∞ =1 be a partition of the natural numbers, Z + . In fact all we need is the disjointness, that is, 1 and define the following countable union of starred segments in C 2 : where we assume that the lengths of the star rays were chosen to satisfy disjointness in C 2 ; namely, We let or if we need a closed (compact) set, the closure of this union. Proof. Let 1 , 2 ∈ Γ satisfy 1 ( ) = 2 ( ). Then each starred line segment and each ( ) -star on , ( ) is mapped onto a holomorphic ( ) -star This is because the valence sequences of the stars are pairwise disjoint natural numbers, and 1 , 2 are local homeomorphisms and hence preserve the star valencies ( ) . The centers of the holomorphic stars form a countable and a dense subset of the curves 1 ({ } × ) = 2 ({ } × ). By continuity this implies that the restrictions coincide. Since 1 and 2 are holomorphic, this implies by Proposition 20 (which is a variant of the identity theorem for entire functions C 2 → C 2 ) that 1 ≡ 2 .
Remark 26. Proposition 25 holds true for any rigid family of local homeomorphisms. Rigidity here means that So the proposition holds true for holomorphic mappings, for harmonic mappings, and in particular for et(C 2 ).
We recall that Definition 16 required also two additional topological properties; namely, the open set should satisfy int( ) = ; is compact (all in the strong topology). These automatically exclude the set that was constructed in Definition 24. However, we can modify this construction to get at least an open set. Proof. Since can not be mapped in the smooth by an holomorphic local homeomorphism, we have for any 1 , 2 ∈ Γ for which 1 ( − ) = 2 ( − ) that also 1 ( ) = 2 ( ). Now the result follows by Proposition 25.
Remark 28. We note that if is a compact then − satisfies, at least, the requirement that − is compact. However, the "no-slit" condition int( − ) = int( ) ̸ = − fails.
Now that we gained some experience with the topological construction of we are going to make one more step and fix its shortcomings that were mentioned above. We need to construct a domain of C 2 which has the following three properties.
(2) is a compact subset of C 2 relative to the strong topology. ( The complex topology and the strong topology are the same. Our construction will be a modification of the construction of the domain that was constructed in Proposition 27. We start by modifying the notion of an -star that was introduced in Definition 23. Definition 29. Let be a natural number and ∈ C . A thick -star at is a union of 2 triangles, so that any pair intersects exactly at one vertex, and this vertex (i.e., common to all the 2 triangles) is .
Definition 30. Let be the construction of Definition 24 that uses thick -stars.

Proposition 31. Let Γ be any family of holomorphic local homeomorphisms
: Then is a characteristic set of Γ.
Proof. The proof is the same word by word as that of Proposition 25 where we replace -star by thick -star .
We finally obtain our construction. Proof. The proof is the same as that of Proposition 27 where we replace -star by thick -star .

Injectivity of the Left Composition Operator
We would like our natural mappings, the right mapping and the left mapping , to be say bi-Lipschitz with respect to the metric (that reflects the fact that our mappings et(C 2 ) Algebra satisfy the Jacobian condition). Considering first the right mapping , it would mean that given threeétale mappings 1 , 2 , ∈ et(C 2 ) and a characteristic set of et(C 2 ) we need to compare the volume of 1 ( )Δ 2 ( ) (multiplicities of 1 and of 2 are taken into account) with the volume of the deformed set ( 1 ∘ )( )Δ( 2 ∘ )( ). A short reflection shows that the two volumes are not comparable (in the sense of bi-Lipschitz). The situation is completely different when we replace the right mapping, by the left mapping, . For example, we have the following.
We now drop the restrictive assumption that ∈ Aut(C 2 ). Thus we merely have ∈ et(C 2 ) and we still want to compare ( 1 , 2 ) with ( ∘ 1 , ∘ 2 ), for any pair 1 , 2 ∈ et(C 2 ). We only know that is a local diffeomorphism of C 2 and (by the Jacobian condition) that it preserves (locally) the volume. In this case the geometrical degree of , can be larger than 1. We have the identity = | −1 ({( , )})| which holds generically (in the Zariski sense) in ( , ) ∈ C 2 . Hence the (complex) dimension of the set {( , ) ∈ C 2 | | −1 ( , )| < } is at most 1. The Jacobian condition det ≡ 1 implies (as we noticed before) that preserves volume taking into account the multiplicity. The multiplicity is a result of the possibility that is not injective and hence the deformation of the characteristic set by convolves (i.e., might overlap at certain locations). However, this overlapping is bounded above by . So if ⊆ C 2 is a measurable subset of C 2 and we compare the volume of with the volume of its image ( ), then the volume of ( ) ≤ the volume of This can be rewritten as follows: 1 ⋅ {the volume of } ≤ the volume of ( ) ≤ the volume of .
This is the place to emphasize also the following conclusion (that follows by the generic identity = | −1 ({( , )})|); namely, lim → C 2 the volume of ( ) the volume of = 1 , provided that the set tends to cover the whole of the complex space C 2 in an appropriate manner. To better understand why the quotient tends to the lower limit 1/ rather than to any number in the interval [1/ , 1] (if at all) we recall that our mapping belongs to et(C 2 ) and so is a polynomialétale mapping. So any point ( , ) ∈ C 2 for which | −1 ( , )| < is an asymptotic value of and hence belongs to the curve which is the asymptotic variety of . In other words the identity = | −1 ( , )| is satisfied exactly on the semialgebraic set C 2 − which is the complement of an algebraic curve. We now state and prove the main result of this paper.
(ii) Suppose that is a family of characteristic sets of et (C 2 ) such that → C 2 , then ∀ > 0 one has for being large enough.
(ii) and (iii). Here the proof is not just set theoretic. We will elaborate more in the remark that follows this proof. We recall that , 1 , 2 ∈ et(C 2 ). This implies that ∀( , ) ∈ C 2 we have | −1 ( , )| ≤ [C( , ) : C( )], the extension degree of ; see [3]. This is the so-called Fiber theorem forétale mappings. Moreover the image is cofinite; |C 2 − (C 2 )| < ∞ [3]. Also has a finite set of exactly maximal domains {Ω 1 , . . . , Ω }. This means that is injective on each maximal domain Ω , and ̸ = ⇒ Ω ∩ Ω = 0, and C 2 = ⋃ =1 Ω and the boundaries Ω are piecewise smooth (even piecewise analytic). For the theory of maximal domains of entire functions in one complex variable see [9], and for that theory for meromorphic functions in one complex variable see [10,11]. Here we use only basic facts of the theory which are valid also for more than complex variable. If is a family of characteristic sets of et(C 2 ) such that → C 2 , then by the Algebra 9 above 1 ( ), 2 ( ) → C 2 − , where is a finite set, and if 1 ̸ ≡ 2 , then we have the identity Recalling that ( ∘ 1 )( )Δ( ∘ 2 ( )) ⊆ ( 1 ( )Δ 2 ( ))) we write the last identity as follows: Taking any two points ∈ 1 ( ) − 2 ( ) and ∈ 2 ( ) − 1 ( ) (as in the defining equation of the set on the righthand side in the last identity), we note that there are ̸ = , 1 ≤ , ≤ such that ∈ Ω ∧ ∈ Ω (for ( ) = ( )!). For̃a large enough characteristic set of et(C 2 ), we will have ∈ 1 (̃) and ∈ 2 (̃) and so , ∈ 1 (̃) ∩ 2 (̃) (since 1 ( ), 2 ( ) → C 2 − {a finite set}). Hence ( 1 (̃)Δ 2 (̃)) − ( ∘ 1 )(̃)Δ( ∘ 2 )(̃) will not include the point . We conclude that if and are -equivalent ( ( ) = ( )), then = ( ) = ( ) will not belong to ( 1 ( )Δ 2 ( )) − ( ∘ 1 )( )Δ( ∘ 2 )( ) for large enough . We obtain the following crude estimate: One can think of as a large open ball centered at the origin of R 4 , ≈ ( ) and with the radius and look at the images of the two polynomialétale mappings ( ∘ 1 )( ( )) and ( ∘ Hence Remark 35. The facts we used in proving (ii) and (iii) forétale mappings are in fact true in any dimension , that is, in C . In dimension = 2 it turns out that the codimension of the image of the mapping is 0 and in fact the coimage is a finite set. Also the fibers are finite and have a uniform bound on their cardinality (one can get a less tight uniform bound by the Bezout theorem). Here are few well-known facts (which one can find in Hartshorne's book on Algebraic Geometry, [13]).
(1) The following two conditions are equivalent.
(a) The Jacobian condition: the determinant det is a nonzero constant. (b) The map * isétale (in standard sense of algebraic geometry). In particular it is flat.
is open in .
(6) The ring homomorphism → is injective, and the induced field extension → is finite.
There is a nonempty open subset fin ⊆ such that on letting fin := ( * ) −1 ( fin ) ⊆ , the map of schemes * | fin : fin → fin is finite. For any point ∈ fin (C) we have the equality = * the geometrical degree of * .  Let be the topological space which is the set (C) ≅ C given the classical topology. Similarly for , the map of Algebra schemes * : → induces a map of topological spaces : → ( = * | (C) ).

Extending the Notion of Geometrical Degree
In this section we will outline the fact that some of the notions and results that are related to geometrical degree of anétale mapping originate, in fact, in the more basic topological spaces (no algebraic or holomorphic structure is needed). We will skip most of the proofs (that are elementary).
Definition 37. Let be a topological space. The semigroup of all the continuous mappings, : → , will be denoted by ( ). Here, as usual, the binary operation is composition of mappings. ≡ ; that is, any ∈ ( ) is determined by its restriction | ( ) .
We are ready to discuss the notion of the geometrical degree, , of appropriate mappings in ( ).
Example 44. If = C 2 with the complex topology and ∈ et(C 2 ), then we know that exists. We also know that the set = { ∈ C 2 | | −1 ( )| < } is a plane algebraic curve (possibly empty). Thus it is closed in C 2 . Moreover it is also small because dim < 2 = dim C 2 .
We need one more property to hold for our mappings, namely, that the fiber size will generically be , that is, that the set of all ∈ for which = | −1 ( )| will be a large set measured in the topology of . This leads us to the following.
Definition 45. Let be a topological space. We will denote by ( ) the set of all the mappings : → that have the following properties.
Proposition 46. Let be an Hausdorff space. Then one has the following.
(1) ( ) is a semigroup with an identity (where the binary operation is composition of mappings). In fact Aut ( ) ⊆ ( ).
A second example is given in Section 4 of this paper.
Example 53. In Definition 45 we take = C 2 with the complex topology, and ( ) = et(C 2 ). Then the theory that was outlined in Propositions 12 and 14 and Theorem 15 is a special case of the above more general topological theory.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.