The m-Möbius transformations are generated (see [1]) starting with

f 2 ( z 1 , z 2 ) = ω s 2 − s 1 + 1 s 2 − s 1 + ω = ( ω z 2 − 1 ) z 1 + 1 − z 2 ( z 2 − 1 ) z 1 + ω − z 2 = ( ω z 1 − 1 ) z 2 + 1 − z 1 ( z 1 − 1 ) z 2 + ω − z 1 (1)

where ω ∈ ℂ ¯ \ { 1 } and s 1 = z 1 + z 2 , s 2 = z 1 z 2 .

Sometimes it will be preferable to use instead of the parameter ωthe parameter

a, where ω = a + 1 a − 1 , a ∈ ℂ ¯ \ { 1 } .

Applying recursively f 2 we get f 3 ( z 1 , z 2 , z 3 ) = f 2 ( f 2 ( z 1 , z 2 ) , z 3 ) = f 2 ( z 1 , f 2 ( z 2 , z 3 ) ) .

An easy computation shows that

f 3 ( z 1 , z 2 , z 3 ) = ( ω + 1 ) s 3 − s 2 + 1 s 3 − s 1 + ( ω + 1 ) , s 3 = z 1 z 2 z 3 , s 2 = z 1 z 2 + z 1 z 3 + z 2 z 3 , s 1 = z 1 + z 2 + z 3 (2)

If for arbitrary m we set

f m ( z 1 , z 2 , ⋯ , z m ) = f 2 ( f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) , z m ) = ( ω z m − 1 ) f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) + 1 − z m ( z m − 1 ) f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) + ω − z m (3)

this will allow the computation of f m when f m − 1 is known, i.e. when all f k from k = 2 to k = m − 1 have been computed.

We have proved in [1] and [2] that

f m ( z 1 , z 2 , ⋯ , z m ) = a 0 ( ω ) s m + a 1 ( ω ) s m − 1 + ⋯ + a m ( ω ) a m ( ω ) s m + a m − 1 ( ω ) s m − 1 + ⋯ + a 0 ( ω ) , where a k ( ω ) are

polynomials and s j are symmetric sums of order j of z 1 , z 2 , ⋯ , z m . Moreover, we have shown that for m = 2 k and m = 2 k + 1 we have that a 0 ( ω ) are polynomials of degree k and a m ( ω ) are polynomials of degrees k − 1 . Let us denote by p k , q k , respectively p k − 1 and q k − 1 these polynomials, i.e.

f 2 k ( z 1 , z 2 , ⋯ , z 2 k ) = p k ( ω ) s 2 k + ⋯ + p k − 1 ( ω ) p k − 1 ( ω ) s 2 k + ⋯ + p k ( ω ) and

f 2 k + 1 ( z 1 , z 2 , ⋯ , z 2 k + 1 ) = q k ( ω ) s 2 k + 1 + ⋯ + q k − 1 ( ω ) q k − 1 ( ω ) s 2 k + 1 + ⋯ + q k ( ω ) .

Let us notice that it is not obvious what should be in the blanks of these formulas and there is no way to proceed further without knowing it. The help comes from the formula (3) which implies:

p k + 1 ( ω ) = ω q k ( ω ) − q k − 1 ( ω ) (4)

and

( ω − 1 ) q k ( ω ) = ω p k ( ω ) − p k − 1 ( ω ) (5)

These formulas allow us to compute recursively p k and q k for every k. Indeed, by (1) and (2) we have: p 1 = ω , p 0 = 1 , q 1 = ω + 1 , q 0 = 1 . Using (4) we get:

p 2 = ω ( ω + 1 ) − 1 = ω 2 + ω − 1 (6)

Using (5) we get: ( ω − 1 ) q 2 = ω ( ω 2 + ω − 1 ) − ω = ω 3 + ω 2 − 2 ω , which gives:

q 2 = ω 2 + 2 ω (7)

Using (4) again we obtain: p 3 = ω ( ω 2 + 2 ω ) − ω − 1 , hence

p 3 = ω 3 + 2 ω 2 − ω − 1 (8)

Using (5) again we have:

( ω − 1 ) q 3 = ω ( ω 3 + 2 ω 2 − ω − 1 ) − ( ω 2 + ω − 1 ) = ω 4 + 2 ω 3 − 2 ω 2 − 2 ω + 1 , thus

q 3 = ω 3 + 3 ω 2 + ω − 1 (9)

Analogously, we compute:

p 4 = ω 4 + 3 ω 3 − 3 ω (10)

q 4 = ω 4 + 4 ω 3 + 3 ω 2 − 2 ω − 1 (11)

These expressions agree with those found in [1] for f k , k = 2 , 3 , ⋯ , 9 . Moreover, with the notation f m ( z ) instead of f m ( z 1 , z 2 , ⋯ , z m ) we have:

f 4 ( z ) = p 2 s 4 − p 1 s 3 + s 2 − s 1 + p 1 p 1 s 4 − s 3 + s 2 − p 1 s 1 + p 2 (12)

f 5 ( z ) = q 2 s 5 − q 1 s 4 + s 3 − s 1 + q 1 q 1 s 5 − s 4 + s 2 − q 1 s 1 + q 2 (13)

f 6 ( z ) = p 3 s 6 − p 2 s 5 + p 1 s 4 − s 3 + s 2 − p 1 s 1 + p 2 p 2 s 6 − p 1 s 5 + s 4 − s 3 + p 1 s 2 − p 2 s 1 + p 3 (14)

f 7 ( z ) = q 3 s 7 − q 2 s 6 + q 1 s 5 − s 4 + s 2 − q 1 s 1 + q 2 q 2 s 7 − q 1 s 6 + s 5 − s 3 + q 1 s 2 − q 2 s 1 + q 3 (15)

f 8 ( z ) = p 4 s 8 − p 3 s 7 + p 2 s 6 − p 1 s 5 + s 4 − s 3 + p 1 s 2 − p 2 s 1 + p 3 p 3 s 8 − p 2 s 7 + p 1 s 6 − s 5 + s 4 − p 1 s 3 + p 2 s 2 − p 3 s 1 + p 4 (16)

f 9 ( z ) = q 4 s 9 − q 3 s 8 + q 2 s 7 − q 1 s 6 + s 5 − s 3 + q 1 s 2 − q 2 s 1 + q 3 q 3 s 9 − q 2 s 8 + q 1 s 7 − s 6 + s 4 − q 1 s 3 + q 2 s 2 − q 3 s 1 + q 4 (17)

The general forms of f 2 k ( z ) and f 2 k + 1 ( z ) can be easily guessed from here and then by using induction we can prove them rigorously with the help of (3):

f 2 k ( z ) = p k s 2 k − p k − 1 s 2 k − 1 + ⋯ + ( − 1 ) k ( s k − s k − 1 ) + ( − 1 ) k p 1 s k − 2 + ⋯ − p k − 2 s 1 + p k − 1 p k − 1 s 2 k − p k − 2 s 2 k − 1 + ⋯ + ( − 1 ) k + 1 ( s k + 1 − s k ) + ( − 1 ) k + 1 p 1 s k − 1 + ⋯ + p k (18)

f 2 k + 1 ( z ) = q k s 2 k + 1 − q k − 1 s 2 k + ⋯ + ( − 1 ) k ( s k + 1 − s k − 1 ) + q 1 s k − 1 + ⋯ + q k − 1 q k − 1 s 2 k + 1 − q k − 2 s 2 k + ⋯ + ( − 1 ) k + 1 ( s k + 2 − s k ) + q 1 s k − 1 + ⋯ + q k (19)

We skip the induction step, which is elementary.

The functions (1) have been used in the theory of Lie groups (see [3] [4]) related to actions of those groups on non orientable Klein surfaces and, in general, on non orientable n-dimensional complex manifolds.

We have proved in [1] that, considered as Möbius transformations in each one of its variables, the functions f 2 , f 3 , f 4 and f 5 have all the same fixed points ξ 1 and ξ 2 , the roots of the equation z 2 − ( ω + 1 ) z + 1 = 0 , thus ξ 1 + ξ 2 = ω + 1 and ξ 1 ξ 2 = 1 . By using Formulas (18) and (19) we can now prove that this is true for any m-Möbius transformation.

Theorem 1. For every k ≥ 2 the following identities are true:

p k + 1 + p k − 1 = ( ω + 1 ) p k (20)

q k + 1 + q k − 1 = ( ω + 1 ) q k (21)

Proof: The relations (20) and (21) are obvious for k = 2 . For an arbitrary k we proceed by induction supposing that (20) and (21) are true for every subscript j < k . Then we have by (4):

p k + 1 + p k − 1 = ω ( q k + q k − 2 ) − ( q k − 1 + q k − 3 ) = ω ( ω + 1 ) q k − 1 − ( ω + 1 ) q k − 2 = ( ω + 1 ) ( ω q k − 1 − q k − 2 ) = ( ω + 1 ) p k .

Also, we have by (5):

( ω − 1 ) ( q k + 1 + q k − 1 ) = ω ( p k + 1 + p k − 1 ) − ( p k + p k − 2 ) = ω ( ω + 1 ) p k − ( ω + 1 ) p k − 1 = ( ω + 1 ) ( ω p k − p k − 1 ) = ( ω + 1 ) ( ω − 1 ) q k ,

which implies (21). Hence (20) and (21) are true for every k.

Theorem 2 (The Main Theorem). For every m ≥ 2 the function f m , considered as a Möbius transformation in any one of its variables, has the same fixed points ξ 1 and ξ 2 , which are the solutions of the equation z 2 − ( 1 + ω ) z + 1 = 0 .

Proof: The affirmation of this theorem may come as a surprise: there is no obvious reason why these functions depending on m independent complex variables should display such a strong property. As it will appear next, this is a result of the symmetry of coefficients appearing in Formulas (18) and (19). By (18), the equality

f 2 k ( z 1 , z 2 , ⋯ , z 2 k ) = z 2 k (22)

is true if and only if

p k s 2 k − p k − 1 s 2 k − 1 + ⋯ + ( − 1 ) k ( s k − s k − 1 ) − p 1 s k − 2 + ⋯ + p k − 1 = z 2 k [ p k − 1 s 2 k − p k − 2 s 2 k − 1 + ⋯ + ( − 1 ) k + 1 ( s k + 1 − s k ) + p 1 s k − 1 − p 2 s k − 2 + ⋯ + p k ] .

Taking into account the fact that s 2 k = s 2 k − 1 z 2 k and for every j < 2 k we can replace s j by z 2 k s j − 1 + s j , where on the left hand side s j are the symmetric sums of order j of z 1 , z 2 , ⋯ , z 2 k and on the right hand side s j − 1 are the symmetric sums of order j − 1 of z 1 , z 2 , ⋯ , z 2 k − 1 , this last equality is:

p k s 2 k − 1 z 2 k − p k − 1 ( s 2 k − 2 z 2 k + s 2 k − 1 ) + ⋯ + ( − 1 ) k [ ( s k − s k − 1 ) z 2 k + s k − 1 − s k − 2 ] + ( − 1 ) k p 1 ( s k − 3 z 2 k + s k − 2 ) + ⋯ − p k − 2 ( z 2 k + s 1 ) + p k − 1 = z 2 k { p k − 1 s 2 k − 1 z 2 k − p k − 2 ( s 2 k − 2 z 2 k + s 2 k − 1 ) + ⋯ + ( − 1 ) k p 1 ( s k + 1 z 2 k + s k + 2 )       + ( − 1 ) k [ ( s k − s k − 1 ) z 2 k + s k + 1 − s k ] + ⋯ + p k − 2 ( z 2 k s 1 + s 2 )     − p k − 1 ( z 2 k + s 1 ) + p k }

This is a second degree equation in z 2 k . An easy computation shows that the coefficient of z 2 k 2 is:

P ( z ) = p k − 1 ( s 2 k − 1 − 1 ) − p k − 2 ( s 2 k − 2 − s 1 ) ⋯ + ( − 1 ) k p 1 ( s k + 1 − s k − 2 )   + ( − 1 ) k + 1 ( s k − s k − 1 ) ,

and it is the same as the constant term of the equation.

Taking into account the Equality (20) we obtain for the coefficient of z 2 k the expression − ( ω + 1 ) P ( z ) therefore we have for the fixed points of f 2 k ( z ) when considered as a function of z 2 k the equation:

P ( z ) [ z 2 k 2 − ( ω + 1 ) z 2 k + 1 ] = 0 (23)

where z = ( z 1 , z 2 , ⋯ , z 2 k − 1 ) .

If P ( z ) = 0 then (22) is satisfied independently of the values of z 2 k , hence every point z 2 k is a fixed point of f 2 m ( z ) considered as Möbius transformation in z 2 k and this is a trivial situation. Otherwise, if P ( z ) ≠ 0 then the fixed points are ξ 1 and ξ 2 such that ξ 1 + ξ 2 = ω + 1 and ξ 1 ξ 2 = 1 . Due to the symmetry of f 2 k (it depends only of symmetric sums of z 1 , z 2 , ⋯ , z 2 k ), this is true for every variable z j .

Now, taking into account (21) we can draw the same conclusion for f 2 k + 1 , except that this time instead of P ( z ) we have Q ( z ) = q k ( s 2 k − 1 ) − q k − 1 ( s 2 k − 1 − s 1 ) + ⋯ + ( − 1 ) k − 1 ( s k + 1 − s k − 1 ) , which completely proves the theorem.

This theorem states that for every j = 1 , 2 , ⋯ , 2 k , if P ( z ) ≠ 0 , z = ( z 1 , z 2 , ⋯ , z j − 1 , z j + 1 , ⋯ , z 2 k ) then we have f 2 k ( z 1 , z 2 , ⋯ , z j − 1 , ξ l , z j + 1 , ⋯ , z 2 k ) = ξ l , l = 1 , 2 and for every j = 1 , 2 , ⋯ , 2 k + 1 , if Q ( z ) ≠ 0 , z = ( z 1 , z 2 , ⋯ , z j − 1 , z j + 1 , ⋯ , z 2 k + 1 ) then we have f 2 k + 1 ( z 1 , z 2 , ⋯ , z j − 1 , ξ l , z j + 1 , ⋯ , z 2 k ) = ξ l , l = 1 , 2 . In other words, if we let constant the variables z 1 , z 2 , ⋯ , z j − 1 , z j + 1 , ⋯ , z m then w = f m ( z ) is a Möbius transformation of the (z_j)-plane having the fixed points ξ 1 and ξ 2 . Then the Steiner net (see [1]) determined by these fixed points is mapped by f m onto a Steiner net in the (w)-plane. The pre-image by f m of this last Seiner net is an object in ℂ ¯ m whose projections on every (z_k)-plane, k = 1 , 2 , ⋯ , m is the Steiner net determined by the same points ξ 1 and ξ 2 from the respective plane. We will prove next that there is a unique Mö bius transformation of the (z_j)-plane onto the (z_k)-plane which carries those Steiner nets one into the other.

It can be easily checked (see [2]) that

f 2 ( z 1 , a ) = a , f 2 ( z 1 , 1 / a ) = 1 / a (24)

for every z 1 ∈ ℂ ¯ , where a + 1 / a = ω + 1 , hence if we want f 2 to be a Möbius transformation in z 1 we need to require that z 2 is omitting the values a and 1/a. Analogously, f 2 ( a , z 2 ) = a and f 2 ( 1 / a , z 2 ) = 1 / a for every z 2 ∈ ℂ ¯ , thus f 2 is a Möbius transformation in z 2 if and only if z 1 is different of a and 1/a. However, the two functions are defined for every z 1 ∈ ℂ ¯ , respectively for every z 2 ∈ ℂ ¯ . Moreover, these equalities show that a and 1/a are in fact the fixed points of f 2 and we can choose, for example ξ 1 = a and ξ 2 = 1 / a . Obviously, the fixed points belong to the domain of each function and they are omitted only when the variable is considered as a parameter. On the other hand, the identity f 2 ( a , 1 / a ) = f 2 ( 1 / a , a ) = 1 shows that indeed we need to omit those values for the parameter z 2 , since otherwise the relations (24) cannot be true. It also results from the Equation (3) that f m ( z 1 , z 2 , ⋯ , z m ) = a if and only if at least one of the variables is a and no other variable is 1/a. Similarly, f m ( z 1 , z 2 , ⋯ , z m ) = 1 / a if and only if at least one of the variables is 1/a and no other variable is a.

Let us deal first with the bi-Möbius transformations

w = f 2 ( z 1 , z 2 ) = ( ω z 2 − 1 ) z 1 + 1 − z 2 ( z 2 − 1 ) z 1 + ω − z 2 , which is a Möbius transformations in z 1 for every z 2 ∈ ℂ ¯ \ { a , 1 / a } and w = f 2 ( z 1 , z 2 ) = ( ω z 1 − 1 ) z 2 + 1 − z 1 ( z 1 − 1 ) z 2 + ω − z 1 , which is a

Möbius transformation in z 2 for every z 1 ∈ ℂ ¯ \ { a , 1 / a } . By Theorem 2 they have the same fixed points ξ 1 and ξ 2 , solutions of the equation z 2 − ( 1 + ω ) z + 1 = 0 .

Theorem 3. If w = f 2 ( z 1 , z 2 ) has distinct fixed points ξ 1 and ξ 2 , then for

every z 2 ∈ ℂ ¯ \ { a , 1 / a } there is a number μ 1 = μ 1 ( z 2 ) ∈ ℂ such that w − ξ 1 w − ξ 2 = μ 1 z 1 − ξ 1 z 1 − ξ 2 and for every z 1 ∈ ℂ ¯ \ { a , 1 / a } there is a number μ 2 = μ 2 ( z 1 ) ∈ ℂ such that w − ξ 1 w − ξ 2 = μ 2 z 2 − ξ 1 z 2 − ξ 2 .

Proof: The existence of those numbers is guaranteed by the following: If we set ζ = φ ( z 1 ) = z 1 − ξ 1 z 1 − ξ 2 , which carries z 1 = ξ 1 into 0 and z 1 = ξ 2 into ∞ , then

M 1 ( ζ ) = φ ∘ f 2 ∘ φ − 1 ( ζ ) , where f 2 is considered as a function of z 1 depending on the parameter z 2 , is a Möbius transformation having the fixed points 0 and ∞ . The only Möbius transformations satisfying such a property are those of the form M 1 ( ζ ) = μ 1 ζ for some complex number μ 1 . Then:

μ 1 z 1 − ξ 1 z 1 − ξ 2 = φ ∘ f 2 ∘ φ − 1 ( ζ ) = f 2 ∘ φ − 1 ( ζ ) − ξ 1 f 2 ∘ φ − 1 ( ζ ) − ξ 2 = f 2 ( z 1 , z 2 ) − ξ 1 f 2 ( z 1 , z 2 ) − ξ 2 = w − ξ 1 w − ξ 2 (25)

We call the number μ 1 the multiplier of this transformation associated with ξ 1 (see [5], p. 166). This is the multiplier of f 2 as a function of z 1 and it

depends on z 2 . Similarly, for ζ = ψ ( z 2 ) = z 2 − ξ 1 z 2 − ξ 2 , which carries z 2 = ξ 1 into

0 and z 2 = ξ 2 into ∞ , we have again that M 2 ( ζ ) = ψ ∘ f 2 ∘ ψ − 1 ( ζ ) is a Möbius transformation having the fixed points 0 and ∞ and therefore M 2 ( ζ ) = μ 2 ζ , where this time μ 2 = μ 2 ( z 1 ) . It is obvious that the multipliers of f 2 associated to ξ 2 are respectively 1 / μ 1 and 1 / μ 2 .

Since f 2 ( ∞ , z 2 ) = ω z 2 − 1 z 2 − 1 , the multiplier of f 2 (as a Möbius transformation

in z 1 ) associated with ξ 1 is obtained replacing z 1 = ∞ in (25), i.e.

μ 1 = μ 1 ( z 2 ) = [ ω z 2 − 1 z 2 − 1 − ξ 1 ] / [ ω z 2 − 1 z 2 − 1 − ξ 2 ] , or

μ 1 ( z 2 ) = ( ω z 2 − 1 ) − ( z 2 − 1 ) ξ 1 ( ω z 2 − 1 ) − ( z 2 − 1 ) ξ 2 = ( ω − ξ 1 ) z 2 + ( ξ 1 − 1 ) ( ω − ξ 2 ) z 2 + ( ξ 2 − 1 ) (26)

and analogously,

μ 2 ( z 1 ) = ( ω z 1 − 1 ) − ( z 1 − 1 ) ξ 1 ( ω z 1 − 1 ) − ( z 1 − 1 ) ξ 2 = ( ω − ξ 1 ) z 1 + ( ξ 1 − 1 ) ( ω − ξ 2 ) z 1 + ( ξ 2 − 1 ) (27)

Let us notice that the discriminant of the linear-fractional functions μ 1 ( z 2 ) and μ 2 ( z 1 ) is ( ω − 1 ) ( ξ 2 − ξ 1 ) and it is different of 0 since ω ≠ 1 and ξ 1 ≠ ξ 2 , hence they are Möbius transformations, which means that there is a one-to-one correspondence between μ 1 and z 2 , respectively μ 2 and z 1 . On the other hand, it is known that the nature of a non parabolic Möbius transformation is completely characterized by the values of its multiplier (see [5], page 164), namely it is elliptic, hyperbolic or loxodromic when the multiplier is respectively e i θ , θ ∈ ℝ , or it is a real number ρ ≠ 1 , or the product ρ e i θ , ρ ≠ 1 . Since the inverse transformations of (26) and (27) are

z 2 = ( ξ 2 − 1 ) μ 1 − ( ξ 1 − 1 ) ( ξ 2 − ω ) μ 1 + ( ω − ξ 1 ) (28)

z 1 = ( ξ 2 − 1 ) μ 2 − ( ξ 1 − 1 ) ( ξ 2 − ω ) μ 2 + ( ω − ξ 1 ) (29)

we can state the following:

Theorem 4. The non parabolic bi-Möbius transformation (1) having distinct fixed points ξ 1 and ξ 2 regarded as a Möbius transformation in each one of its variables is elliptic, hyperbolic or loxodromic when the other variable takes the values

χ ( μ ) = [ ( ξ 2 − 1 ) μ − ( ξ 1 − 1 ) ] / [ ( ξ 2 − ω ) μ + ( ω − ξ 1 ) ] (30)

where μ is respectively e i θ , θ ∈ ℝ , or ρ ≠ 1 , ρ ∈ ℝ , or the product of two such numbers.

Proof: This is a straightforward result from Felix Klein classification (see [5], page 163) of classical Möbius transformations. We notice that χ ( μ ) is a Möbius transformation and therefore it carries the unit circle μ = e i θ into a circle or a straight line. An easy computation shows that χ ( 1 ) = 1 and χ ( − 1 ) = − 1 , therefore if the image of the unit circle by χ ( μ ) is a strait line, this line should be the real axis, otherwise it is a circle passing through 1 and −1. The function χ ( μ ) also carries the real axis into a circle or a straight line. Checking if it is a straight line comes to see if the denominator of χ ( μ ) cancels for a real μ . It cancels for μ = ( ξ 1 − ω ) / ( ξ 2 − ω ) = ξ 1 ξ 2 − ( ξ 1 + ξ 2 ) ω + ω 2 = 1 − ω and this is real when ω is real. Otherwise, the image by χ ( μ ) of the real axis is a circle passing through −1 and 1. We conclude that the non parabolic bi-Möbius transformation (1) is elliptic in every variable on a given generalized circle in the plane of the other variable and it is a hyperbolic bi-Möbius transformation in one variable when the other variable describes another given generalized circle. In all the other cases (1) is loxodromic.

Formula (30) describes the way the Steiner nets from the (z₁)-plane and respectively the (z₂)-plane (see [1]) corresponding to the fixed points ξ 1 and ξ 2 are moved by f 2 ( z 1 , z 2 ) into a Stainer net in the (w)-plane, when this last one is identified with the (z₁)-plane, respectively with the (z₂)-plane. Namely, when μ = e i θ , θ ∈ ℝ the net is moved alongside every Apollonius circle (clockwise for the circles around ξ 2 and counterclockwise for the circles around ξ 1 , see Figure 1 below), while when μ ∈ ℝ , μ ≠ 1 it is moved alongside every circle passing through ξ 1 and ξ 2 (see Figure 2 below). For a loxodromic transformation the motion is spiral-like alongside a double spiral issuing from ξ 1 and ending in ξ 2 (see [5], page 165 and 166).

When representing these motions on the Riemann sphere, it can be easily seen that the loxodromic motion is obtained by composing in any order the elliptic

and the hyperbolic corresponding motions (see [5], page 153). On the other hand, we have seen in [1] that given w ∈ ℂ ¯ there is a unique Möbius transformation

z 2 = h ( z 1 ) = ( 1 − w ) z 1 + ω w − 1 ( ω − w ) z 1 + w − 1 such that f 2 ( z 1 , h ( z 1 ) ) = f 2 ( h − 1 ( z 2 ) , z 2 ) = w ,

therefore every Steiner net from the (w)-plane is the image by f 2 of a couple of Steiner nets from the (z₁)-plane, respectively (z₂)-plane. These last nets are the image of each other by h, respectively by h − 1 (see the figures below).

The generalization of this theory to m-Möbius transformations for m > 2 can be easily done by using the recurrence formula (3).

Theorem 5. If the m-Möbius transformation (3) has distinct fixed points ξ 1 and ξ 2 , then it is elliptic, hyperbolic or loxodromic in each one of its variables when the multiplier μ is e i θ , θ ∈ ℝ , or it is real different of 1, or respectively the product of two such numbers.

Proof: By Theorem 2, f m ( z 1 , z 2 , ⋯ , z m ) has the same fixed points ξ 1 and ξ 2 when treated as a Möbius transformation in any one of its variables. For the

sake of simplicity, we choose the variable z m . Let us denote ζ = φ ( z m ) = z m − ξ 1 z m − ξ 2

which carries z m = ξ 1 into 0 and z m = ξ 2 into ∞, then M ( ζ ) = φ ∘ f m ∘ φ − 1 ( ζ ) (where f m stands for f m ( z 1 , z 2 , ⋯ , z m ) in which all the variables except z m are fixed) is a Möbius transformation having the fixed points 0 and ∞ , hence M ( ζ ) = μ ζ for some complex number μ = μ ( z 1 , z 2 , ⋯ , z m − 1 ) . This means

μ z m − ξ 1 z m − ξ 2 = φ ∘ f m ∘ φ − 1 ( ζ ) = f m ∘ φ − 1 ( ζ ) − ξ 1 f m ∘ φ − 1 ( ζ ) − ξ 2 = f m ( z 1 , z 2 , ⋯ , z m ) − ξ 1 f m ( z 1 , z 2 , ⋯ , z m ) − ξ 2 = w − ξ 1 w − ξ 2 (31)

By writing repeatedly the recursive formula (3) we obtain f m under the form

f m ( z 1 , z 2 , ⋯ , z m ) = [ ω f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] z m + 1 − f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) [ f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] z m + ω − f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) (32)

It can be easily seen from here that

f m ( z 1 , z 2 , ⋯ , z m − 1 , ∞ ) = ω f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 (33)

Thus, by (31) we have

μ = [ ω f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] − [ f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] ξ 1 [ ω f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] − [ f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) − 1 ] ξ 2 (34)

This formula shows how the multiplier μ depends on z 1 , z 2 , ⋯ , z m − 1 . It can be written also under the form:

f m − 1 ( z 1 , z 2 , ⋯ , z m − 1 ) = ( ξ 2 − 1 ) μ + ( 1 − ξ 1 ) ( ξ 2 − ω ) μ + ( ω − ξ 1 ) (35)

which agrees with (28) and (29). We have seen in Theorem 4 that when μ = e i θ describes the unit circle, the right hand side in (35) describes a generalized circle. The pre-image by f m − 1 of this circle is an object C e in ℂ ¯ m − 1 . When ( z 1 , z 2 , ⋯ , z m − 1 ) ∈ C e , the function f m ( z 1 , z 2 , ⋯ , z m ) is an elliptic Möbius transformation in z m . Similarly, when μ ∈ ℝ \ { 1 } , the right hand side in (35) describes a circle or a straight line, the pre-image by f m − 1 of which is an object C h in ℂ ¯ m − 1 . When ( z 1 , z 2 , ⋯ , z m − 1 ) ∈ C h , the function f m ( z 1 , z 2 , ⋯ , z m ) is a hyperbolic Möbius transformation in z m . In all the other cases this Möbius transformation is loxodromic. Due to the symmetry of f m this is true for any other variable z k instead of z m .

Now, let us project onto the (z_l)-plane sections of C e and C h obtained by keeping z j constant for all j ≠ l . These projections are in turn generalized circles C e , l respectively C h , l in the (z_l)-plane. Obviously, z l ∈ C e , l for all l ≠ k if and only if ( z 1 , z 2 , ⋯ , z k − 1 , z k + 1 , ⋯ , z m ) ∈ C e , therefore, a non parabolic m-Möbius transformation f m is an elliptic Möbius transformation in z k if and only if z l ∈ C e , l for every l ≠ k . A similar property is true for C h , l . In all the other cases f m is loxodromic.

The function f 2 , considered as a Möbius transformation in any one of its variables, is parabolic if and only if it has a unique fixed point, i.e. the equation z 2 − ( ω + 1 ) z + 1 = 0 has a double root. Since ω ≠ 1 , this can happen if and only if ω = − 3 and then the fixed point is −1, i.e. for ω = − 3 we have f 2 ( − 1 , z 2 ) = f 2 ( z 1 , − 1 ) = − 1 .

For ζ = φ ( z 1 ) = 1 / ( z 1 + 1 ) , let us define M 1 ( ζ ) = φ ∘ f 2 ∘ φ − 1 ( ζ ) , where

f 2 ( z 1 , z 2 ) = ( 3 z 2 + 1 ) z 1 + ( z 2 − 1 ) ( 1 − z 2 ) z 1 + ( z 2 + 3 ) (36)

Here f 2 is considered as a function of z 1 depending on the parameter z 2 . The function M 1 ( ζ ) is a Möbius transformation having the unique fixed point ζ = ∞ .

Then, necessarily,

M 1 ( ζ ) = ζ + μ 1 (37)

for a complex number μ 1 = μ 1 ( z 2 ) . This number can be determined knowing M 1 ( 0 ) , i.e. f 2 ( ∞ , z 2 ) . We have

f 2 ( ∞ , z 2 ) = 3 z 2 + 1 1 − z 2 (38)

and then

μ 1 ( z 2 ) = M 1 ( 0 ) = φ ( f 2 ( ∞ , z 2 ) ) = φ ( 3 z 2 + 1 1 − z 2 ) = 1 / ( 3 z 2 + 1 1 − z 2 + 1 ) = 1 − z 2 2 ( z 2 + 1 ) , z 2 = 1 − 2 μ 1 1 + 2 μ 1 (39)

It can be easily checked that (39) implies 1 f 2 ( z 1 , z 2 ) + 1 = 1 z 1 + 1 + μ 1 ( z 2 )

We find analogously:

μ 2 ( z 1 ) = 1 − z 1 2 ( z 1 + 1 ) , z 1 = 1 − 2 μ 2 1 + 2 μ 2 , 1 f 2 ( z 1 , z 2 ) + 1 = 1 z 2 + 1 + μ 2 ( z 1 ) (40)

Having in view (37), every straight line parallel to the vector μ 1 is mapped by M 1 onto itself and every orthogonal line to μ 1 is mapped onto another orthogonal line. On the other hand, we have z 1 = φ − 1 ( ζ ) = 1 / ζ − 1 , which shows that φ − 1 maps those orthogonal lines into two families of orthogonal circles passing through z 1 = − 1 , for every z 2 ∈ ℂ ¯ . An analogous result is obtained if we switch z 1 and z 2 . These nets are mapped by f 2 ( z 1 , z 2 ) into a similar net in the (w)-plane passing through w = − 1 . Figure 3 below illustrates this phenomenon.

For the general case, we notice that for ω = − 3 we have

f m ( z 1 , z 2 , ⋯ , z k − 1 , − 1, z k + 1 , ⋯ , z m ) = − 1 , hence if ζ = φ ( z k ) = 1 / ( z k + 1 ) , then M k ( ζ ) = φ ∘ f m ∘ φ − 1 ( ζ ) , where the argument of f m is z k , is a Möbius transformation having the only fixed point ζ = ∞ . Therefore M k ( ζ ) = ζ + μ k ,

where this time

μ k = φ ( f m ( z 1 , z 2 , ⋯ , z k − 1 , ∞ , z k , ⋯ , z m ) ) = 1 − f m − 1 ( z 1 , z 2 , ⋯ , z k − 1 , z k + 1 , ⋯ , z m ) 2 [ f m − 1 ( z 1 , z 2 , ⋯ , z k − 1 , z k + 1 , ⋯ , z m ) + 1 ] . (41)

Again, every straight line parallel to the vector μ k in the (ζ)-plane is mapped by M k into itself and every straight line orthogonal to μ k is mapped by M k

into another line orthogonal to μ k . On the other hand, since z k = 1 ζ − 1 , the

function φ − 1 ( ζ ) maps those orthogonal lines into two families orthogonal circles passing through z k = − 1 . The image by f m of this net is a similar net in the (w)-plane. The pre-image by f m of this last net is an object in ℂ ¯ m whose every section obtained by keeping z k fixed, k ≠ j is projected onto the (z_j)-plane into a similar net.

Given ω 1 , ω 2 ∈ ℂ \ { 1 } , ω k = a k + 1 / a k − 1 , let us define M : ℂ ¯ 2 → ℂ ¯ 2 by

M ( z 1 , z 2 ) = ( w 1 , w 2 ) = ( f ω 1 ( z 1 , z 2 ) , f ω 2 ( z 1 , z 2 ) ) (42)

where

w k = f ω k ( z 1 , z 2 ) = ω k z 1 z 2 − z 1 − z 2 + 1 z 1 z 2 − z 1 − z 2 + ω k , k = 1 , 2. (43)

We will stick with this harmless change of notation in what follows since we need to specify the parameter on which every bi-Möbius transformation (42) depends.

We notice that f ω k ( z 1 , z 2 ) = f ω k ( z 2 , z 1 ) , hence M ( z 1 , z 2 ) = M ( z 2 , z 1 ) , which implies that M is not injective. However, we can choose a sub-domain of ℂ ¯ 2 in which M is injective, as for example G 1 × G 2 where G 1 = { z 1 | R e z 1 ≥ 0 and if R e z 1 = 0 then I m z 1 > 0 } and G 2 = ℂ ¯ \ G 1 . Let us notice that z ∈ G 1 if and only if 1 / z ∈ G 2 . In the following we will deal with the function M : G 1 × G 2 → ℂ ¯ 2 defined by M ( z 1 , z 2 ) = ( w 1 , w 2 ) , where w 1 and w 2 are given by (43).

Theorem 6. The function M maps G 1 × G 2 one to one and onto ℂ ¯ 2 .

Proof: Indeed, let ( w 1 , w 2 ) ∈ ℂ ¯ 2 . We are looking for ( z 1 , z 2 ) ∈ G 1 × G 2 such that ( w 1 , w 2 ) = M ( z 1 , z 2 ) . For arbitrary z 2 ∈ G 2 \ { a 2 , 1 / a 2 } , solving the first

Equation (43) for z 1 we get z 1 = ( w 1 − 1 ) z 2 − ω 1 w 1 + 1 ( w 1 − ω 1 ) z 2 − w 1 + 1 and dividing both the

denominator and numerator by − z 2 we obtain

z 1 = ω 1 w 1 / z 2 − w 1 − 1 / z 2 + 1 w 1 / z 2 − w 1 − 1 / z 2 + ω 1 = f ω 1 ( 1 / z 2 , w 1 ) (44)

Similarly, solving the second Equation (43) for z 2 , with z 1 already found, we get

z 2 = f ω 2 ( 1 / z 1 , w 2 ) (45)

and both Equation (43) are satisfied with these values of z 1 and z 2 . Hence we have found a couple ( z 1 , z 2 ) ∈ G 1 × G 2 such that M ( z 1 , z 2 ) = ( w 1 , w 2 ) , which means that M maps G 1 × G 2 onto ℂ ¯ 2 . Moreover, both z 1 and z 2 have been uniquely determined since the first Equation (43) is a Möbius transformation in z 1 for every z 2 ∈ G 2 \ { a 2 , 1 / a 2 } and the second Equation (43) is a Möbius transformation in z 2 for every z 1 ∈ G 1 \ { a 1 , 1 / a 1 } , therefore M is injective, which completely proves the theorem.

The mapping ( w 1 , w 2 ) → ( z 1 , z 2 ) given by (44) and (45) is the inverse mapping M − 1 of M. We notice that although f ω 1 ( 1 / z 2 , w 1 ) is a Möbius transformation in w 1 for every z 2 ∈ G 2 \ { a 1 , 1 / a 1 } and f ω 2 ( 1 / z 1 , w 2 ) is a Möbius transformation in w 2 for every z 1 ∈ G 1 \ { a 2 , 1 / a 2 } , the mapping M − 1 is not of the same nature as M. To avoid this inconvenience, let us redefine M in the following way. With ω 1 and ω 2 , as previously given, we choose two other parameters ζ 1 ∈ ℂ ¯ \ { a 1 , 1 / a 1 } and ζ 2 ∈ ℂ ¯ \ { a 2 , 1 / a 2 } and set:

w 1 = f ω 1 ( z 1 , ζ 1 ) = ω 1 ζ 1 z 1 − ζ 1 − z 1 + 1 ζ 1 z 1 − ζ 1 − z 1 + ω 1 (46)

w 2 = f ω 2 ( z 2 , ζ 2 ) = ω 2 ζ 2 z 2 − ζ 2 − z 2 + 1 ζ 2 z 2 − ζ 2 − z 2 + ω 2 (47)

The functions f ω 1 and f ω 2 are Möbius transformations in z 1 and respectively z 2 , hence we can solve (46) and (47) for these variables and we get:

z 1 = f ω 1 ( w 1 , 1 / ζ 1 ) (48)

z 2 = f w 2 ( w 2 , 1 / ζ 2 ) (49)

This time

( w 1 , w 2 ) = M ( z 1 , z 2 ) = ( f ω 1 ( z 1 , ζ 1 ) , f ω 2 ( z 2 , ζ 2 ) ) : ℂ ¯ 2 → ℂ ¯ 2 (50)

is a bijective function for every ζ 1 ∈ ℂ ¯ \ { a 1 , 1 / a 1 } and ζ 2 ∈ ℂ ¯ \ { a 2 , 1 / a 2 } and

M − 1 ( w 1 , w 2 ) = ( f ω 1 ( w 1 , 1 / ζ 1 ) , f ω 2 ( w 2 , 1 / ζ 2 ) ) . (51)

Thus, M and M − 1 are functions of the same nature depending on the parameters ζ 1 ∈ ℂ ¯ \ { a 1 , 1 / a 1 } , ζ 2 ∈ ℂ ¯ \ { a 2 , 1 / a 2 } and respectively 1 / ζ 1 ∈ ℂ ¯ \ { a 1 , 1 / a 1 } , 1 / ζ 2 ∈ ℂ ¯ \ { a 2 , 1 / a 2 } .

Let us denote by L 2 = L 2 ( ω 1 , ω 2 ) the class of these functions, where ω 1 and ω 2 are fixed and notice that M ∈ L 2 if and only if M − 1 ∈ L 2 . Different values of the parameters ζ 1 and ζ 2 define different functions M ∈ L 2 . Two of them can be composed following the usual rule of function composition. We will show next that the result is an element of L 2 .

Theorem 7. If M , M ′ ∈ L 2 then M ′ ∘ M ∈ L 2 .

Proof: Let ζ 1 , ζ ′ 1 ∈ ℂ ¯ \ { a 1 , 1 / a 1 } and ζ 2 , ζ ′ 2 ∈ ℂ ¯ \ { a 2 , 1 / a 2 } and let ( w 1 , w 2 ) = M ( z 1 , z 2 ) = ( f ω 1 ( z 1 , ζ 1 ) , f ω 2 ( z 2 , ζ 2 ) ) , ( η 1 , η 2 ) = M ′ ( w 1 , w 2 ) = ( f ω 1 ( w 1 , ζ ′ 1 ) , f ω 2 ( w 2 , ζ ′ 2 ) ) . Then

M ′ ∘ M ( z 1 , z 2 ) = ( f ω 1 ( f ω 1 ( z 1 , ζ 1 ) , ζ ′ 1 ) , f ω 2 ( f ω 2 ( z 2 , ζ 2 ) , ζ ′ 2 ) ) = ( f ω 1 ( z 1 , f ω 1 ( ζ 1 , ζ ′ 1 ) ) , f ω 2 ( z 2 , f ω 2 ( ζ 2 , ζ ′ 2 ) ) ) = ( f ω 1 ( z 1 , ζ ″ 1 ) , f ω 2 ( z 2 , ζ ″ 2 ) )

where ζ ″ 1 = f ω 1 ( ζ 1 , ζ ′ 1 ) ∈ ℂ ¯ \ { a 1 , 1 / a 1 } and ζ ″ 2 = f ω 2 ( ζ 2 , ζ ′ 2 ) ∈ ℂ ¯ \ { a 2 , 1 / a 2 } , which shows that indeed M ′ ∘ M ∈ L 2 .

When M ′ = M − 1 then ζ ′ 1 = 1 / ζ 1 and ζ ′ 2 = 1 / ζ 2 , thus ζ ″ 1 = f ω 1 ( ζ 1 , 1 / ζ 1 ) = 1 and ζ ″ 2 = f ω 2 ( ζ 2 , 1 / ζ 2 ) = 1 (see [2]), which means that M − 1 ∘ M ( z 1 , z 2 ) = ( f ω 1 ( z 1 , 1 ) , f ω 2 ( z 2 , 1 ) ) = ( z 1 , z 2 ) (see [2]), hence the unit element of L 2 is M 0 : ℂ ¯ 2 → ℂ ¯ 2 , defined by M 0 ( z 1 , z 2 ) = ( z 1 , z 2 ) for every z 1 , z 2 ∈ ℂ ¯ .

Since f ω k ( ζ k , ζ ′ k ) = f ω k ( ζ ′ k , ζ k ) , this composition law in L 2 is commutative. Finally, with the proper notations, we have

M ″ ∘ ( M ′ ∘ M ) = M ″ ∘ ( f ω 1 ( z 1 , f ω 1 ( ζ 1 , ζ ′ 1 ) ) , f ω 2 ( z 2 , f ω 2 ( ζ 2 , ζ ′ 2 ) ) ) = ( f ω 1 ( z 1 , f ω 1 ( f ω 1 ( ζ 1 , ζ ′ 1 ) , ζ ″ 1 ) ) , f ω 2 ( z 2 , f ω 2 ( f ω 2 ( ζ 2 , ζ ′ 2 ) , ζ ″ 2 ) ) ) = ( f ω 1 ( z 1 , ( f ω 1 ( ζ 1 , f ω 1 ( ζ ′ 1 , ζ ″ 1 ) ) ) ) , f ω 2 ( z 2 , ( f ω 2 ( ζ 2 , f ω 2 ( ζ ′ 2 , ζ ″ 2 ) ) ) ) ) = ( M ″ ∘ M ′ ) ∘ M

hence the composition law in L 2 is associative.

Corollary 1. The function composition law in L 2 defines a structure of Abelian group on L 2 .

This result is in contrast with the case of ordinary Möbius transformations in the plane for which the composition law is not commutative.

The generalization of this theory to the dimension m is straightforward. Let a k ∈ ℂ \ { 0,1 } , k = 1 , 2 , ⋯ , m be arbitrary complex numbers and let

ω k = a k + 1 / a k − 1 . For every k and a parameter ζ k ∈ ℂ ¯ \ { a k , 1 / a k } we define the Möbius transformation in z k depending on the parameter ζ k ,

w k = f ω k ( z k , ζ k ) = ω k ζ k z k − z k − ζ k + 1 ζ k z k − z k − ζ k + ω k . These transformations define a bijective

mapping M : ℂ ¯ m → ℂ ¯ m ( w 1 , w 2 , ⋯ , w m ) = ( f ω 1 ( z 1 , ζ 1 ) , f ω 2 ( z 2 , ζ 2 ) , ⋯ , f ω m ( z m , ζ m ) ) . Let L m be the set of these functions endowed with the usual function composition law. Proceeding as for L 2 , it can be easily proved that L m is an Abelian group.

The m-Möbius transformations have been introduced in connection with Lie groups’ actions on complex manifolds (see [3] and [4]). They represent an interesting mathematical topic in itself and we dedicated ourselves to performing in this paper a study of these transformations parallel to that of classical Möbius transformations of the complex plane. The geometric properties of m-Möbius transformations revealed in [1] have been expanded in this paper by using the tool of multipliers. This became possible after proving that regarded as an ordinary Möbius transformation in any one of its variables, a m-Möbius transformation has the same fixed points. This is the main result and it was instrumental in the classification of these transformations. We ended the study with group properties of m-Möbius transformations by showing that they form Abelian groups.

The topic we dealt with here is a new one and it has been studied just in [1] [2] [3] [4]. No other reference was needed. In [5] one can find everything about ordinary Möbius transformations.

We thank Aneta Costin for her support with technical matters.

The authors declare no conflicts of interest regarding the publication of this paper.

Ghisa, D. and Mikulin, E. (2022) Multipliers and Classification of m-Möbius Transformations. Advances in Pure Mathematics, 12, 436-450. https://doi.org/10.4236/apm.2022.126033