Invariants in separated variables: Yang-Baxter, entwining and transfer maps

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called {\it triad family of maps} and we propose a multi-field generalisation of the later. We show that by imposing separability of variables to the invariants of this family of maps, the $H_I, H_{II}$ and $H_{III}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_I, H_{II}$ and $H_{III}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H-$list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H-$list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as {\it extended transfer maps} and in turn they lead to $k-$point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlev\'e equations.


Introduction
The quantum Yang-Baxter equation originates from the theory of exactly solvable models in statistical mechanics [73,11]. It reads: where R : V ⊗ V → V ⊗ V a linear operator and R lm , l = m ∈ {1, 2, 3} the operators that acts as R on the l−th and m−th factors of the tensor product V ⊗ V ⊗ V . For the history of the later and for the early developments on the theory see [37]. Replacing the vector space V with any set X and the tensor product with the cartesian product, Drinfeld [21] introduced the set theoretical version of (1). Solutions of the later appeared under the name of set theoretical solutions of the quantum Yang-Baxter equation. The first instance of such solutions, appeared in [65,24]. The term Yang-Baxter maps was proposed by Veselov [71] as an alternative name to the Drinfeld's one. Early results on the context of Yang-Baxter maps were provided in [2,57,41]. In the recent years, many results arose in the interplay between studies on Yang-Baxter maps and the theory of discrete integrable systems [8,12,10,18,19,31,20,9]. In [23] it was considered a special type of set theoretical solutions of the quantum Yang-Baxter equation, the so called non degenerate rational maps. Nowadays, this type of solutions is referred to as quadrirational Yang-Baxter maps. Note that the notion of quadrirational maps, was extended in [46] to the notion of 2 n −rational maps, where highly symmetric higher dimensional maps were considered. Under the assumption of quadrirationality and modulo conjugation (see Definition 3.1), in [6,59] a list of ten families of maps was obtained. Five of them were given in [6], which constitute the so-called F −list of quadrirational Yang-Baxter maps and five more in [59], which constitute the so-called H−list of quadrirational Yang-Baxter maps. For their explicit form see Appendix A. The Yang-Baxter maps of the F −list and the H−list can also be obtained from some of the integrable lattice equations in the classification scheme of [5], by using the invariants of the generators of the Lie point symmetry group of the later [60]. In the series of papers [45,44,56], from the Yang-Baxter maps of the F −list and of the H−list, integrable lattice equations and correspondences (relations) were systematically constructed. Invariant, under the maps, functions where the variables appeared in separated form, played an important role to this construction. The cornerstone of this manuscript are invariant functions where the variables appear in separated form.
In [4], it was introduced a family rational of maps in 3 variables that preserves two rational functions the so-called the triad map. The triad map serves as a generalisation of the QRT map [61] (cf. [22]). In Section 2 we present an explicit formula for Adler's triad map as well as we prove the Liouville integrability of the later. We also propose an extension of the triad map in k ≥ 3 number of variables. If one imposes separability to the variables of the invariants of the triad map, the H I , the H II and the H A III Yang-Baxter maps in general positions of singularities, emerge. This is presented in Section 3 together with the explicit formulae for these maps.
In Section 4, we develop two methods to obtain non-equivalent entwining maps [52] i.e. maps R, S, T that satisfy the relation R 12 S 13 T 23 = T 23 S 13 R 12 . The first method gives us entwining maps associated with the H I , H II and the H A III members of the H−list of Yang-Baxter maps. The second one produces entwining maps for the whole F −list and the H−list. In this manuscript we present the entwining maps associated with the H−list of quadrirational Yang-Baxter maps only.
In Section 5, we re-factorise the transfer maps [71] associated with the H−list of Yang-Baxter maps. We show that the transfer maps coincide with the (k − 1)−iteration of some maps of simpler form that we refer to as extended transfer maps. Moreover, we show that the extended transfer maps, after a change of variables followed by an integration, are written as k−point recurrences, which some of them can be considered as alternating versions of discrete Painlevé hierarchies [57,15,33]. In Section 6 we end this manuscript with some conclusions and perspectives.

The Adler's Triad family of maps
In [4], Adler proposed the so-called triad family of maps. The triad map is a family of maps in 3 variables that consists of the composition of involutions which preserve two rational invariants of a specific form. In what follows we present the explicit form of the later in terms of its invariants.
Consider the polynomials Where x 1 , x 2 , x 3 are considered as variables and α i j,k,l , β i j,k,l as parameters. We consider also 3 maps R ij , i < j, i, j ∈ {1, 2, 3}. These maps can be build out of the polynomials n i , d i and they read: D xi n 2 · d 2 ∂ xi D xj n 1 · d 1 + ∂ xj D xi n 1 · d 1 ∂ xi D xj n 2 · d 2 + ∂ xj D xi n 2 · d 2 , X j = x j + 2 D xi n 1 · d 1 D xi n 2 · d 2 D xj n 1 · d 1 D xj n 2 · d 2 D xj n 1 · d 1 D xj n 2 · d 2 ∂ xi D xj n 1 · d 1 + ∂ xj D xi n 1 · d 1 ∂ xi D xj n 2 · d 2 + ∂ xj D xi n 2 · d 2 , X k = x k for k = i, j. (2) Here with ∂ z we denote the partial derivative operator wrt. to z i.e. ∂ z h = ∂h ∂z . D z is the Hirota's bilinear operator i.e. D z h · k = (∂ z h) k − h ∂ z k.
(2) The functions Mappings R ij are anti-measure preserving with respect to the measures m 1 = n 1 d 2 , m 2 = n 2 d 1 . Proof.
(1) The invariants H 1 , H 2 depend on 3 variables and they include 32 parameters. Acting with a different Möbius transformation to each of the variables, 9 parameters can be removed. A Möbius transformation of an invariant remains an invariant, since we have 2 invariants, 6 more parameters can be removed. Finally, since any multiple of an invariant remains an invariant, 2 more parameters can be removed. That leaves us with 32 − 9 − 6 − 2 = 15 essential parameters for the invariants H 1 , H 2 and hence for the maps R ij .
(2) The functions H 1 = n 1 /d 1 , H 2 = n 2 /d 2 , reads where a, a 1 , b, b 1 , k, k 1 , . . . are linear functions of x 3 (note we have suppressed the dependency on x 3 of the functions H 1 , H 2 ). From the set of equations by eliminating X 2 or by eliminating X 1 the resulting equations respectively factorize as: The factor A is linear in X 1 and the factor B is linear in X 2 . By solving these equations (we omit the trivial solution X 1 = x 1 , X 2 = x 2 ) we obtain where γ kl ij := u ij u kl v ij v kl , with u ij the determinants of a matrix generated by the ith and jth column of the matrix and v kl the determinants of a matrix generated by the kth and lth column of the matrix v = k l m n k 1 l 1 m 1 n 1 Now it is a matter of long and tedious calculation to prove that the map φ : ( where X 1 , X 2 are given by (4) coincides with the map R 12 of (2). Similarly we can work on R 13 and R 23 . (3) Since the map R 12 : (x 1 , x 2 , x 3 ) → (X 1 , X 2 , x 3 ) satisfies (3), the proof of involutivity follows. (4) It is enough to prove that the map R 12 anti-preserves the measure m 1 = n 1 d 2 i.e. the Jacobian Since the functions H i = n i /d i , i = 1, 2 are invariant under the action of the map R 12 , it holds: where κ, λ are rational functions of x 1 , x 2 , x 3 . So, We differentiate equations (6) with respect to x 1 and we eliminate here we have suppressed the dependency of κ, λ, n i , d i on x 1 , x 2 , x 3 . Byñ i we denoteñ i := n i (X 1 , X 2 , x 3 ), i = 1, 2, and similarly ford i . Also if we differentiate the equations (6) with respect to x 2 and eliminate ∂κ ∂x 2 and ∂λ ∂x 2 we obtain Due to the form of n i , d i , i = 1, 2, equations (8), (9) are linear in , κ, λ and by using (4), the Jacobian determinant reads: (7) we have: that completes the proof. Note that the same holds true for the remaining maps R ij . (5) The proof of this statement is given in [4].
Remark 2.2. Any map that can be build out of the involutions R 12 , R 13 , R 23 can be considered as an Adler's triad map. Hence here we have provided the explicit form of Adler's Triad family of maps.
Among all the maps that can be constructed by the involutions R ij , the following maps are of special interest since they are not periodic and moreover they satisfy [4] ( (1) they preserve the functions H 1 , H 2 , (2) they are measure-preserving with respect to the measures m 1 , m 2 (3) they preserve the following degenerate Poisson tensors, where it holds , Ω j 2 ∇H 2 = 0, j = 1, 2 (4) they are Liouville integrable maps Proof. The statements (1), (2) follows from Proposition 2.1. To prove the statement (3), (4), first note that since the maps T i are measure preserving, they preserve the following volume forms Hence, the contractions V j ⌋dH i , i, j ∈ {1, 2}, (see [32,29]) are degenerate Poisson tensors. Namely: The maps T i preserve the Poisson tensors Ω j i and the 2 invariants H 1 , H 2 , so they are Liouville integrable maps [53,14,70].
Note that on the level surfaces H 2 (x 1 , x 2 , x 3 ) = c, maps T 1 , T 2 , T 3 reduce to pair-wise commuting maps on the plane which preserve the functionĤ 1 (x 1 , x 2 ; c). One of these reduced maps is the associated with the invariant H 1 (x 1 , x 2 ; c) QRT map. Examples of commuting maps with specific members of the QRT family of maps were also constructed in [30].
The involution R 12 under the reduction x 2 = x 1 , H 2 = H 1 = H, so H = n d = ax 2 1 + bx 1 + c kx 2 1 + lx 1 + m , reads: that coincides with the QRT involution i x that preserves the invariant H. This formulae for the QRT involution i x was firstly given in [38], where an elegant presentation of the QRT map was considered.
2.1. A generalisation of the triad family of maps. Following the same generalisation procedures introduced for the QRT family of maps [16,36,67,62,29], the triad family of maps can be generalised in similar manners.
Here, in order to generalise the triad family of maps, we mimic the generalisation of the QRT family of maps introduced in [67]. Consider the following polynomials Where x 1 , x 2 , . . . , x k are considered as variables and α i j1,j2,...,j k , β i j1,j2,...,j k as parameters. We consider the k 2 maps R ij , i < j, i, j ∈ {1, 2, . . . , k}. These maps can be build out of the polynomials n i , d i and they read: , where X l = x l ∀ l = i, j and X i , X j are given by the formulae (2), where n i , d i , i = 1, 2 are given by (11). Proposition 2.1 is straight forward extended to the k−variables case. Namely for the mappings R ij it holds: • Mappings R ij depend on 4 · 2 n parameters α i j1,j2,...,j k , β i j1,j2,...,j k , i = 1, 2, j 1 , j 2 , . . . , j k ∈ {0, 1}. Only 4 · 2 n − 3n − 8 of them are essential. • Mappings R ij are involutions i.e. R 2 ij = id. • Mappings R ij are anti-measure preserving with respect to the measures m 1 = n 1 d 2 , m 2 = n 2 d 1 .
We take a stand here to comment that for k = 3 the construction above coincides with the Adler's triad family of maps hence we have Liouville integrability. For k > 3 we have a generalisation of the later and since always we will have maps in k variables with 2 invariants, Liouville integrability is not expected for generic choice of the parameters α i j1,j2,...,j k , β i j1,j2,...,j k . For a specific but quite general choice of the parameters though, one can associate a Lax pair to these maps and recover the additional integrals which are required for the Liouville integrability to emerge.
We also have to note that the case k = 4 was firstly introduced in [48]. Although for k = 4 we have mappings in 4 variables with 2 invariants, Liouville integrability is not apparent unless we specify the parameters. A specific choice of the parameters which leads to integrability is presented to the following example.
Example 2.4 (The Adler-Yamilov map [7]). Consider the following special form of the functions n i , d i Then the functions H i = n i /d i , i = 1, 2 are preserved by construction by the maps R ij as well as by the following elementary involutions: The Adler-Yamilov map (ξ) is considered by the following composition: The Adler-Yamilov map is Liouville integrable since it preserves, and the invariants H 1 , H 2 are in involution with respect to the canonical Poisson Bracket. For further discussions on the Adler-Yamilov map see [49,30].

Invariants in separated variables and Yang-Baxter maps
Mappings R mn , m < n ∈ {1, 2, . . . , k}, presented in subsection 2.1, satisfy the identities R ij R il R jl = R jl R il R ij , nevertheless as they stand they are not Yang-Baxter. Take for example the map R 12 : (x 1 , x 2 , x 3 , . . . , x k ) → (X 1 , X 2 , x 3 , . . . , x k ). The formulae for X 1 is fraction linear in x 1 with coefficients that depend on all the remaining variables and X 2 is fraction linear in x 2 with coefficients that depend on all the remaining variables. In order for R 12 to be a Yang-Baxter map the coefficients of x 1 in the formulae of X 1 should depend only on x 2 and the coefficients of x 2 in the formulae of X 2 should depend only on x 1 . This "separability" requirement can be easily achieved by requiring separability of variables on the level of the invariants of the map R 12 . We have two invariants H 1 = n 1 /d 1 , H 2 = n 2 /d 2 , so we can have three different kinds of separability. (I) Both H 1 and H 2 to be multiplicative separable on the variables x 1 and x 2 . (II) H 1 to be multiplicative and H 2 to be additive separable and finally (III) both H 1 and H 2 to be additive separable on the variables x 1 and x 2 . In what follows we explicitly present these three different kinds of separability in all variables of the invariants H 1 and H 2 .
(III) Additive/additive separability of variables. Where . . k, parameters, 8k in total. In all three cases above, the number of essential parameters is 3k − 6. This argument can be proven by the following reasoning. Since the invariants H 1 , H 2 depends on k variables, by a Möbius transformation on each of the k variables 3k parameters can be removed. Also any Möbius transformation of an invariant remains an invariant so since we have two invariants 2×3 more parameters can be removed. Finally, for each one of the 2k in number functions ai−bixi ci−dixi , Ai−Bixi Ci−Dixi , i = 1, . . . k, one non-zero parameter can be absorbed simply by dividing with it (and reparametrise), so 2k in number more parameters can be removed. In total we have 8k − 3k − 2 × 3 − 2k = 3k − 6 essential parameters.
3.0.1. Multiplicative/multiplicative separability of variables. Let us first introduce some definitions.
. . , 4 known polynomials of v and u respectively, will be said to be of subclass [γ : δ], if the highest degree that appears in the polynomials a i is γ and the higher degree that appears in the polynomials b i is δ.
Clearly, maps that belong to different subclasses are not (Möb) 2 equivalent. Proposition 3.3. Consider the multiplicative/multiplicative separability of variables of the invariants H 1 and H 2 (see (12)). Consider also the following sets of parameters and the functions The following holds: , etc. Note that in the expressions of X i , X j appears only the coordinates x i , x j and the parameters p ij . From further on we denote the maps R ij as R p ij ij , in order to stress this separability feature.
(3) Mappings R ij are anti-measure preserving with respect to the measures m 1 = n 1 d 2 , m 2 = n 2 d 1 , where n i , d i the numerators and the denominators respectively, of the invariants ij are involutions with the sets of singularities and the sets of fixed points where in the formulae for P m ij and Q m ij , m = 1, . . . , 4, we have suppressed the dependency on the remaining variables. For example, with P 1 ij = ai bi , cj dj we denote Note that the values of the invariants H i at the singular points P m ij are undetermined i.e. H 1 (P m ij ) = 0 0 , m = 1, 2, after a re-parametrization mappings R ij gets exactly the form of the H I map. Here, with CR[a, b, c, d] we denote the cross-ratio of 4 points, namely Each one of the maps R ij has a set of singularities which consists of 4 distinct points. With appropriate limits we are allowed to merge some of the singularities and obtain Yang-Baxter maps which are not (Möb) 2 equivalent with the original one.
By setting C i = ǫA i , D i = ǫB i , A j = ǫC j , B j = ǫD j and letting ǫ → 0 the singular points P 4 ij and P 3 ij merge. The resulting maps, under a re-parametrization, coincide with the ones obtained in the multiplicative/additive case (see subsection 3.0.2), hence are (Möb) 2 equivalent with the H II Yang-Baxter map. The same result can be obtained by merging P 2 ij and P 1 ij . Note that merging P 4 ij with P 2 ij or P 4 ij with P 1 ij is not of interest since the resulting maps are trivial.
By further setting c i = ǫa i , d i = ǫb i , a j = ǫc j , b j = ǫd j and letting ǫ → 0 the singular points P 2 ij and P 1 ij merge as well. The resulting maps, under a re-parametrization, coincide with the ones obtained in the additive/additive case (see subsection 3.0.3), hence are (Möb) 2 equivalent with the H A III Yang-Baxter map. Any further merging of singularities leads to trivial maps. ij are more general than the H I map since they include degenerate cases as well. In the same respect Q V [72], the rational version of the discrete Krichever-Novikov equation Q 4 [3], is more general.
Example 3.6 (k = 3). For k = 3, the invariants H 1 = f 1 f 2 f 3 , H 2 = g 1 g 2 g 3 are functions of 3 variables with 24 parameters, 3 of them are essential. Without loss of generality, after removing the redundancy of the parameters, the invariants H 1 , H 2 can be cast into the form: Then each of the mappings R ij , i = j ∈ {1, 2, 3} is exactly the H I Yang-Baxter map. The H I Yang-Baxter map explicitly reads: The maps φ i and ψ i have a special role in [59] since though them the H I map was derived out of the F I Yang-Baxter map. We will discuss more about these maps in the next Section. We just quickly recall that φ 1 R 12 φ 2 is exactly the F I Yang-Baxter map.
Remark 3.7. We have to remark that with loss of generality, mappings R ij can belong on a different subclasses than the [2 : 2] subclass of maps that the H I map belongs to. For example, for R 12 is the Hirota's KdV map (see [44]) that belongs on the subclass The Hirota's KdV map entwines with S 13 and T 23 , since R 12 S 13 T 23 = T 23 S 13 R 12 holds.
Example 3.8 (k ≥ 4). For k = 4 the invariants depend on 32 parameters and only 6 of them are essential. Without loss of generality they can be cast into the form For k > 4 the invariants depend on 8k parameters and only 3k − 6 of them are essential. Without loss of generality they can be cast into the form 3.0.2. Multiplicative/additive separability of variables.
Proposition 3.9. Consider the multiplicative/additive separability of variables of the invariants H 1 and H 2 (see (13)). Consider also the following sets of parameters and the functions The following holds: gi depend on 8k parameters. Only 3k−6 of them are essential. (2) Mappings R ij explicitly read: where X l = x l ∀ l = i, j and X i , X j are given by the formulae , etc. Note that in the expressions of X i , X j appears only the coordinates x i , x j and the parameters p ij . From further on we denote the maps R ij as R p ij ij , in order to stress this separability feature.
(3) Mappings R ij are anti-measure preserving with respect to the measures m 1 = n 1 d 2 , m 2 = n 2 d 1 , where n i , d i the numerators and the denominators respectively, of the invariants H i , i = 1, 2.
ij are involutions with the sets of singularities where the superscript 2 in P 3 ij denotes that these singular points appears with multiplicity 2. In the formulae for P m ij , m = 1, . . . , 3, we have suppressed the dependency on the remaining variables. For example, with P 1 ij = ai bi , cj dj we denote x 1 , . . . , x i−1 , ai bi , x i+1 , . . . , x j−1 , cj dj , x j+1 , . . . , x k and similarly for the remaining P m ij .
Then each of the mappings R ij , i = j ∈ {1, 2, 3} is exactly the H II Yang-Baxter map. For k > 3 the invariants depend on 8k parameters and only 3k − 6 of them are essential. Without loss of generality they can be cast into the form 3.0.3. Additive/additive separability of variables.
Proposition 3.11. Consider the additive/additive separability of variables of the invariants H 1 and H 2 (see (14)). Consider also the following sets of parameters and the functions The following holds: gi depend on 8k parameters. Only 3k−6 of them are essential. (2) Mappings R ij explicitly read: where X l = x l ∀ l = i, j and X i , X j are given by the formulae , etc. Note that in the expressions of X i , X j appears only the coordinates x i , x j and the parameters p ij . From further on we denote the maps R ij as R p ij ij , in order to stress this separability feature.
(3) Mappings R ij are anti-measure preserving with respect to the measures m 1 = n 1 d 2 , m 2 = n 2 d 1 , where n i , d i the numerators and the denominators respectively, of the invariants H i , i = 1, 2.
ij are involutions with the sets of singularities where the superscript 2 in P 1 ij and P 2 ij denotes that these singular points appears with multiplicity 2. In the formulae for P m ij , m = 1, . . . , 2, we have suppressed the dependency on the remaining variables. For example, with P 1 Example 3.12 (k ≥ 3). For k = 3, the invariants H 1 = f 1 + f 2 + f 3 , H 2 = g 1 + g 2 + g 3 are functions of 3 variables with 24 parameters, 3 of them are essential. Without loss of generality, after removing the redundancy of the parameters, the invariants H 1 , H 2 can be cast into the form: Then each of the mappings R ij , i = j ∈ {1, 2, 3} is exactly the H A III Yang-Baxter map. For k > 3 the invariants depend on 8k parameters and only 3k − 6 of them are essential. Without loss of generality they can be cast into the form

Entwining Yang-Baxter maps
Following [52], three different maps S, T, U are called entwining Yang-Baxter maps if they satisfy: We consider two maps to be different if they are not (Möb) 2 equivalent. Hence, in order to ensure that we have different maps we require that at least one of the maps S, T, U either belongs to a different subclass than the remaining ones or it has different singularity pattern (even if it belongs to the same subclass with the remaining ones) or it has different periodicity. In what follows we present two methods to obtain entwining maps. The first one is based on degeneracy i.e. we construct maps which belong to different subclasses and we obtain entwining maps associated with the H I , H II and H A III families of maps. The second one is based on the symmetries of the H−list of Yang-Baxter maps and we obtain entwining maps for all members of the H−list.

4.1.
Degeneracy and entwining Yang-Baxter maps. In subsection 3.0.1 it was shown that for k = 3 and for the multiplicative/multiplicative case, the invariants H 1 , H 2 depend on 3 essential parameters. Without loss of generality they read The associated maps R 12 , R 13 and R 23 which preserve the invariants have exactly the form of the H I map. In order to obtain entwining maps associated with the H I map, we consider: For these invariants, R 12 is exactly the H I map and for generic α 3 , β 3 , γ 3 mappings R 13 and R 23 are (Möb) 2 equivalent to the H I . In order to obtain entwining maps we need to violate this (Möb) 2 equivalency of the maps R 13 and R 23 with the H I map. This is achieved by violating the generality, f.i. setting α 3 = 0 or β 3 = 0, the maps R 13 and R 23 , belongs to different subclasses than the H I map does. Working similarly for the H II map we find 1 family of maps which entwine with the later without being (Möb) 2 equivalent. Finally, for H A III we find also 1 family of entwining maps which are not (Möb) 2 equivalent with the later. Our results are presented in Propositions 4.1 − 4.3.  Table 1 according to   Table 1. Entwining maps associated with the H I Yang-Baxter map through degeneracy the entwining relation Proof. Starting with the invariants the map R 12 is exactly the H I map. By setting a = 0, R 13 and R 23 takes the form of e a H I of according to the entwining relation where S 12 is the H II map acting on the (1, 2)−coordinates, T 13 and T 23 are e b H II acting on (1, 3) and (2, 3) coordinates respectively.

Proof. Starting with the invariants
where φ 1 is the involution that acts as φ to the first factor of the cartesian product CP 1 × CP 1 and φ 2 is the involution that acts as φ to the second factor of the cartesian product.
Let m < n ∈ {1, . . . , k}, k ≥ 3 fixed. A direct consequence of the previous definition is that if φ is a symmetry of the Yang-Baxter map R, then the map φ m R mn φ n is a new Yang-Baxter map since it is not (Möb) 2 equivalent with R mn . By finding the symmetries of the F −list of Yang-Baxter maps, the authors of [59] apart the Yang-Baxter relation that holds, only the following three entwining relations holds Proof. To show that only the entwining relations (18), (19), (20) holds, we start with since φ 1 commutes with R 23 and the Yang-Baxter relation R 12 R 13 R 23 = R 23 R 13 R 12 holds. But due to the symmetry we have R 12 φ 1 φ 2 = φ 1 φ 2 R 12 so (21) reads: and that completes the proof that (18) holds. For the remaining relations we work similarly for their proof.
In Table 4, we present the entwining maps S, T, U that correspond to the entwining relations (18)− (20), where R is any Yang-Baxter map. In what follows, we specify R to be any member of the H−list 1 of quadrirational Yang-Baxter maps.
where α a complex parameter, are symmetries for the H I map (see [59]), since it holds where R 12 is the H I map acting on the 12−coordinates and Note that the symmetries φ and τ can be derived from our considerations (see example 3.6) since for k = 3 it holds Remark 4.6. By using similar arguments as in the proof of the Theorem 4.5, entwining relations where the symmetries φ and ψ of the H I map interlace do not exist i.e. it does not exists for example any relation of entwining type (φ i , φ j , ψ k ).
In Table 5 we present the entwining maps associated with the H I map which are generated by using the symmetries φ and ψ. In Table 5 it appears the H I map, the companion of the H I map that is denoted as cH I , as well ascF I which is the companion map of the mapF I that was derived in [59]. We also have four novel maps which are not (Möb) 2 equivalent to H I , which we refer to as Φ a I , Φ b I , Ψ a I and Ψ b I . In the proposition that follows we present their explicit form.
entwine with the H I Yang-Baxter map according to the entwining relations of Table 5.

Entwining maps associated with the H II Yang-Baxter map. The invariants
generate the maps R ij , i < j ∈ {1, 2, 3} which are exactly the H II map acting on the (ij)−coordinates.
Explicitly the H II map reads A symmetry of the H II map is φ : u → α − u, since it holds φ 1 φ 2 R 12 = R 12 φ 1 φ 2 , where R 12 is the H II map acting on the (12)−coordinates and Table 6. Entwining maps S, T, U associated with H II Yang-Baxter map using the symmetry φ.
Entwining type entwine with the H II Yang-Baxter map according to the entwining relations of Table 6.
The map cH II denotes the companion map of the H II map.

4.2.3.
Entwining maps associated with the H A III Yang-Baxter map. The invariants generate the maps R ij , i < j ∈ {1, 2, 3} which are exactly the H A III map acting on the (ij)−coordinates. Explicitly the H A III map reads III map acting on the (12)−coordinates and Note that the map φ 1 R 12 φ 2 is exactly the H B III Yang-Baxter map. Proposition 4.9. The following non-periodic maps (u, v) → (U, V ) where entwine with the H A III Yang-Baxter map according to the entwining relations of Table 7.
The map cH A III denotes the companion map of the H A III map and withĤ A III we denote a (Möb) 2 equivalent map to the H A III .

4.2.4.
Entwining maps associated with the H B III Yang-Baxter map. The invariants that were derived in [45,44,56,47], generate the maps R ij , i < j ∈ {1, 2, 3} which are exactly the H B III map acting on the (ij)−coordinates. Explicitly the H B III map reads The symmetries φ, ψ of the H A III map are symmetries of H B III as well. Proposition 4.10. The following non-periodic maps (u, v) → (U, V ), where entwine with the H B III Yang-Baxter map according to the entwining relations of Table 8.
The mapsĤ B III ,H B III that appear in Table 8 [45,44,56,47], The involution ψ : u → −u is a symmetry of the H V map.
entwine with the H V Yang-Baxter map according to the entwining relations of Table 9. Table 9. Entwining maps S, T, U associated with H V Yang-Baxter map using the symmetry ψ Entwining type S 12 T 13 U 23 The map cH V denotes the companion map of the H V map.

Transfer maps
The notion of transfer maps associated with Yang-Baxter maps was introduced by Veselov in [71]. In [68] dynamical aspects of the later were discussed. The transfer maps associated with any reversible Yang-Baxter map are defined as where the indices are considered modulo k. There is: For example for k = 4 we have T (1) they preserve the invariants H 1 , H 2 , presented in the propositions 3.3, 3.9, 3.11 (2) for k = 2n + 1 they preserve the measures given in the propositions 3.3, 3.9, 3.11 (3) for k = 2n they anti-preserve the measures given in the propositions 3.3, 3.9, 3.11 (4) they possess Lax pairs (5) for generic values of the parameter sets p ij , are equivalent by conjugation to the transfer maps associated with H I , H II and H A III Yang-Baxter maps respectively (6) for non-generic values of the parameter sets p ij , we have novel transfer maps Proof. The statements (1) − (3) have already been proven (see propositions 2.1, 3.3, 3.9, 3.11). As for the statement (4), one can construct a Lax matrix for the Yang-Baxter map R following [66]. Then the Lax equations associated with the transfer maps T (k) i , correspond to certain factorizations of the monodromy matrix (see [71]). We will show the statement (5) for the transfer maps associated with R Then the mapsR ij µ i µ j are exactly the H I map acting on the (ij)−coordinates (see proposition 3.3). For the transfer mapT Note that we have omitted the parameter sets p ij that the maps depends on for simplicity. (6). For non-generic choice of the parameter sets p ij , the conjugation equivalence (22) does not holds.
Remark 5.2. Note that π 0 = π 12 π 13 . . . π 1k and the maps π 0 , π ij ∀ i, j ∈ {1, . . . , k}, preserve the invariants H 1 , H 2 of the propositions 3.3, 3.9, 3.11. Moreover, the maps S i := π ii+1 R ii+1 , i ∈ {1, . . . , k}, also preserve the invariants H 1 , H 2 . The following relations holds The group g =< π 0 , S 1 , S 2 , . . . , S k > generated by these maps provides a bi-rational realization of the extended Weyl group of type A Since π 12 R p12 12 : L(x k , p k ; λ)L(x k−1 , p k−1 ; λ) . . . L(x 2 , p 2 ; λ)L(x 1 , p 1 ; λ) → L(x k , p k ; λ)L(x k−1 , p k−1 ; λ) . . . L(x 2 , p 2 ; λ)L(x 1 , p 1 ; λ), and π 0 : L(x k , p k ; λ)L(x k−1 , p k−1 ; λ) . . . L(x 2 , p 2 ; λ)L(x 1 , p 1 ; λ) → L(x 1 , p 1 ; λ)L(x k , p k ; λ) . . . L(x 3 , p 3 ; λ)L(x 2 , p 2 ; λ), So the map t (k) 1 has the following Lax equation But the map t (k) 1 acts on the parameter sets p i as follows t (k) 1 : (p 1 , . . . , p k ) → (P 1 , . . . , P k ), where P 1 = p 1 , P k = p 2 and ∀i = 1, k P i = p i+1 , that is periodic with period k − 1, so the Lax equation of the map (t Proof. Let us first prove that t where we have the composition of m in number expressions of the form π 0 S i π 0 S i+1 , and for each one of them (using Remark 5.2) it holds π 0 S i π 0 S i+1 = π 0 S 2 i π 0 = π 2 0 . So Let us now prove that (t We have: where we have used the fact that Remark 5.5. Note that for k odd, it holds the more general condition Here, we associate k−point recurrences with the maps t . Let us first introduce the shift operator T as follows: The maps t and t HV (k) 2 , explicitly read: (x 1 , . . . , x k ; p 1 , . . . , p k ) → (T x 1 , . . . , T x k ; T p 1 , . . . , T p k ), and the indices are considered modulo k. Clearly we have, x 3 = T 2−k x 1 , p 3 = T 2−k p 1 , So we obtain: Adding the first two equations from above we get the following invariance condition 4 So it is guaranteed the existence of a potential function f such that In terms of f, (24) becomes the following (k + 1)−point recurrence In terms of a new variable h defined as h := λ + (T 1 − T 0 )f, there is, where we chose λ = 2c k−2 to simplify the formulae. , in terms of the corresponding variables h defined in Table 12, get the form of the following k−point recurrences and for each recurrence presented above we have that the parameters vary as follows: T p 2 = p 2 , T k−1 p 1 = p 1 . So p 2 is constant and p 1 is periodic with period k − 1.
Remark 5.8. As for the recurrencesrt H A III (k) i ,rt HII (k) i , one could consider T k−1 p 1 = p 1 + (k − 1)a and forrt HI (k) i T k−1 p 1 = p 1 a k−1 , in order to de-autonomise them. We anticipate that this is a proper de-autonomisation, although we have no proof yet. The finding of the Poisson structures that the later recurrences we anticipate that preserve, will sort this issue out. Remark 5.9. As a final remark, we note that the k−point recurrences associated with the extended transfer maps of the Yang-Baxter map F V , are exactly the same as the k−point recurrences associated with the extended transfer maps of the Yang-Baxter map H V which (one of them) were presented in Corollary 5.7. Since the (k − 1)−iteration of the extended transfer maps of any Yang-Baxter map coincides with its transfer maps, we conclude that the dynamics of the transfer maps of the Yang-Baxter maps F V and H V , are the same. The same holds true for the transfer maps associated with the Yang-Baxter maps F III and H A III . As for the remaining members of the F and the H lists of Yang-Baxter maps, further investigation is required in order to prove the equivalence of their transfer dynamics.

Conclusions
In Section 2 we have presented a family of maps in k variables which preserve 2 rational invariants of a specific form. One could mimic the procedures introduced in [29] to obtain rational maps in k variables which preserve m rational invariants where m < k. For example, there are 2k k rational maps (x 1 , . . . , x k , y 1 , . . . , y k ) → (X 1 , . . . , X k , Y 1 , . . . , Y k ) which preserve k invariants of the form: where the indices are considered modulo k and α i , β i , κ i , λ i , etc. are given functions of the variables y i , y i+1 . If separability of variables on the invariants is imposed, then higher rank analogues of the Yang-Baxter maps of propositions 3.3, 3.9 and 3.11 are expected. Moreover, solutions of the functional tetrahedron equation [50,42,43,64], or even of higher simplex equations [54,55,17] are anticipated. For example if we consider the following, different than the (32), choice of invariants: They are exactly the Hirota's map [42,43,64], i.e. the map R : (u, v, w) → (U, V, W ), where U = uv u + w , V = u + w, W = vw u + w , acting on (123), (145), (246) and (356) coordinates respectively. For the involution φ : u → −u, it holds φ 1 φ 2 φ 3 R 123 = R 123 φ 1 φ 2 φ 3 . So φ is a symmetry of the Hirota's map R and it can be easily proven that the following entwining relation holds: Hence we have obtained a solution of the following entwining functional tetrahedron relation The complete set of entwining relations and maps associated with the Hirota's map as well as with the Hirota-Miwa's map, will be considered elsewhere.
In Section 4, we considered two methods to obtain entwining maps. The first method uses degeneracy arguments and produces entwining maps associated with the H I , H II and H A III Yang-Baxter maps. The entwining maps of this method belongs to different subclasses than the [2 : 2] subclass of maps that the H I , H II and H A III Yang-Baxter maps belongs to so they are not (Möb) 2 equivalent to the later. The outcomes of the second method are non-periodic 6 entwining maps of subclass [2 : 2] associated with the whole H−list. The fact that the entwining maps which were presented in this Section preserve two invariants in separated variables, enable us to introduce appropriate potentials (as shown in [45,44,56]) to obtain integrable lattice equations. Actually we obtain integrable triplets of lattice equations (in some cases even correspondences). Note that integrable triplets of lattice equations were systematically derived in [13] and more recently in [34]. We plan to consider the integrable triplets of lattice equations derived from entwining maps, elsewhere. In Section 6, we have proved that the transfer maps associated with the H list of Yang-Baxter maps can be considered as the (k−1)-iteration of some maps of simpler form. As a consequence of this re-factorisation we have obtained (k+1)−point (see proposition 5.6) and k−point (see corollary 5.7) alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations. Moreover, the autonomous versions of some of the k−point recurrences presented in corollary 5.7, can be obtained by periodic reductions ( [58], c.f. [35]) of integrable lattice equations. Here we have obtained alternating k−point recurrences from Yang-Baxter maps without performing periodic reductions. Hence, our results might be compared/extended to the novel and independent frameworks introduced in [8,10] and [39,40], where by using symmetry arguments, integrable lattice equations and discrete Painlevé equations of 2nd order were linked.