Bi-rational maps in four dimensions with two invariants

TL;DR

This paper introduces a class of four-dimensional bi-rational maps with two invariants, analyzing their integrability through degree growth and Liouville theory, contributing to the understanding of complex dynamical systems.

Contribution

It presents a new class of four-dimensional bi-rational maps with specific invariants and explores their integrability properties, advancing the study of higher-dimensional discrete integrable systems.

Findings

Maps exhibit controlled degree growth indicating integrability

Liouville integrability confirmed for the proposed maps

Provides new examples of higher-dimensional integrable maps

Abstract

In this paper we present a class of four-dimensional bi-rational maps with two invariants satisfying certain constraints on degrees. We discuss the integrability properties of these maps from the point of view of degree growth and Liouville integrability.

Tables

Table 1. Table 1. Integrability properties of the (P,Q) maps.

Equation	Degree pattern of invariants	Degree of growth	Variational
(P.i)*	(1,3,3,1), (2,4,4,2)	cubic	no
(Q.i)*	(1,2,2,1), (2,4,4,2)	cubic	no
(P.ii)	(1,2,2,1), (1,3,3,1)	exponential	no
(Q.ii)	(2,4,4,2)	exponential	no
(P.iii)	(1,3,3,1), (2,4,4,2), (2,5,5,2)	quadratic	no
(Q.iii)	(1,2,2,1), (2,4,4,2)	exponential	no
(P.iv) ($d{P}_{\text{I}}^{\left(2\right)}$)	(1,3,3,1), (2,4,4,2)	quadratic	yes
(Q.iv)	(1,2,2,1), (1,2,2,1), (2,4,4,2)	quadratic	no
(P.v)	(1,3,3,1), (2,4,4,2)	quadratic	yes
(Q.v)	(1,2,2,1), (1,2,2,1), (2,4,4,2)	quadratic	no
(P.vi) ($d{P}_{\text{II}}^{\left(2\right)}$)	(1,3,3,1), (2,4,4,2)	quadratic	yes
(Q.vi)	(1,2,2,1), (1,2,2,1), (2,4,4,2)	quadratic	no
* Deflatable to a three-dimensional Liouville integrable map [21].

Equations

$$\phi :[\mathbf{x}]\in {\mathbb{CP}}^n\to [{\mathbf{x}}^{\prime }]\in {\mathbb{CP}}^n,$$

$$w_{k+n}=f\left(w_k,\dots ,w_{k+n-1}\right).$$

$$\varpi :\mathbf{w}\in {\mathbb{C}}^n\to {\mathbf{w}}^{\prime }\in {\mathbb{C}}^n,$$

$${\mathbf{w}}^{\prime }=\left(f\left(\mathbf{w}\right),w_n,w_{n-1},\dots ,w_1\right).$$

$${\phi }^{\ast }\left(I\right)=I,$$

$$\frac{\partial I}{\partial x_1}\frac{\partial I}{\partial x_n}\ne 0.$$

$$I\left([\mathbf{x}]\right)=\frac{I^{\prime }\left([\mathbf{x}]\right)}{x_{n+1}^d},d=\mathrm{deg}{I}^{\prime }\left([\mathbf{x}]\right),$$

$$\mathrm{dp}F=\left({\mathrm{deg}}_{x_1}F,{\mathrm{deg}}_{x_2}F,\dots ,{\mathrm{deg}}_{x_n}F\right).$$

$$\iota :\left[x_1:x_2:\cdots :x_n:x_{n+1}\right]\to \left[x_n:x_{n-1}:\cdots :x_1:x_{n+1}\right],$$

$$I_j={\kappa }_j,$$

$$\sum _{l=1}^n\left(J_{li}\frac{\partial J_{jk}}{\partial w_{l-1}}+J_{lj}\frac{\partial J_{ki}}{\partial w_{l-1}}+J_{lk}\frac{\partial J_{ij}}{\partial w_{l-1}}\right)=0,\forall i,j,k.$$

$$\{f,g\}=\nabla fJ\left(\mathbf{w}\right)\nabla g^T,$$

$$d\phi J(\mathbf{w})d{\phi }^T=J({\mathbf{w}}^{\prime }),$$

$$\sum _{i=0}^N\frac{\partial L}{\partial w_k}\left(w_{k+N-i},\dots ,w_{k-i}\right)=0.$$

$$\frac{{\partial }^{2}L}{\partial w_k\partial w_{k+N}}\ne 0.$$

$$\mathcal{O}:\mathbf{w}\to \left(\mathbf{q},\mathbf{p}\right)$$

$$q_i$$

$$p_i$$

$$q_i^{\prime }$$

$$q_N^{\prime }$$

$$p_i^{\prime }$$

$$p_N^{\prime }$$

$$p_1=-\frac{\partial L}{\partial q_1}(\mathbf{q},q_N^{\prime }),$$

$$\Omega =\left(\begin{array}{cc}{\mathbb{O}}_N& {\mathbb{I}}_N\\ -{\mathbb{I}}_N& {\mathbb{O}}_N\end{array}\right),$$

$$J\left(\mathbf{w}\right)=d{\mathcal{O}}^{-1}{\Omega }^{-1}(d{\mathcal{O}}^{-1}{)}^T,$$

$$\epsilon =\underset{k\to \infty }{\lim }\frac{1}{k}\log {\mathrm{deg}}_{[{\mathbf{x}}_0]}{\phi }^k,$$

$$d_k={\mathrm{deg}}_{[{\mathbf{x}}_0]}{\phi }^k,k\in \mathbb{N}.$$

$$g\left(z\right)=\sum _{n=0}^{\infty }{d}_k{z}^k.$$

$$H={\alpha }_1{I}_1+\cdots +{\alpha }_L{I}_L.$$

$$H=P_d\left(I_1,I_2,\dots ,I_L\right)$$

$$\left[x_1:x_2:\cdots :x_{n+1}\right]\mapsto \left[x_1^{\prime }:x_2^{\prime }:\cdots :x_{n+1}^{\prime }\right]=\left[x_1^{\prime }:x_1:\cdots :x_{n+1}\right]$$

$$\hat{H}(x_1^{\prime },[\mathbf{x}])=H\left([{\mathbf{x}}^{\prime }]\right)-H\left([\mathbf{x}]\right)=0.$$

Full text

1. Introduction

In this paper we classify maps of $\mathbb{CP}^{4}$ to itself, which possess two polynomial invariants, under certain conditions. The outcomes of our classification include eight new classes of maps, which have surprising properties. Despite the existence of two invariants, there turn out to be non-integrable cases, with exponential growth. Other cases are integrable, with cubic and quadratic growth. The cases of cubic growth are only possible in dimension greater than two [11, 12]. We discuss the geometric properties of these systems [4].

In dimension two it is well-known that integrable bi-rational maps can be characterized by the existence of a rational invariant. For instance most of the integrable maps on the plane fall in the class of QRT maps [33, 34], even though there are some notable exceptions [45, 13, 36]. The integrability of these maps can be explained geometrically and has led to many interesting developments [37, 13, 41].

In higher dimension an analogous general framework does not exist. In particular, for mappings in four dimension, a generalisation of the QRT class [33, 34] was given in [7]. However, this generalisation does not cover all possible integrable maps in four dimensions. Indeed, some of the new maps obtained in [7] turn out to be autonomous versions of Painlevé hierarchies [22] which are multiplicative equations in Sakai’s scheme [37]. On the other hand, there exists hierarchies of additive discrete Painlevé equations too [10]. Equations coming from the hierarchies of additive Painlevé equations are naturally outside the framework of [7]. Other examples of four-dimensional maps falling outside the class presented in [7] are given in [25, 21, 31, 9, 8, 30].

Our starting point is [25], where the authors considered the autonomous limit of the second member of the $\mathrm{d}P_{\text{I}}$ and $\mathrm{d}P_{\text{II}}$ hierarchies [10]. We will denote these equations as $\mathrm{d}P_{\text{I}}^{(2)}$ and $\mathrm{d}P_{\text{II}}^{(2)}$ equations. These $\mathrm{d}P_{\text{I}}^{(2)}$ and $\mathrm{d}P_{\text{II}}^{(2)}$ equations are given by recurrence relations of order four, and shown to be integrable according to the algebraic entropy approach. Therein the authors showed that both maps possess two polynomial invariants. Using these invariants, they produced the dual maps of the $\mathrm{d}P_{\text{I}}^{(2)}$ and $\mathrm{d}P_{\text{II}}^{(2)}$ equations in the sense of [35]. Moreover, they showed that these dual maps are integrable according to the algebraic entropy test and also possess invariants. In fact, the number of invariants showed that the dual maps are actually superintegrable. Finally they gave a scheme to construct recurrence relations of an assigned form. Using this scheme in [25] some new examples, with no classification purposed were presented. Starting from these considerations in this paper we consider and solve the problem of finding all fourth-order bi-rational maps possessing two polynomial invariants of general enough form to contain those of the $\mathrm{d}P_{\text{I}}^{(2)}$ and $\mathrm{d}P_{\text{II}}^{(2)}$ equations.

The structure of the paper is the following: in section 2 we give a concise explanation of the background material we need. In particular we discuss the various definitions of integrability for mapping we are going to use throughout the paper. In section 3 we present the motivations for our search and we present our the search method and we state the general result. In section 4 we give the explicit form of the maps we derived with the method of section 3 and we discuss their integrability properties following the discussion of section 2. Finally, in section 5 we make some general comments on the maps we obtained, and we underline the possible future development.

2. Setting

In this Section we give the fundamental definitions we need to explain how our list of equations is found and what kind of integrability we are going to consider within this paper. One can also find this setting in our shot communication [21].

2.1. Bi-rational maps and invariants

The main subject of this paper are bi-rational maps of the complex projective space into itself:

[TABLE]

where $n>1$111Bi-rational maps in $\mathbb{CP}^{1}$ are just Möbius transformations so everything is trivial. and $\mathbf{[x]}=\left[x_{1}:x_{2}:\dots:x_{n+1}\right]$ and $\mathbf{[x^{\prime}]}=\left[x_{1}^{\prime}:x_{2}^{\prime}:\dots:x_{n+1}^{\prime}\right]$ to be homogeneous coordinates on $\mathbb{CP}^{n}$. Moreover, we recall that a bi-rational map is a rational map $\varphi\colon V\to W$ of algebraic varieties $V$ and $W$ such that there exists a rational map $\psi\colon W\to V$, which is the inverse of $\varphi$ in the dense subset where both maps are defined [38].

Bi-rational maps of the form (1) are the natural mathematical object needed to study autonomous single-valued invertible rational recurrence relations. Indeed, an autonomous recurrence relation of order $n$ is a relation where the $(n+1)$-th element of a sequence is defined in terms of the preceding $n$, i.e. an expression of the form:

[TABLE]

A recurrence relation is autonomous if the function $f$ in (2) does not depend explicitly on $k$. Moreover, we say that a recurrence relation is rational if the function $f$ in (2) is a rational function. Finally, the recurrence is invertible and single-valued if equation (2) is solvable uniquely with respect to $w_{0}$. All the terms of the sequence $w_{k}$ for $k>n$ are then obtained by iterated substitution of the previous one into equation (2). For this reason it is possible to interpret the recurrence relation (2) as a map of the complex space of dimension $n$ into itself as:

[TABLE]

where $\mathbf{w}=\left(w_{n},w_{n-1},\dots,w_{0}\right)$ are the initial conditions and the map acts as:

[TABLE]

The recurrence relation (2) is then given by the repeated application of the map $\varpi$, namely $w_{n+k}$ is the first component of $\varpi^{k}$. Interpreting the coordinates $\mathbf{w}\in\mathbb{C}^{n}$ as an affine chart in $\mathbb{CP}^{n}$, i.e. assuming that $\left(w_{n-1},\dots,w_{0}\right)=\left[w_{n-1}:\dots:w_{0}:1\right]$ we have that the map (3) can be brought to a bi-rational map of $\mathbb{CP}^{n}$ into itself of the form (1).

Throughout the paper we will often make use of the correspondence between bi-rational maps and recurrence relations. This is due to the fact that some definitions are easier to state and use in the projective setting, while others are easier to state and use in the affine one. In any case for us “bi-rational map” and “recurrence relation” will be completely equivalent terms.

One characteristic of integrability is the existence of first integrals. In the continuous context, for finite dimensional systems, integrability refers to the existence of a “sufficiently” high number of first integrals, i.e. of non-trivial functions constant along the solution of the differential system. In particular, for a Hamiltonian system, the number of first integrals is less as its integrability was given by Liouville [27]. In discrete setting, the analogue of first integrals for maps are the invariants which is defined as follows.

Definition 1**.**

An invariant of a bi-rational map $\varphi\colon\mathbb{CP}^{n}\to\mathbb{CP}^{n}$ is a homogeneous function $I\colon\mathbb{CP}^{n}\to\mathbb{C}$ such that it is preserved under the action of the map, i.e.

[TABLE]

where $\varphi^{*}\left(I\right)$ means the pullback of $I$ through the map $\varphi$, i.e. $\varphi^{*}\left(I\right)=I\left(\varphi\left(\mathbf{[x]}\right)\right)$. For $n>1$, an invariant is said to be non-degenerate if:

[TABLE]

Otherwise an invariant is said to be degenerate.

In what follows we will concentrate on a particular class of invariants:

Definition 2**.**

An invariant $I$ is said to be polynomial, if in the affine chart $\left[x_{1}:\dots:x_{n}:1\right]$ the function $I$ is a polynomial function.

In definition 2 we use $x_{n+1}$ as homogenising variable to go from an affine (polynomial) form to a projective (rational) form of the invariants. A polynomial invariant in the sense of definition 2 written in homogeneous variables is always a rational function homogeneous of degree 0. The form of the polynomial invariant in homogeneous coordinates is then given by:

[TABLE]

where $\deg$ is the total degree.

To better characterize the properties of these invariants we introduce the following:

Definition 3**.**

Given a polynomial function $F\colon\mathbb{CP}^{n}\to V$, where $V$ can be either $\mathbb{CP}^{n}$ or $\mathbb{C}$, we define the degree pattern of $F$ to be:

[TABLE]

Finally we will consider invariants which are not of generic shape, but satisfy the following condition:

Definition 4**.**

We say that a function $I\colon\mathbb{CP}^{n}\to\mathbb{C}$ is symmetric if it is invariant under the following involution:

[TABLE]

i.e. $\iota^{*}\left(I\right)=I$.

2.2. Integrability of bi-rational maps

Integrability both for continuous and discrete systems can be defined in different ways, see [46, 23] for a complete discussion of the continuous and the discrete case. Different ways of defining integrability do not always necessarily agree, even though most of the time they do. We underline that the list we are going to make is not meant to be completely exhaustive of all the possible definitions of integrability. We will discuss only the definitions for autonomous recurrence relations we will need throughout the rest of the paper. We mention that additional definitions of integrability have been proposed for non-autonomous systems.

In general the solution of a recurrence relation of order $n$ will depend on $n$ arbitrary constants. This means that if a recurrence relation defined by the map $\varphi\colon\mathbb{CP}^{n}\to\mathbb{CP}^{n}$ possesses $n-1$ invariants $I_{j}$, $j=1,\dots,n-1$, then, in principle, it is possible to reduce it to a map $\hat{\varphi}\colon\mathbb{CP}^{1}\to\mathbb{CP}^{1}$ by solving the relations:

[TABLE]

where $\kappa_{j}$ are the value of the invariants on a set of initial data. This stimulates the simplest and most natural definition of integrability for maps:

Definition 5** (Existence of invariants).**

An $n$-dimensional map is (super)integrable if it admits $n-1$ functionally independent invariants.

Remark 1*.*

We underline that, in general, the reduction to a lower-dimensional map solving the system of equations (10) can break the bi-rationality.

Definition 5 is very general, and works for arbitrary maps. If some additional structure are present, then the number of invariants needed for integrability can be significantly reduced. A special, but relevant case is the one of Poisson maps.

Definition 6** (Poisson structures and Poisson maps [7, 29]).**

In affine coordinates $\mathbf{w}$ a Poisson structure of rank $2r$ is a skew-symmetric matrix $J=J\left(\mathbf{w}\right)$ of constant rank $2r$ such that the Jacobi identity holds:

[TABLE]

A Poisson structure defines a Poisson bracket through the identity:

[TABLE]

where $\nabla f$ is the gradient of $f$. Two functions $f$ and $g$ are said to be in involution with respect to the Poisson structure $J\left(\mathbf{w}\right)$ if $\left\{f,g\right\}=0$. We can easily see that $\{w_{i-1},w_{j-1}\}=J_{ij}$. A map of the affine coordinates $\varphi\colon\mathbf{w}\mapsto\mathbf{w^{\prime}}$ is a Poisson map if it preserves the Poisson structure $J\left(\mathbf{w}\right)$, i.e. if:

[TABLE]

where $\operatorname{d}\varphi$ is the Jacobian matrix of the map $\varphi$.

Then we have the following characterisation of integrability for Poisson maps:

Definition 7** (Liouville integrability [42, 5, 28]).**

An $n$-dimensional Poisson map is integrable if it possesses $n-r$ functionally independent invariants in involution with respect to this Poisson structure.

Remark 2*.*

A Poisson structure of full rank, i.e. $n=2r$ is invertible. The inverse matrix of the matrix $J\left(\mathbf{w}\right)$, i.e. $\Omega\left(\mathbf{w}\right)=J^{-1}\left(\mathbf{w}\right)$ is said to be a symplectic structure. We note that in the symplectic case we only need $n/2$ invariants in involution to claim integrability.

Symplectic structures are quite important in the theory of integrable maps. For instance, the classification made in [7] was carried out assuming of the existence of linear Poisson structure and of two invariants.

A difficult problem is, given a map, to find if there exists a symplectic structure for which this map is symplectic. In [6] it was proved that there exists a pre-symplectic structure (a degenerate sympectic structure) for any $n$-dimensional volume-preserving map possessing $n-2$ invariants. The rank of the obtained pre-symplectic structure is $n-2$ which implies that to claim integrability in the sense of Liouville one must be able to find another invariant. On the other hand, when the map comes from a discrete variational principle, i.e. it is variational, to find a symplectic structure is easy. We recall that an even-order recurrence relation (2) is said to be variational if there exists a function, called Lagrangian, $L=L\left(w_{k+N},\dots,w_{n}\right)$ such that the recurrence relation (2) is equivalent to the Euler-Lagrange equations:

[TABLE]

Here $N=n/2$ in the recurrence (2). A Lagrangian is called normal if

[TABLE]

Let $T$ be a shift operator, i.e $T^{j}(w_{k+i})=w_{k+i+j}$. Then due to the normality condition the discrete Ostrogradsky transformation [40]:

[TABLE]

where the new coordinates $\left(\mathbf{q},\mathbf{p}\right)=\left(q_{1},\dots,q_{N},p_{1},\dots,p_{N}\right)$ are defined through the formula:

[TABLE]

is well defined and invertible. Then the following result holds true [5]:

Lemma 1**.**

The map given by $\Phi=\mathcal{O}\circ\varpi\circ\mathcal{O}^{-1}\colon\left(\mathbf{q},\mathbf{p}\right)\to\left(\mathbf{q^{\prime}},\mathbf{p^{\prime}}\right)$, where $\varpi$ is the map corresponding to the Euler-Lagrange equations (14) has the following form

[TABLE]

where $\alpha(\mathbf{q},p_{1})$ is the solution with respect to $q^{N^{\prime}}$ of the equation:

[TABLE]

and $\widetilde{L}(\mathbf{q},p_{1})=L(\mathbf{q},\alpha(\mathbf{q},p_{1}))$. Moreover, the map (18) is symplectic with respect to the canonical symplectic structure:

[TABLE]

where $\mathbb{O}_{N}$ is the zero $N\times N$ matrix and $\mathbb{I}_{N}$ the $N\times N$ identity matrix.

Lemma 1 has the following corollary:

Corollary 2**.**

The Euler–Lagrange equations (14) admit the following non-degenerate Poisson bracket:

[TABLE]

where the differential of the Ostrogradsky transformation $\mathcal{O}$ must be evaluated on the original coordinates.

Therefore we have that corollary 2 allows us to construct a non-degenerate Poisson structure, and hence a symplectic structure, for every variational map.

Lagrangians for $2N$-order recurrence relations can be found following [24] or [20] for $N>1$. The method presented in [20] allows also to disprove the existence of a Lagragian for a given $2N$-order recurrence relation for $N>1$.

Moreover, bi-rational maps possess another definition of integrability: the low growth condition [43, 15, 3]. To be more precise we say that an $n$-dimensional bi-rational map is integrable if the degree of growth of the iterated map $\varphi^{k}$ is polynomial with respect to the initial conditions $\mathbf{[x_{0}]}$. Therefore we have the following characterisation of integrability:

Definition 8** (Algebraic entropy [3]).**

An $n$-dimensional bi-rational map is integrable in the sense of the algebraic entropy if the following limit

[TABLE]

called the algebraic entropy is zero for every initial condition $\mathbf{[x_{0}]}\in\mathbb{CP}^{n}$.

Algebraic entropy is an invariant of bi-rational maps, meaning that its value is unchanged up to bi-rational equivalence. Practically algebraic entropy is a measure of the complexity of a map, analogous to the one introduced by Arnol’d [2] for diffeomorphisms. In this sense growth is given by computing the number of intersections of the successive images of a straight line with a generic hyperplane in complex projective space [43].

The value of the degree of the iterates of the map is conditioned by its singularity structure. Some hypersurfaces are blown down by the map. If one of the successive images of these hypersurfaces coincide with a singular variety, there is a drop in the degree [3, 44, 39]. Therefore, from a heuristic point of view we can say that the singularity makes the entropy. This actually also applies to non-autonomous cases like the discrete Painlevé equations [37].

In principle, the definition of algebraic entropy in equation (22) requires us to compute all the iterates of a bi-rational map $\varphi$ to obtain the sequence

[TABLE]

Fortunately, for the majority of applications the form of the sequence can be inferred by using generating functions [26]:

[TABLE]

A generating function is a predictive tool which can be used to test the successive members of a finite sequence. It follows that the algebraic entropy is given by the logarithm of the smallest pole of the generating function, see [19, 18].

Several results are known about the relationship of the above definitions of integrability. First of all, the low growth condition means that the complexity of the map is very low, and it is known that invariants help in reducing the complexity of a map. Indeed the growth of a map possessing invariants cannot be generic since the motion is constrained to take place on the intersection of hypersurfaces defined by the invariants. However, the drop in complexity must be big enough to reduce the growth to a polynomial one. On the other hand it is known that the existence of invariants can give some bounds on the growth of bi-rational maps. Indeed, it is known that the orbits of superintegrable maps with rational invariant are confined to elliptic curves and the growth is at most quadratic [4, 17]. In low dimension some explicit results on the growth of bi-rational maps are known. For maps in $\mathbb{CP}^{2}$, it was proved in [12] that the growth can be only bounded, linear, quadratic or exponential. Linear cases are trivially integrable in the sense of invariants. We note that for polynomial maps in $\mathbb{C}^{2}$, it was already known from [43] that the growth can be only linear or exponential. It is known that QRT mappings and other maps with invariants in $\mathbb{CP}^{2}$ possess quadratic growth [13], so the two notions are actually equivalent for large class of integrable systems.

2.3. Duality

Now we discuss briefly the concept of duality for rational maps, which was introduced in [35]. Let us assume that our map $\varphi$ possesses $L$ invariants, i.e.$I_{j}$ for $j\in\left\{1,\dots,L\right\}$. Then we can form the linear combination:

[TABLE]

Being a function of invariants it follows that $H$ defined by (25) is itself an invariant of the map.

Remark 3*.*

We note that in principle more general combinations of invariants can be considered:

[TABLE]

where $P_{d}$ is a homogeneous polynomial of total degree $d$ in $L$ variables. Again even in this generalized case $H$ defined by (26) is an invariant of the map. However, in this paper we won’t consider this case, following the original definition of [35].

For an unspecified recurrence relation

[TABLE]

we can write down the invariant condition for $H$ (25):

[TABLE]

Since we know that $\mathbf{[x^{\prime}]}=\varphi\left(\mathbf{[x]}\right)$ is a solution of (28) we have the following factorization:

[TABLE]

We can assume without loss of generality that the map $\varphi$ corresponds to the annihilation of $A$ in (29). Now since $\deg_{x_{1}^{\prime}}\widehat{H}=\deg_{x_{1}}H$ and $\deg_{x_{n}}\widehat{H}=\deg_{x_{n}}H$ we have that if if $\deg_{x_{1}}H,\deg_{x_{n}}H>1$ the factor $B$ in (29) is non constant222We remark that this assertion is possible because we are assuming that all the invariants are non-degenerate. It is easy to see that degenerate invariants can violate this property.. In general, since the map $\varphi$ is bi-rational, we have the following equalities:

[TABLE]

Therefore we have that, in general, if $\deg_{x_{1}}H,\deg_{x_{n}}H>2$, the annihilation of $B$ does not define a bi-rational map, but an algebraic one. However when $\deg_{x_{1}}H,\deg_{x_{n}}H=2$ the annihilation of $B$ defines a bi-rational projective map. We call this map the dual map and we denote it by $\varphi^{\vee}$.

Remark 4*.*

We note that in principle for $\deg_{x_{1}}H=\deg_{x_{n}}H=d>2$, more general factorizations can be considered:

[TABLE]

but in this paper we won’t consider this case.

Now assume that the invariants (and hence the map $\varphi$) depends on some arbitrary constants $I_{i}=I_{i}\left(\mathbf{[x]};a_{i}\right)$, for $i=1,\dots,K$. Choosing some of the $a_{i}$ in such a way that there remains $M$ arbitrary constants and such that for a subset $a_{i_{k}}$ we can write equation (25) in the following way:

[TABLE]

where $J_{i}=J_{i}\left(\mathbf{[x]}\right)$, $i=1,2,\dots,K$ are new functions. Then using the factorization (29) we have that the $J_{i}$ functions are invariants for the dual maps.

Remark 5*.*

It is clear from equation (32) that even though the dual map is naturally equipped with some invariants, it is not necessarily equipped with a sufficient number of invariants to claim integrability. In fact there exists examples of dual maps with any possible behaviour, integrable, superintegrable and non-integrable [25, 21].

3. Derivation of the class of 4D maps

In this Section we explain how we derive the class of 4D maps with two invariants we are going to present in Section 4.

Our starting points are the maps corresponding to the autonomous $\mathrm{d}P_{\text{I}}^{2}$ and the $\mathrm{d}P_{\text{II}}^{2}$ equations and their invariants as presented in [25]. These two are maps of $\mathbb{CP}^{4}$ into itself with coordinates $\left[x:y:z:u:t\right]$. Their components are given by:

[TABLE]

and by:

[TABLE]

It can be checked that the map $\mathrm{d}P_{\text{I}}^{2}$ has two invariants $I^{\scriptscriptstyle(\rm I)}_{4}$ and $I^{\scriptscriptstyle(\rm I)}_{5}$ which are:

[TABLE]

while the map $\mathrm{d}P_{\text{II}}^{2}$ possesses two invariants $I^{\scriptscriptstyle(\rm II)}_{6}$ and $I^{\scriptscriptstyle(\rm II)}_{8}$ given by

[TABLE]

The invariants of the maps $\mathrm{d}P_{\text{I}}^{2}$ and $\mathrm{d}P_{\text{II}}^{2}$ have the following properties:

1. Property A:

The invariants are symmetric in the sense of definition 9. 2. Property B:

The lowest order invariants (33a) and (34a) have degree pattern $\left(1,3,3,1\right)$ and are particular instances of the homogeneous polynomial in $\mathbb{C}\left[x,y,z,u,t\right]$:

[TABLE]

depending parametrically on 11 coefficients, namely $s_{i}$, $i=1,\dots,11$. 3. Property C:

The highest order invariants (33b) and (34b) have degree pattern $\left(2,4,4,2\right)$. The most general homogeneous polynomial in $\mathbb{C}\left[x,y,z,u,t\right]$ depends parametrically on 1820 coefficients. Taking into account the symmetry with respect to the involution (9) the number of coefficients 121. Since, one of this coefficients is just an additional constant then we can lower the number of independent coefficient to 120. We denote this invariant by $I_{\text{high}}$, but we do not present the general form of this polynomial here, since it will be too cumbersome to write down.

Based on the above consideration it is natural to address the following problem:

Problem 1**.**

Find all the bi-rational maps $\varphi\colon\mathbb{CP}^{4}\to\mathbb{CP}^{4}$ and their dual maps $\varphi^{\vee}\colon\mathbb{CP}^{4}\to\mathbb{CP}^{4}$ having two non-degenerate, functionally independent invariants with properties A, B and C.

Solving this problem amount to obtain a list of equations which are expected to behave like the two fourth-order Painlevé equations $\mathrm{d}P_{\text{I}}^{2}$ and $\mathrm{d}P_{\text{II}}^{2}$. Before going to the solution of this problem, let us remark the following general result on the dual map of a map with two invariants possessing properties A, B and C:

Lemma 3**.**

Assume that a map $\varphi\colon\mathbb{CP}^{4}\to\mathbb{CP}^{4}$ possesses two invariants with properties A, B and C. Then we have the map $\varphi$ has degree pattern $\operatorname{dp}\varphi=\left(2,3,2,1\right)$ and the maximal degree pattern of the dual $\varphi^{\vee}\colon\mathbb{CP}^{4}\to\mathbb{CP}^{4}$ is $\operatorname{dp}\varphi^{\vee}=\left(2,1,2,1\right)$.

Proof.

By direct computation is it possible to check that if an invariant $I_{\text{low}}$ has the form (35) then the invariant condition (5) implies the following factorisation:

[TABLE]

Equation (36) means that we have the following degree distribution:

[TABLE]

The second part of the statement comes from an analogous consideration applied to equation (29). Since the degree pattern of $A$ is fixed, the degree pattern of $B$ is maximal when there are no factors depending only on $\mathbf{[x]}$. Under this assumptions we find the following distribution of the degrees:

[TABLE]

This ends the proof. ∎

Corollary 4**.**

Bi-rational maps possessing two invariants satisfying properties A, B, and C in general are not self-dual.

We sketch now the procedure we used to solve problem 1. We underline that this procedure is based on the one proposed in [25] to find bi-rational maps with invariants of assigned degree pattern.

1. (1)

Find the value of $x^{\prime}$ from (36) where $I_{\text{low}}$ is given by equation (35). 2. (2)

Substitute the obtained form of $x^{\prime}$ into the invariant condition (5) for $I_{\text{high}}$. Geometrically this describes the intersection of the two hypersurfaces given by $I_{\text{low}}=I_{\text{low}}^{(0)}$ and $I_{\text{high}}=I_{\text{hight}}^{(0)}$, where $I_{\text{low}}^{(0)}$ and $I_{\text{hight}}^{(0)}$ are arbitrary constants. 3. (3)

We can take coefficients with respect to the independent variables. This yield a system of nonlinear homogeneous equations. We put this system in a collection of systems that we call. 4. (4)

We convert this system to a set of simpler systems by solving iteratively all the monomial equations of each system. At each stage we exclude the systems originating invariants contraddicting properties A, B and C. 5. (5)

This yields 117 different smaller systems. 6. (6)

Solving these systems we found 25 solutions respecting the properties A, B and C.

Through a degeneration scheme the 25 solutions we obtain can be cast into six different maps along with their duals. We proved the following:

Theorem 5**.**

The solutions of problem 1, up to degeneration and identification of the free parameters, is given by six pairs of main/dual maps which we denote by (P.x) with x small roman number for the main maps and by (Q.x) for the dual map.

We call this class of maps the (P,Q) class. In the next section we present the explicit form of these maps and we discuss their integrability properties.

4. Maps of the (P,Q) class and their integrability properties

In this section we show the explicit form of the maps of the class (P,Q). We denote the pairs of main/dual maps by (x) where x is a small roman number. Moreover we discuss their integrability properties from the point of view of the existence of invariants, the degree growth of their iterates and the existence of Lagrangians. For the cases admitting Lagrangian following corollary 2 we present the form of their symplectic structure.

4.1. Maps (i)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{i}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on four parameters $a,d$ and $\mu,\nu$. The map (P.i) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

All the poles of the generating function (40) lie on the unit circle, so that the map (P.i) is integrable according to the algebraic entropy criterion. Moreover, due to the presence of $\left(s-1\right)^{4}$ in the denominator of the generating function (40) we have that the main map (P.i) has cubic growth.

The map (P.i) has the following invariants:

[TABLE]

The two invariants (41) alone cannot explain why the map (P.i) is integrable according to the algebraic entropy criterion. Indeed, as we stressed in section 2 two invariants are not enough to claim Liouville integrability, nor to claim integrability in the sense of definition 5. We can show using the method presented in [20] that the map (P.i) is not variational. However, as we shown in [21], the recurrence associated to the map (P.i) can be deflated to a three dimensional map via the transformation $v_{k}=w_{k}w_{k+1}$. The map obtained in this way is then integrable in the sense of Liouville. For all the details we refer to [21].

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{i}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on three parameters $\alpha,\beta$, and $\mu,\nu$. The parameters $\mu$ and $\nu$ are shared with the main map (P.i). The map (Q.i) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the dual map is integrable according to the algebraic entropy test with cubic growth, just like the main map.

The main map (P.i) possesses two invariants and depends on $a$ and $d$ whereas the dual map (Q.i) do not depend on them. Then according to (32) we can write down the first invariants for the dual map (Q.i) as:

[TABLE]

Therefore we obtain the following expressions:

[TABLE]

The invariant (45a) has degree pattern $\left(1,2,2,1\right)$.

The properties of the dual map (Q.i) are very similar to those of the main map (P.i). Again, the two invariants (45) alone cannot explain the low growth and following [20] no Lagrangian exists. However, as was shown in [21], the recurrence associated to the map (Q.i) can be deflated to a three dimensional via the transformation $v_{k}=w_{k}w_{k+1}$. Again, the map obtained in this way is integrable in the sense of Liouville. For all the details we refer again to [21].

4.2. Maps (ii)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{ii}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on the parameter $\mu$. The map (P.ii) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is non-integrable according to the algebraic entropy test with positive entropy $\varepsilon=\log 2$.

Despite being non-integrable the main map (P.ii) has, by construction, the following invariants:

[TABLE]

Moreover, using the test of [20] we have that the map (P.ii) is not variational.

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{ii}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ is given by the following components:

[TABLE]

This map depends on three parameters $\alpha,\beta$ and $\mu$. The parameter $\mu$ is shared with the main map (P.ii). The map given by (Q.ii) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is non-integrable according to the algebraic entropy test with positive entropy $\varepsilon=\log 2$. We remark that the growth is exactly the same as the main map.

Since the main map (P.ii) possesses two invariants, but it has only one parameter $\mu$ shared with the dual map. Then according to (32) we can only write down a single invariant for the dual map (Q.ii) as:

[TABLE]

The invariant (51) has degree pattern $\left(2,4,4,2\right)$.

Finally, using the test of [20] we have that the map (Q.ii) is not variational, as the main map (P.ii).

For additional comments about the maps (P.ii) and (Q.ii) we refer to [21].

4.3. Maps (iii)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{iii}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ is has the following components:

[TABLE]

This map depends on four parameters $a,d$ and $\mu,\nu$. The map (P.iii) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth.

The main map (P.iii) has the following invariants:

[TABLE]

Moreover, we note that according to the test in [20] the main map (P.iii) does not posses a Lagrangian. However, by direct search, we can prove that this map has an additional functionally independent invariant of degree pattern $(2,5,5,2)$. This means that the low growth of the main map (P.iii) is explained in terms of integrability as existence of invariants, as given by definition 5. More specifically, the quadratic growth it explained by the fact that if a map in $\mathbb{CP}^{4}$ has three rational invariants, the orbits are confined to elliptic curves and the growth is at most quadratic [4].

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{iii}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on four parameters $\alpha,\beta$ and $\mu.\nu$. The parameters $\mu,\nu$ are shared with the main map (P.iii). The map (Q.iii) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

where

[TABLE]

The growth of the map (Q.iii) is given by the inverse of the smallest pole of the generating function (56). These poles are given by the zeroes of the function $Q_{\text{\ref{eq:Diii}}}(s)$ in (57b). Clearly the zeroes of $1-s$ and $s^{2}+1$, lie on the unit circle, therefore we have to look at the location of the zeroes of the polynomial:

[TABLE]

Defining $q_{1}\left(s\right)=-2s+1$ and $q_{2}\left(s\right)=s^{10}-s^{9}$ we have that on the circle $C_{\rho}:=\left\{s\in\mathbb{C}\middle|\absolutevalue{s}=\rho\right\}$ with $\rho\in\left(1/2,1\right)$ the following inequality holds:

[TABLE]

By Rouche’s theorem [1] this implies that $q_{1}\left(s\right)$ and $q_{1}\left(s\right)+q_{2}\left(s\right)=q\left(s\right)$ have the same number of zeroes inside the circle $C_{\rho}$, i.e. the polynomial $Q_{\text{\ref{eq:Diii}}}(s)$ has a unique zero inside the circle $C_{\rho}$. This zero is the smallest one of $Q_{\text{\ref{eq:Diii}}}(s)$ and due to the fact that $Q_{\text{\ref{eq:Diii}}}(s)$ has real coefficients this zero is real. This implies the growth of the dual map (Q.iii) is exponential. The approximate value of the zero of $Q_{\text{\ref{eq:Diii}}}(s)$ inside $C_{\rho}$ is $s_{0}=0.49857104591719819\dots$. This implies that the algebraic entropy of the dual map (Q.iii) is:

[TABLE]

The growth of the sequence of degrees of equation (Q.iii) is then slightly greater than $2^{n}$.

Since the main map (P.iii) possesses two invariants and depends on $a$ and $d$ whereas the dual map (Q.iii) do not depend on them according to (32) we can write down the invariants for the dual map (Q.iii) as:

[TABLE]

Therefore we obtain the following expressions:

[TABLE]

The first invariant (62a) has degree pattern $\left(1,2,2,1\right)$ and the second invariant has degree pattern $\left(2,4,4,2\right)$. However, the degree pattern of the second invariant is not minimal: we can reduce the degree pattern of the second invariant to $(1,3,3,1)$ by replacing $I_{\text{high}}$ with $2\beta I_{\text{high}}-I_{\text{low}}^{2}$. Moreover, we can see that the existence of these two invariants is not sufficient to ensure the low growth of the dual map (Q.iii). Finally, we note that that the dual map (Q.iii) according to the test in [20] does not possess a Lagrangian.

4.4. Maps (iv)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{iv}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on four parameters $a,b,d$ and $\nu$. We note that the map (P.iv) is the autonomous $\mathrm{d}P_{\text{I}}^{2}$, derived in [10] and whose invariants, duality and growth properties where studied in [25]. For sake of completeness we repeat these properties here. The map (P.iv) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth.

The map (P.iv) has the following invariants:

[TABLE]

Using the methods of [20] we have that the map (P.iv) is variational. In affine coordinates $w_{n}$ its Lagrangian is given by:

[TABLE]

Using Corollary 2, we obtain the following non-degenerate Poisson bracket333Asterisked entries are placed to avoid the repetitions of entries, since a Poisson-bracket is skew-symmetric $J_{i,j}=-J_{j,i}$.

[TABLE]

One can check that the invariants (65) are in involution with respect to the Poisson bracket (67). Therefore, the map (P.iv) is Liouville integrable.

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{iv}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on three parameters $\alpha,\beta$, and $\nu$. The parameter $\nu$ is shared with the main map (P.iv). The map (Q.iv) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the dual map is integrable according to the algebraic entropy test with quadratic growth, just like the main map.

Since the main map (P.iv) possesses two invariants and depends on $a,b$ and $d$ whereas the dual map (Q.iv) do not depend on them according to (32) we can write down the invariants for the dual map (Q.iv) as:

[TABLE]

Therefore we obtain the following expressions:

[TABLE]

The invariants (71a) and (71c) both have degree pattern $\left(1,2,2,1\right)$. However, the second invariant is not minimal and it can be replaced with an invariant of degree pattern $(1,3,3,1)$. Moreover, using the test of [20], we obtain that the map (Q.iv) is not variational. Therefore, we conclude that the dual map (Q.iv) is integrable in the sense of the existence of invariants, i.e. according to definition 5.

4.5. Maps (v)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{v}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on the parameters $a,c,d$ and $\nu$. The map (P.v) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth.

The map (P.v) has the following invariants:

[TABLE]

Using the methods of [20] we have that the map (P.v) is variational. In affine coordinates $w_{n}$ its Lagrangian is given by:

[TABLE]

Using Corollary 2 we obtain the following non-degenerate Poisson structure

[TABLE]

One can check that the invariants (74) are in involution with respect to the Poisson bracket (76). Hence, the map (P.v) is Liouville integrable.

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{v}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on three parameters $\alpha,\beta$ and $\nu$. The parameter $\nu$ is shared with the main map (P.v). The map given by (Q.v) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth like the main map.

The main map (P.v) possesses two invariants and depends on $a,c$ and $d$ whereas the dual map (Q.v) do not depend on them. Then according to (32) we can write down the invariants for the dual map (Q.v) as:

[TABLE]

Therefore we obtain the following expressions:

[TABLE]

We note that the degree pattern of these invariants is $\left(1,2,2,1\right)$, $\left(1,2,2,1\right)$ and $\left(2,4,4,2\right)$ respectively. Finally, using the test of [20], we obtain that the map (Q.v) is not variational. Therefore, we conclude that the dual map (Q.v) is integrable in the sense of the existence of invariants, i.e. according to definition 5.

4.6. Maps (vi)

The main map $\mathbf{[x]}\mapsto\varphi_{\text{vi}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on the five parameters $a,c,d$ and $\mu,\delta$. The maps (P.vi) it is a slight generalization of $\mathrm{d}P_{\text{II}}^{2}$ equation which was discussed in [25]. Here, we recall its properties and we discuss its duality in the parameter space. First, the map (P.vi) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth.

The map (P.vi) has the following invariants:

[TABLE]

Using the methods of [20] we have that the map (P.vi) is variational. In affine coordinates $w_{n}$ its Lagrangian is given by:

[TABLE]

Using Corollary 2, we obtain the following non-degenerate Poisson structure for the map (P.vi)

[TABLE]

One can check that the invariants (83) are in involution with respect to the Poisson bracket (85). Therefore, the map (P.vi) is Liouville integrable.

The dual map $\mathbf{[x]}\mapsto\varphi^{\vee}_{\text{vi}}\left(\mathbf{[x]}\right)=\mathbf{[x^{\prime}]}$ has the following components:

[TABLE]

This map depends on three parameters $\alpha,\beta$ and $\delta$. The parameter $\delta$ is shared with the main map (P.vi). The map given by (Q.vi) has the following degrees of iterates:

[TABLE]

with generating function:

[TABLE]

This means that the main map is integrable according to the algebraic entropy test with quadratic growth like the main map.

The main map (P.vi) possesses two invariants and depends on $a,c$ and $d$ whereas the dual map (Q.vi) do not depend on them. Then according to (32) we can write down the invariants for the dual map (Q.vi) as:

[TABLE]

Therefore we obtain the following expressions:

[TABLE]

We note that the degree pattern of these invariants is $\left(2,4,4,2\right)$, $\left(1,2,2,1\right)$ and $\left(1,2,2,1\right)$ respectively. Finally, using the test of [20], we obtain that the map (Q.vi) is not variational. Therefore, we conclude that the dual map (Q.vi) is integrable in the sense of the existence of invariants, i.e. according to definition 5.

5. Summary and outlook

In this paper we presented the (P,Q) class of four-dimensional maps. These maps were obtained by assuming they possess two invariants satisfying the conditions A, B and C given in section 3. In section 4 we discussed the integrability properties of these maps.

Integrability in the (P,Q) list can arise in different ways depending weather the map is variational or not. Variational maps are all Liovuille integrable, as remarked in section 2. The only additional structure needed for integrability was then the Lagrangian, constructed using the method in [20]. On the other hand integrability in the non-variational maps can arise in two different ways. The pair of maps (P.i) and (Q.i) possessing cubic growth is deflatable. This means that the two maps arise as non-invertible non-local transformation from two lower-dimensional maps. In [21] we proved that the invariants are preserved in this process and that the integrability of the three-dimensional maps can be understood using the definition of Liouville integrability with a rank two Poisson structure. All the other maps possess quadratic growth and possess a third invariant of motion. In the case of the map (P.iii) the third invariant was found by direct inspection, while in all the other cases it was produced directly from the duality approach. As last remark, we note that the maps with three invariants admit three different degenerate Poisson structure constructed using the method of [6], but this construction does not yield Liouville integrability.

All the remaining maps have exponential growth and are therefore non integrable in the sense of the algebraic entropy. Direct search of invariants for these maps excluded the their existence up to order 14. Moreover, using the test of [20], we proved that these exponentially-growing maps are not variational. Therefore we have a strong evidence of the fact that these maps do not possess any non-degenerate Poisson structure, and therefore these cannot be Liouville integrable. Unfortunately, this result is not enough for a complete proof of the fact that no non-degenerate Poisson structure exists at all. This is because, in principle, a fourth-order recurrence relation can be cast into a system of two second-order recurrence relations which can be variational. Therefore, as we did in [21], we conjecture that the maps (P.ii) and (Q.iii) either do not admit any full-rank Poisson structure, or for all full-rank Poisson structure they admit their invariants do not commute.

In table 1 we give a schematic resume of all the above considerations.

The search procedure carried out in this paper has been very fruitful giving some interesting and non-trivial examples of four-dimensional maps. Indeed, all the maps, but four are new. Particularly interesting is the variety of behaviours we encountered in the maps of the class (P,Q). Work is in progress to characterize the surfaces generated by the invariants in both integrable and non-integrable cases. We expect this to give some hints on how the integrability arises from purely geometrical considerations. This is well known for maps in two dimension with the theory of elliptic fibrations applied to the QRT mapping [13]. However, it was discussed in [25, 21] how examples with cubic growth can go beyond the existence of elliptic fibrations making the underlying geometrical structure more complex and richier.

Finally, we believe that the direct search of maps with invariants alongside with the algorithmic tests available in the discrete setting may produce many new results and integrable maps in the next years. Analogous procedure in the continuous case still yield many new result after more that fifty years of their introduction [16, 32, 14]. Work is in progress to extend the present class by considering invariants of more general form.

6. Acknowledgments

This research was supported by an Australian Laureate Fellowship #FL120100094 and grant #DP160101728 from the Australian Research Council.