Spectral theorem for convex monotone homogeneous maps, and ergodic control

Abstract

We consider convex maps $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ that are monotone (i.e., that preserve the product ordering of $\text{R}{}^{\mathrm{n}}$), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of $\text{R}{}^{\mathrm{n}}$, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

Introduction

We say that a map $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ is monotone if for all $\text{x,y∈}\text{R}{}^{\mathrm{n}}$, x⩽y⇒f(x)⩽f(y), where ⩽ denotes the product ordering of $\text{R}{}^{\mathrm{n}}$ (x⩽y if x_i⩽y_i, for all 1⩽i⩽n). We say that f is additively homogeneous if for all $\text{λ∈}\text{R}$, $\text{x∈}\text{R}{}^{\mathrm{n}}$, f(λ+x)=λ+f(x), where λ+x=(λ+x₁,…,λ+x_n). It is easy to see that a monotone homogeneous map is nonexpansive for the sup-norm: for all $\text{x,y∈}\text{R}{}^{\mathrm{n}}$, |f(x)−f(y)|⩽|x−y|, where $\text{|x|=}\text{max}{}_{\mathrm{1⩽i⩽n}}\text{|x}{}_{\mathrm{i}}\text{|}$ (see [17]).

Monotone homogeneous maps arise classically in optimal control and game theory (see for instance [10], [31], [71], [32], [20], [61]), in the modeling of discrete events systems (see [6], [25], [14], [15], [68], [21], [26], [27]), and in nonlinear potential theory [19], as nonlinear extension of Markov transitions. They also arise in nonlinear Perron–Frobenius theory, when one considers multiplicatively homogeneous maps F acting on a cone and preserving the order of the cone: in the simplest case, when the cone is $\text{(}\text{R}{}_{\mathrm{+}}{}^{\mathrm{\ast }}\text{)}{}^{\mathrm{n}}$ (where $\text{R}{}_{\mathrm{+}}{}^{\mathrm{\ast }}\text{={x∈}\text{R}\text{|}\text{x>0}}$), the transformation $\text{F↦}\text{log}\text{∘F∘}\text{exp}$ (where $\text{log}\text{:}\text{(}\text{R}{}_{\mathrm{+}}{}^{\mathrm{\ast }}\text{)}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ is the map which does log entrywise, and exp=log⁻¹) sends the set of monotone multiplicatively homogeneous maps to the set of monotone additively homogeneous maps. See for instance [11], [38], [44], [41], [46], [47], [62], [72] for various studies and applications.

A basic problem, for a monotone homogeneous map f, is the existence, and uniqueness (up to an additive constant), of the additive eigenvectors of f, which are the $\text{v∈}\text{R}{}^{\mathrm{n}}$ such that f(v)=λ+v, for some additive eigenvalue $\text{λ∈}\text{R}$. In the sequel, we will omit the term “additive”, when the additive nature of the objects will be clear from the context. When f has an eigenvector v with eigenvalue λ, by homogeneity of f, f^k(v)=kλ+v holds for all k⩾0, and by nonexpansiveness of f, |f^k(x)−kλ−v|=|f^k(x)−f^k(v)|⩽|x−v|, hence,

$$\text{f}{}^{\mathrm{k}}\text{(x)=kλ+O(1)}\text{when}\text{k→∞}$$

for all $\text{x∈}\text{R}{}^{\mathrm{n}}$ (all the orbits of f have a linear growth rate of λ). This implies in particular that the eigenvalue λ is unique. Hence, we can speak without ambiguity of the eigenspace of f, which is the set $\text{E}\text{(f)={x∈}\text{R}{}^{\mathrm{n}}\text{|}\text{f(x)=λ+x}}$. In many applications, the eigenvalue and eigenvector are fundamental objects: for instance, in stochastic control, the eigenvalue gives the optimal reward per time unit, and eigenvectors give stationary rewards (we explain this in detail in Section 7). In discrete event systems applications, the eigenvalue gives the throughput, and eigenvectors give stationary schedules.

Several Perron–Frobenius like theorems guarantee the existence of eigenvectors of monotone (additively) homogeneous maps $\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$. Such results go back at least to Kreı̆n and Rutman [34, Section 7], in the context of monotone multiplicatively homogeneous maps leaving a cone in a Banach space invariant, and to Morishima, whose book [44] contains a complete study of finite dimensional nonlinear Perron–Frobenius theory. A modern overview appears in the memoirs of Nussbaum [46], [47], which contain in particular general existence results for eigenvectors, following [45]. Different existence conditions appeared in [53]. General results on the geometry of the eigenspaces are available, for instance, the result of Bruck [12] shows in particular that $\text{E}\text{(f)}$ is the image of a nonexpansive projector and a fortiori is connected, see also [46, Theorems 4.5–4.7].

In this paper, we describe the eigenspaces of convex monotone homogeneous maps $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$. (We say that a $\text{R}{}^{\mathrm{n}}$-valued map is convex when its coordinates are convex. We refer the reader to [57] for all convexity notions used in the paper: subdifferentials, domain, Fenchel transform, etc.)

To state our main result, we need a few definitions (see Section 2 for details). We first generalize the notion of subdifferential to maps $\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ by setting, for $\text{x∈}\text{R}{}^{\mathrm{n}}$, $\text{∂f(x)={P∈}\text{R}{}^{\mathrm{n\times n}}\text{|}\text{f(y)−f(x)⩾P(y−x),}\text{∀y∈}\text{R}{}^{\mathrm{n}}\text{}}$. It is easy to see (Corollary 2.2 and Eq. (4) below) that by monotonicity and homogeneity of f, the elements of ∂f(x) are stochastic matrices. If v is an eigenvector of f, we call critical graph of f, the (directed) graph $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ which is the union of final graphs of stochastic matrices P∈∂f(v) (we call final graph of a stochastic matrix the restriction of its graph to the set of final classes, see 2.2 Convex rectangular sets of stochastic matrices, 2.3 Critical graph of convex monotone homogeneous maps). The graph $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ is independent of the choice of the eigenvector v (Proposition 2.5 below). We call critical nodes of f, the nodes of $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$, and denote by $\text{N}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ the set of critical nodes. We call critical classes of f the sets of nodes C₁,…,C_s of the strongly connected components of $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$, $\text{G}{}_1\text{,…,}\text{G}{}_{\mathrm{s}}$. We call cyclicity of $\text{G}{}_{\mathrm{i}}$, and denote by $\text{c}\text{(}\text{G}{}_{\mathrm{i}}\text{)}$, the gcd of the lengths of the circuits of $\text{G}{}_{\mathrm{i}}$, and we define the cyclicity of f by $\text{c}\text{(f)=}\text{lcm}\text{(}\text{c}\text{(}\text{G}{}_1\text{),…,}\text{c}\text{(}\text{G}{}_{\mathrm{s}}\text{))}$. We say that a monotone homogeneous map $\text{g}\text{:}\text{U⊂}\text{R}{}^{\mathrm{n}}\text{→V⊂}\text{R}{}^{\mathrm{p}}$ is a monotone homogeneous isomorphism if it has a monotone homogeneous inverse. The following theorem (already announced in [2]), gathers results from Theorem 3.4, Corollary 3.6, Corollary 5.7, and Theorem 6.6 below.

Theorem 1.1 Convex spectral theorem

Let $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ denote a convex monotone homogeneous map that has an eigenvector. Denote by $\text{C=}\text{N}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ the set of critical nodes of $\text{f,}\text{c=}\text{c}\text{(f)}$ the cyclicity of f, and λ the unique eigenvalue of f. Then,

• the restriction $\text{r}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{C}}\text{,}\text{x↦(x}{}_{\mathrm{i}}\text{)}{}_{\mathrm{i\in C}}$, is a monotone homogeneous isomorphism from $\text{E}\text{(f)}$ to its image $\text{E}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$;

• $\text{E}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ is an inf-subsemilattice of $\text{(}\text{R}{}^{\mathrm{C}}\text{,⩽)}$;

• $\text{E}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ is a convex set whose dimension is at most equal to the number of critical classes of f, and this bound is attained when f is piecewise affine;

• for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, f^kc(x)−kcλ has a limit when k→∞.

In particular, when f has only one critical class, the eigenvector of f is unique (up to an additive constant). It also follows from the last assertion of the theorem that the set of limit points of f^k(x)−kλ when k→∞, and $\text{x∈}\text{R}{}^{\mathrm{n}}$, is precisely $\text{E}\text{(f}{}^{\mathrm{c}}\text{)}$. Theorem 1.1 also allows us to bound the dimension of this set. Indeed, we shall see in Theorem 4.1 and Proposition 5.3 below that the set of critical nodes is the same for f and f^c, and that f^c has $\text{c}\text{(}\text{G}{}_1\text{)+⋯+}\text{c}\text{(}\text{G}{}_{\mathrm{s}}\text{)}$ critical classes. Hence, applying Theorem 1.1 to f^c, we get that the restriction $\text{r}$ is a monotone homogeneous isomorphism from $\text{E}\text{(f}{}^{\mathrm{c}}\text{)}$ to a convex set, $\text{E}{}^{\mathrm{<mtext>c</mtext>}}\text{(f}{}^{\mathrm{c}}\text{)}$, of dimension at most $\text{c}\text{(}\text{G}{}_1\text{)+⋯+}\text{c}\text{(}\text{G}{}_{\mathrm{s}}\text{)}$, the bound being attained when f is piecewise affine.

The paper is devoted to the proof (2 Class structure of convex monotone homogeneous maps, 3 Structure of eigenspaces, 4 Critical graph of, 5 Cyclicity theorem for convex monotone homogeneous maps, 6 Piecewise affine convex monotone homogeneous maps) and to the stochastic control interpretation (Section 7) of Theorem 1.1. In Section 2, we detail the definitions and properties of subdifferentials and critical graph of convex monotone homogeneous maps. An important element of the proofs is the maximum principle for Markov chains (Lemma 2.9). In Section 3, we establish the first part of Theorem 1.1 concerning the structure of the eigenspace: points 1, 2 and the first assertion in point 3. The main argument is again the maximum principle. Section 4 is devoted to further tools and properties used in the remaining sections, which are of independent interest: Theorem 4.1 shows that $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f}{}^{\mathrm{k}}\text{)=}\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}{}^{\mathrm{k}}$ (this will be used in Section 5 for the proof of the cyclicity part of Theorem 1.1); we also introduce directional derivatives (which will be used in Section 6 to connect $\text{E}\text{(f)}$ to $\text{E}\text{(f}{}_{\mathrm{v}}\text{′)}$ for any eigenvector v), additive recession functions (formula (15)), invariant critical classes and the associated decomposition of f (Lemma 4.9), and also a characterization of the set of critical nodes in terms of supports of nonlinear “excessive” measures (Proposition 4.5).

Section 5 is devoted to the proof of point 4 of Theorem 1.1. This result relies on a more general theorem of Nussbaum [48] and Sine [67], which states that if $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ is nonexpansive for the sup-norm and has a fixed point, then, for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, f^kc(x) converges when k→∞, for some minimal constant c which can be bounded by a function of n. When f is convex monotone and homogeneous, the last assertion of the convex spectral theorem shows that the possible values of c are exactly the orders of elements of the symmetric group on n letters. Equivalently, convex monotone homogeneous maps have the same orbit lengths as permutation matrices, a result which was known to be true in the special cases of linear maps associated to nonnegative matrices (see [52, Chapter 9]), of linear maps over the max-plus semiring (see [13], [49]), and also of piecewise affine convex monotone homogeneous maps (which include max-plus linear maps), see the discussion in Section 1.2 below. More generally, computing the orbit lengths of nonexpansive maps for polyhedral norms raises interesting combinatorial and analytical problems (see in particular [1], [70], [63], [64], [48], [50], [51], [52], [36]).

The equality in point 3 of Theorem 1.1 is proved in Section 6. As will be discussed in Section 1.2, this part of the theorem has already been proved by Romanovsky [60] and by Schweitzer and Federgruen [66]. We provide here an independent proof, which emphasizes the qualitative properties of $\text{E}\text{(f)}$, using the tools of Section 4. We also give a polynomial time algorithm to compute $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$.

Convex monotone homogeneous maps $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ are exactly dynamic programming operators associated to stochastic control problems with state space {1,…,n}. Computing the stationary solutions and the asymptotic behavior of solutions of dynamic programming equations is an old problem of stochastic control which is essentially equivalent to that of computing the eigenspace $\text{E}\text{(f)}$ and the asymptotics of f^k when k→∞. This has been much studied in the stochastic control literature, particularly in the case of finite action spaces, which corresponds to piecewise affine maps. In this special case, results equivalent to the third assertion of Theorem 1.1 were obtained by Romanovsky [60] using linear programming techniques, and also by Schweitzer and Federgruen [66] who gave an explicit representation of $\text{E}\text{(f)}$ in terms of resolvents associated to optimal strategies (see [66, Theorem 4.1]). Again in this special case, a result equivalent to the fourth assertion of Theorem 1.1 was stated by Lanery [35]. The arguments of [35] only proved the special case where $\text{N}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)={1,…,n}}$, see the discussion in the introduction and in Note 1 of [65]. A proof valid for a general $\text{N}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)⊂{1,…,n}}$ was given by Schweitzer and Federgruen [65], who also proved the optimality of $\text{c}\text{(f)}$.

The special case of deterministic control problems leads to maps f that are max-plus linear. These maps have been studied independently by the max-plus community. In this context, the dimension of the eigenspace was characterized by Gondran and Minoux [24], and the remaining part of the max-plus spectral theorem, dealing with cyclicity, was obtained by Cohen et al., [13] (see also [6]). (Note however that more precise results—explicit form of the eigenspace, finite time convergence of the iterates—are available in the max-plus case.) The max-plus spectral theorem has a long story, which goes back to Cuninghame–Green (see [18] and the references therein), Romanovsky [59], and Vorobyev [69], to quote the most ancient contributions. See the collection of articles [40], Kolokoltsov and Maslov [33], and the references therein, for generalizations to infinite dimension. See also [23], [7] for surveys.

The present work was inspired by the max-plus spectral theorem and uses nonexpansive maps techniques (we were unaware of the results of [35], [60], [65], [66]). We next emphasize differences with earlier results. We consider general convex monotone homogeneous maps, which correspond to stochastic control problems with finite state space and arbitrary action spaces, whereas the results in [35], [60], [65], [66] require the action space to be finite. Our proof technique, which relies on the maximum principle, can be naturally transposed to other (infinite dimensional) contexts. Another tool in our proof is the critical graph $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$, which generalizes the critical graph that appears in max-plus algebra (see Proposition 2.7). The critical graph already appeared in [60, p. 491], with a different definition in terms of optimal policies. The new definition that we give here in terms of subdifferentials leads in particular to a polynomial time algorithm to compute $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ (see Section 6.3). (The equivalence of both definitions is shown in Proposition 7.2.) It should also be noted that when passing from the case of finite action spaces to arbitrary action spaces, new phenomena occur. For instance, Example 3.9 shows that we cannot hope, in general, to characterize the dimension of eigenspaces in terms of graphs like $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$.

Let us mention in passing that the critical graph has an intuitive interpretation in terms of “recurrence”. For a Markov chain, a node is recurrent if the probability of return to this node is equal to one. For a max-plus matrix with eigenvalue 0, a node is “recurrent”, i.e. belong to a critical class, if we can return to this node with zero reward. When f is a convex monotone homogeneous map with eigenvalue 0, a node i is “recurrent”, i.e. belong to a critical class, if we can find a strategy for which, starting from i, we eventually return to i with probability 1 and zero mean reward. This provides a new illustration of the analogy between probability and optimization developed in [39, Chapter VIII], [43], [55], [3], [42], [56, Section 4.2.], [37], [54].

Ergodic control problems of diffusion processes lead to spectral problems for infinite dimensional monotone homogeneous semigroups which can be expressed in terms of ergodic Hamilton–Jacobi–Bellman (HJB) partial differential equations. In [8], Bensoussan proved uniqueness of the eigenvector (as weak solution of the ergodic HJB equation) under assumptions, which translated in finite dimension imply irreducibility of stochastic matrices P∈∂f(v). Inspired by the results of the present paper, the first author, Sulem and Taksar [4] proved uniqueness of the viscosity solution of a special ergodic HJB equation. This yields an example of concrete situation where some nonoptimal stationary strategies have several final classes, whereas the optimal ones have only one final class (translated to our setting, this means that for some $\text{x∈}\text{R}{}^{\mathrm{n}}$ and P∈∂f(x), P may have several final classes, whereas there exists an eigenvector v such that all elements of ∂f(v) have one final class).

The uniqueness result that follows from Theorem 1.1 can be thought of as a partial extension of the condition of Nussbaum [46, Theorem 2.5]: specialized to convex monotone homogeneous maps $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$, the result of Nussbaum shows that if f is $\text{C}{}^1$ and for all eigenvectors v, f′(v) has only one final class (in this case of course f has a unique critical class), then the eigenvector of f is unique. The idea of all these results is that the dimension of $\text{E}\text{(f)}$ can be bounded by looking at “linearizations” of f near an eigenvector.

It is instructive to note that the uniqueness of eigenvectors in $\text{R}{}^{\mathrm{n}}$ is governed by the same kind of graph properties as the existence of eigenvectors, albeit the graphs are different. For instance, a result of the second author and Gunawardena [22, Theorem 2] guarantees the existence of an eigenvector for a monotone homogeneous map which has a strongly connected graph. Here, the graph $\text{G}\text{(f)}$ of a monotone homogeneous map $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ has nodes {1,…,n} and an arc i→j if $\text{lim}{}_{\mathrm{\nu \to \infty }}\text{f}{}_{\mathrm{i}}\text{(νe}{}_{\mathrm{j}}\text{)=+∞}$, where e_j denotes the jth vector of the canonical basis of $\text{R}{}^{\mathrm{n}}$. Another way to guarantee the existence of an eigenvector is to use the convex spectral theorem itself, thanks to the following observation taken from [22]. We denote by $\text{f}\text{̂}\text{(x)=}\text{lim}{}_{\mathrm{\mu \to \infty }}\text{μ}{}^{\mathrm{-1}}\text{f(μx)}$ the recession function of f (f̂ need not exist when f is monotone and homogeneous, but it does exist when f is convex). We have $\text{f}\text{̂}\text{(0)=0}$, and when f is (additively) homogeneous, so does f̂, so that all points on the diagonal are (trivial) fixed points of f. It is proved in [22] that if the recession function of a monotone homogeneous map f exists and has only fixed points on the diagonal, then, f has an eigenvector. Combining this observation with the convex spectral theorem, we obtain:

Corollary 1.2

A monotone homogeneous map has an eigenvector if its recession function exists, is convex, and has only one critical class.

If f is a convex monotone homogeneous map, it is not difficult to see that the recession function f̂ is exactly the support function of the domain of the Fenchel transform $\text{f}{}^{\mathrm{\ast }}$ of f (defined in Section 2.1 below), that is . In this formula, one can replace $\text{dom}\text{f}{}^{\mathrm{\ast }}$ by its closure $\text{cl}\text{(}\text{dom}\text{f}{}^{\mathrm{\ast }}\text{)}$, which is equal to $\text{∂}\text{f}\text{̂}\text{(0)}$. The graph $\text{G}\text{(f)}$ is the union of the graphs of $\text{P∈}\text{dom}\text{f}{}^{\mathrm{\ast }}$, or equivalently the union of the graphs of $\text{P∈}\text{cl}\text{(}\text{dom}\text{f}{}^{\mathrm{\ast }}\text{)}$, whereas the critical graph $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(}\text{f}\text{̂}\text{)}$ is the union of the final graphs of $\text{P∈}\text{cl}\text{(}\text{dom}\text{f}{}^{\mathrm{\ast }}\text{)}$. If $\text{G}\text{(f)}$ is strongly connected, one can see that $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(}\text{f}\text{̂}\text{)}$ is also strongly connected, so that in the special case of convex monotone homogeneous maps, Corollary 1.2 is stronger than Theorem 2 of [22] (which however holds in a more general context).

Finally, let us mention two immediate extensions of the convex spectral theorem. First, since the map f↦(x↦−f(−x)) sends convex monotone homogeneous maps to concave monotone homogeneous maps, there is of course a dual concave spectral theorem. Another, more interesting, extension, is obtained by considering subhomogeneous maps f, which satisfy f(λ+x)⩽λ+f(x), for all λ⩾0 and $\text{x∈}\text{R}{}^{\mathrm{n}}$. It is easy to see that a monotone map is subhomogeneous if, and only if, it is nonexpansive for the sup-norm. To a monotone subhomogeneous map $\text{f:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$, we associate canonically a monotone homogeneous map $\text{g}\text{:}\text{R}{}^{\mathrm{n+1}}\text{→}\text{R}{}^{\mathrm{n+1}}$,

$$\text{g(x,y)=}\text{y+f(−y+x)}\text{y}\text{,}\text{∀x∈}\text{R}{}^{\mathrm{n}}\text{,}\text{y∈}\text{R}$$

(this is a nonlinear extension of the classical way of passing from a substochastic to a stochastic matrix, by adding a cemetery state, in this nonlinear context, this construction is due to Gunawardena and Keane [28]). A vector $\text{z∈}\text{R}{}^{\mathrm{n}}$ is a fixed point of f, if, and only if, (z,0) is an eigenvector of g (and the eigenvalue is 0). Using this construction, one translates readily the convex spectral theorem to a theorem describing fixed point sets and the asymptotics of the iterates of convex monotone subhomogeneous maps. (We might also use this construction, with −λ+f instead of f, to describe the eigenspace of f for an additive eigenvalue λ, but when f is only monotone and subhomogeneous, λ need not be unique, and it tells little about the asymptotics of f^k, in general.) For a convex monotone subhomogeneous map f with fixed point v, the critical graph $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ of f is defined as the union of the graphs of the matrices P_CC, where P∈∂f(v), C is a final class of P, and the C×C submatrix of P, P_CC, is stochastic (when f is homogeneous, this property is automatically satisfied). Equivalently, $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ (which can be empty) is the restriction of $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(g)}$ to {1,…,n}. The notions of critical classes and cyclicity are defined from $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ as above. When $\text{G}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)}$ is empty, we have $\text{N}{}^{\mathrm{<mtext>c</mtext>}}\text{(f)=∅}$, and we take the convention $\text{R}{}^{\mathrm{\varnothing }}\text{={0}}$, and $\text{c}\text{(f)=1}$.

Corollary 1.3

Let $\text{f}\text{:}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$ denote a convex monotone subhomogeneous map that has a fixed point. Then, all the conclusions of the convex spectral theorem apply to f and λ=0. In particular, if f has no critical classes, then its fixed point is unique.