Stability and convergence in discrete convex monotone dynamical systems
aa r X i v : . [ m a t h . D S ] M a r Stability and convergence in discrete convexmonotone dynamical systems
Marianne Akian, St´ephane Gaubert ∗ INRIA and CMAP, ´Ecole Polytechnique,91128 Palaiseau Cedex, France. [email protected] , [email protected] andBas Lemmens, SMSAS, University of Kent,Canterbury, CT2 7NF, United Kingdom
November 20, 2018
Abstract
We study the stable behaviour of discrete dynamical systems wherethe map is convex and monotone with respect to the standard positivecone. The notion of tangential stability for fixed points and periodicpoints is introduced, which is weaker than Lyapunov stability. Amongothers we show that the set of tangentially stable fixed points is iso-morphic to a convex inf-semilattice, and a criterion is given for theexistence of a unique tangentially stable fixed point. We also showthat periods of tangentially stable periodic points are orders of permu-tations on n letters, where n is the dimension of the underlying space,and a sufficient condition for global convergence to periodic orbits ispresented. ∗ The first two authors were supported by the Arpege programme of the French NationalAgency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005 and by theDigiteo project DIM08 “PASO” number 3389. Introduction
Many natural dynamical systems preserve a type of ordering on the statespace. Such dynamical systems are called monotone and often display rathersimple behaviour. In the last couple of decades monotone dynamical sys-tems have been studied intensively, see [15] for an up-to-date survey. Groundbreaking work on monotone dynamical systems was done by Hirsch [13, 14],who showed, among others, that in a continuous time strongly monotonedynamical system almost all pre-compact orbits converge to the set of equi-librium points. In a discrete time strongly monotone dynamical system onehas generic convergence to periodic orbits under appropriate conditions onthe map, see [9, 12, 24]. Various additional conditions have been studiedto obtain convergence of all orbits instead of almost all orbits. A type ofconcavity condition, also called sub-homogeneity, has received a great dealof attention, see [2, 16, 19, 30, 31]. The concavity condition makes the dy-namical system non-expansive, which allows one to prove strikingly detailedresults concerning their behaviour.In this paper we study discrete time dynamical systems x k +1 = f ( x k ) for k = 0 , , , . . . , (1)where f : D → D is a convex monotone map on
D ⊆ R n preserving thepartial ordering induced by the standard positive cone. Such dynamicalsystems are in general not non-expansive. We introduce the notion of tan-gential stability for fixed points and periodic points, which is weaker thanLyapunov stability. It turns out that this notion is the right one to prove avariety of results concerning the stable behaviour of monotone convex dy-namical systems, which are of comparable detail as the ones for monotonenon-expansive dynamical systems.In particular, we show that the tangentially stable fixed point set isisomorphic to a convex inf-semilattice in R n . We also give a criterion forthe existence of a unique tangentially stable fixed point. In addition, tan-gentially stable periodic orbits are analysed and a condition is presentedunder which there is global convergence to Lyapunov stable periodic orbits.Among others it is shown that the periods of tangentially stable periodicpoints divide the cyclicity of the critical graph, which implies that the peri-ods are orders of permutations on n letters. However, the periods of unstableperiodic orbits can be arbitrary large.The results are a continuation of [1] in which the first two authors stud-ied discrete time dynamical systems (1), where f : R n → R n is not onlyconvex and monotone, but also additively sub-homogeneous. The extra2ub-homogeneity condition makes the dynamical system non-expansive un-der the sup-norm [8]. The non-expansiveness property severely constrainsthe complexity of its behaviour [18, 22] and makes all fixed points and pe-riodic orbits Lyapunov stable. It also ensures that the subdifferential of f at a fixed point consists of row-stochastic matrices. Without the addi-tively homogeneity condition, the subdifferential merely consists of stablenonnegative matrices, which makes the analysis more subtle.Motivating examples of discrete convex monotone dynamical systemsarise in Markov decision processes and game theory as value iteration schemes,see [1] and the references therein. The results in this paper extend resultsfor Markov decision processes with sub-stochastic transition matrices to ar-bitrary nonnegative matrices. In particular, they apply to Markov decisionprocesses with negative discount rates, see [28]. Discrete convex monotonedynamical systems are also used in static analysis of programs by abstractinterpretation [7], i.e., automatic verification of variables in computer pro-grams. They also appear in the theory of discrete event systems [4], statis-tical mechanics [23], and in the analysis of imprecise Markov chains [10]. Atthe end of Section 2 we give several explicit examples.The paper contains nine sections. In Section 2 several basic definitionsand properties of convex monotone maps are collected. Subsequently variousdegrees of stability of fixed points of convex monotone maps are discussedand the notion of tangential stability is introduced. In Section 4 several pre-liminary results concerning stable nonnegative matrices are given. Amongothers rectangular sets of stable nonnegative matrices are studied. In Section5 we analyse tangentially stable fixed points and introduce the critical graphof a monotone convex map. Section 6 is used to collect several preliminaryresults concerning convex monotone positively homogeneous maps that areneeded in the analysis of the geometry of the tangentially stable fixed pointset. Section 7 contains the main result on the geometry of the tangentiallystable fixed point set. Section 8 concerns tangentially stable periodic pointsand their periods. In the final section a criterion is given under which eachorbit of a discrete time convex monotone dynamical systems converges to aLyapunov stable periodic orbit. Let R n + = { x ∈ R n : x i ≥ ≤ i ≤ n } denote the standard positive cone .The cone R n + induces a partial ordering on R n by x ≤ y if y − x ∈ R n + . Wewrite x ≪ y if y − x is in the interior of R n + . In particular, we say that x is3 ositive if 0 ≪ x . For x, y ∈ R n we also use the notation x ≥ y and x ≫ y with the obvious interpretation. A set X ⊆ R n is called bounded from above if there exists u ∈ R n such that x ≤ u for all x ∈ X . Similarly, we say that X ⊆ R n is bounded from below if there exists l ∈ R n such that l ≤ x for all x ∈ X . The partially ordered vector space ( R n , ≤ ) is a vector lattice, wherethe binary relations ∧ and ∨ are defined as follows. For x, y ∈ ( R n , ≤ ), x ∧ y is the greatest lower bound of x and y , so ( x ∧ y ) i = min { x i , y i } for1 ≤ i ≤ n , and x ∨ y is least upper bound of x and y , so ( x ∨ y ) i = max { x i , y i } for 1 ≤ i ≤ n .A map f : D → R n , where D ⊆ R m , is called monotone if for each x, y ∈ D with x ≤ y we have that f ( x ) ≤ f ( y ). It is called strongly monotone if x ≤ y and x = y implies that f ( x ) ≪ f ( y ). A map f : D → R n , where D ⊆ R m is convex, is called convex if f ( λx + (1 − λ ) y ) ≤ λf ( x ) + (1 − λ ) f ( y ) for all 0 ≤ λ ≤ x, y ∈ D . In other words, f : D → R n is convex if each coordinate function is convexin the usual sense. The reader may note that our notion of monotonicity isdifferent from the one commonly used in convex analysis [25].The orbit of x ∈ D under a map f : D → D is given by O ( x ; f ) = { f k ( x ) : k = 0 , , , . . . } . We say that x ∈ D is a periodic point of f : D → D if f p ( x ) = x for some integer p ≥
1, and the minimal such p ≥ period of x under f .Let M m,n denote the set of all m × n real matrices, and let P m,n bethe set of all nonnegative matrices in M m,n . A matrix P ∈ P m,n is called positive if p ij > ≤ i ≤ m and 1 ≤ j ≤ n . Given a matrix M ∈ M m,n we denote its rows by M , . . . , M m ∈ M ,n , and we identify M with the m -tuple ( M , . . . , M m ). So, M m,n is identified with the m -folddirect product M m,n = M ,n × . . . × M ,n . We say that
R ⊆ M m,n is rectangular if R can be written as R = R × . . . ×R m , where R , . . . , R m are non-empty subsets of M ,n . Furthermore it isconvenient to introduce the following matrix notation. Given M ∈ M m,n , I ⊆ { , . . . , m } , and J ⊆ { , . . . , n } we write M IJ to denote the | I | × | J | sub-matrix of M with row indices in I and column indices in J . Likewise,given x ∈ R n and K ⊆ { , . . . , n } we write x K ∈ R K to denote the vectorin R K obtained by restricting x to its coordinates in K .For a convex map f : D → R n , where D ⊆ R m is open and convex, the subdifferential of f at v ∈ D is defined by, ∂f ( v ) = { M ∈ M m,n : f ( x ) − f ( v ) ≥ M ( x − v ) for all x ∈ D} . (2)4n the following proposition several basic facts concerning the subdifferentialare collected, cf. [25, Theorem 23.4]. Proposition 2.1. If f is a convex map from an open convex subset D ⊆ R m to R n , then for each v ∈ D , the set ∂f ( v ) is non-empty, compact, convexand rectangular. We note that the rectangularity of ∂f ( v ) follows directly from the factthat f ( x ) − f ( v ) ≥ M ( x − v ) is equivalent to f i ( x ) − f i ( v ) ≥ M i ( x − v ) forall 1 ≤ i ≤ m . If, in addition, the map is monotone, then ∂f ( v ) consists ofnonnegative matrices as the following proposition shows. Proposition 2.2. If f : D → R n , where D ⊆ R m is open and convex, is aconvex monotone map, then ∂f ( v ) ⊆ P m,n for all v ∈ D . Moreover, if f isstrongly monotone, then each P ∈ ∂f ( v ) is positive.Proof. If v ∈ D , then there exists U open neighbourhood of 0 such that v − u ∈ D for all u ∈ U . Now if u ≥
0, with u ∈ U , and P ∈ ∂f ( v ), then0 ≥ f ( v − u ) − f ( v ) ≥ − P u , as f is monotone. Thus, P x ≥ x ∈ R n + and hence P ∈ P m,n . We note that if f is strongly monotone, u ≥ u = 0, then 0 ≫ f ( v − u ) − f ( v ) ≥ − P u . This implies that
P x ≫ x ∈ R n + \ { } and therefore P is positive.For a convex map f : D → R n , where D ⊆ R m is open and convex, and v ∈ D , the one-sided directional derivative of f at v is given by, f ′ v ( y ) = lim ε ↓ f ( v + εy ) − f ( v ) ε . (3)The map f ′ v : R m → R n is well-defined, convex, finite valued and positivelyhomogeneous, meaning that f ′ v ( λx ) = λf ′ v ( x ) for all λ > x ∈ R m , see[25, Theorem 23.1]. Moreover, f ′ v is monotone (because it is defined as apointwise limit of monotone maps). We shall occasionally need the followingrepresentation of f ′ v : f ′ v ( y ) = sup P ∈ ∂f ( v ) P y for y ∈ R m (4)(see [25, Theorem 23.4]). If f : D → D , where
D ⊆ R n is open and convex,then for each v ∈ D we also have that( f ′ v ) k = f ′ f k − ( v ) ◦ · · · ◦ f ′ f ( v ) ◦ f ′ v (see [1, Lemma 4.3]).Throughout the remainder of the exposition we shall make the followingassumption on the domain of convex monotone maps f : D → D .5 ypothesis 2.3. The set
D ⊆ R n is open and convex. Although in some results more general domains can be treated, we re-strict ourselves to this case, as it simplifies the presentation.To conclude this section we briefly discuss several examples of convexmonotone maps. In the theory of Markov decision processes one considersmonotone convex maps f : R n → R n of the form: f i ( x ) = sup j ∈ A i r ji + p j · x for i = 1 , . . . , n. (5)Here r ji ∈ R and p j is a sub-stochastic vector for each j ∈ A i . The resultsin this paper apply to the case where p j is merely a nonnegative vector.Other examples arise in the study of systems of polynomial equations, x = P ( x ), where P = ( P , . . . , P n ) and each P i is a polynomial with variables x , . . . , x n and nonnegative coefficients. Looking for a positive solution of x = P ( x ) is equivalent to finding a fixed point of the map f ( x ) = Log ◦ P ◦ Exp, where Exp( x , . . . , x n ) = ( e x , . . . , e x n ) and Log denotes its inverse.We can write P i ( x ) = P j ∈ A i a ij x j , where A i ⊆ N n is a finite set and each a ij ≥
0, with the convention that x j = x j · · · x j n n for j = ( j , . . . , j n ) ∈ N n .Then f i ( x ) = log( X j ∈ A i a ij exp( j · x )) . (6)Such “log-exp” functions are not only monotone, but also convex, see [26,Example 2.16]. More generally, we could allow A i to be a subset R n + , insteadof N n . This yields a class of functions P i which are usually called posynomials [6]. Posynonmials play a role in static analysis of programs by abstractinterpretation [7]. Note that the example in (5) can be obtained as a limitof posynomials by setting a ij = e βr ji and f βi ( x ) := β − log( X j ∈ A i exp( β ( r ji + p j · x ))) . If β tends to + ∞ , then f βi converges to the map (5), see [32]. Recall that a fixed point v ∈ D of f : D → D is Lyapunov stable if for eachneighbourhood U of v , there exists a neighbourhood V ⊆ D of v such that6 ∈ V and f k ( V ) ⊆ U for all k ≥
1. An n × n matrix P is called stable if allthe orbits of P are bounded. Stable matrices have the following well-knowncharacterizations. Proposition 3.1.
For a matrix P the following assertions are equivalent:(i) P is stable.(ii) There exists a norm on R n such that the induced matrix norm of P isat most one.(iii) All the eigenvalues of P have modulus at most one and all the eigen-values of modulus one are semi-simple.(iv) The origin is a Lyapunov stable fixed point of P . In the analysis of convex monotone dynamical systems, it is useful todistinguish various notions of stability that are weaker than the classicalLyapunov stability.
Proposition 3.2. If f : D → D is a convex monotone map with a fixedpoint v ∈ D , then the following assertions;(i) v is a Lyapunov stable fixed point,(ii) there exists a neighbourhood V ⊆ D of v such that every orbit of x ∈ V is bounded,(iii) there exists a neighbourhood V ⊆ D of v such that every orbit of x ∈ V is bounded from above,(iv) every orbit of f ′ v : R n → R n is bounded,(v) every orbit of f ′ v : R n → R n is bounded from above,(vi) each P ∈ ∂f ( v ) is stable.(vii) every orbit of f ′ v : R n → R n is bounded from below,(viii) every orbit of f : D → D is bounded from below,satisfy the following implications: ( i ) ⇔ ( ii ) ⇔ ( iii ) ⇒ ( iv ) ⇔ ( v ) ⇒ ( vi ) ⇒ ( vii ) ⇒ ( viii ) . roof. The implications ( i ) ⇒ ( ii ) ⇒ ( iii ) and ( iv ) ⇒ ( v ) are trivial. Westart by showing that ( iii ) implies ( v ). Let x ≥ v + x ∈ V .Remark that f ( v + x ) ≥ f ( v ) + f ′ v ( x ) = v + f ′ v ( x ). Since f is monotone, wededuce that f k ( v + x ) ≥ v + ( f ′ v ) k ( x ) for all k ≥ , (7)and hence O ( x ; f ′ v ) is bounded from above. As f ′ v is monotone and positivelyhomogeneous, there exist for each y ∈ R n , a vector x ≥ λ > y ≤ λx and v + x ∈ V . Thus, we get that f ′ v has all its orbitsbounded from above.Next we prove that ( v ) implies ( vi ). If P ∈ ∂f ( v ), then we know by (4)that f ′ v ( x ) ≥ P x . As f ′ v is monotone, we get that ( f ′ v ) k ( x ) ≥ P k x for all k ≥ x ∈ R n and therefore P has all its orbits bounded from above.This implies that P has all its orbits bounded, since P is linear.Suppose that ∂f ( v ) contains a stable matrix P . We deduce from (4)that ( f ′ v ) k ( x ) ≥ P k x for all k ≥ x ∈ R n . As P is stable, this impliesthat O ( x ; f ′ v ) is bounded from below, which shows that ( vi ) implies ( vii ).To see that ( vii ) implies ( viii ) let x ∈ D and note that, by (7), f k ( x ) ≥ v + ( f ′ v ) k ( x − v ) for all k ≥ . As O ( x − v ; f ′ v ) is bounded from below, we get that O ( x ; f ) is also boundedfrom below.Note that ( v ) implies ( vi ) and ( vi ) implies ( vii ), so that ( iv ) and ( v ) areequivalent. Similarly, ( iii ) implies ( viii ), so that ( ii ) and ( iii ) are equivalent.It remains to be shown that ( ii ) implies ( i ).If ( ii ) holds there exists u ≫ v + λu ∈ D for all | λ | ≤ O ( v + u ; f ) is bounded. Let k ≥ < λ < λ ( f k ( v + u ) − v ) = λ ( f k ( v + u ) − v ) + (1 − λ )( f k ( v ) − v ) ≥ f k ( λ ( v + u ) + (1 − λ ) v ) − v = f k ( v + λu ) − v . Take P ∈ ∂f ( v ) and remark that f ( x ) ≥ P ( x − v ) + v . This implies that f k ( x ) ≥ P k ( x − v ) + v for all x ∈ D . Hence f k ( v − λu ) − v ≥ − λP k u for 0 < λ < . (8)As ( ii ) implies ( vi ), we know that P is stable. Define a norm k · k u on R n by k x k u = inf { α > − αu ≤ x ≤ αu } and let γ = sup k ≥ {k P k u k u , k f k ( v + u ) − v k u } . γ < ∞ , as P is stable and O ( v + u ; f ) is bounded. Let λ := k x − v k u , so that v − λu ≤ x ≤ v + λu . If λ <
1, we get that − λγu ≤ f k ( x ) − v ≤ λγu for each k ≥
1. Thus, k f k ( x ) − v k u ≤ γλ = γ k x − v k u forall x ∈ D such that k x − v k u < k ≥
1. Hence v is a Lyapunovstable fixed point. Example 1.
Let f : R n → R n be given by f ( x ) = max { , x + x } for x ∈ R n . Then f ′ ( x ) = max { , x } and hence every orbit of f ′ is bounded.However, the orbit of each x > f . This show that ( iv )does not imply ( iii ). Example 2.
This example shows that ( vi ) does not imply ( v ). Consider h ( p ) = − p log p − (1 − p ) log(1 − p ) for p ∈ [0 , g : R → R by g ( x ) = sup p ∈ [0 , px + h ( p ) x for x = ( x , x ) ∈ R , which is the Legendre-Fenchel transform of − h . Note that g ( x ) = x log(1 + e x /x ), for x >
0. Indeed, put x = 1 and consider ddp ( px + h ( p )) = 0 . Solving for p gives p = e x / (1 + e x ), so that g ( x ,
1) = log(1 + e x ). As g ispositively homogeneous, g ( x ) = x log(1 + e x /x ) for x > f : R → R by f ( x ) = (cid:26) max { , x } if x ≤ x log(1 + e x /x ) if x > f ( x ) = x for all x = ( x , x ) ∈ R . If x >
0, we get that f k ( x ) = x log( k + e x /x ) → ∞ , as k → ∞ . Thus, not all orbits of f ′ = f arebounded from above. But ∂f (0) consists of matrices of the form (cid:18) p s (cid:19) , where 0 ≤ s ≤ h ( p ) and 0 ≤ p ≤ . As h (1) = 0, all these matrices are stable. Example 3.
To see that ( vii ) does not imply ( vi ) we consider the map f : R → R given by f ( x ) = max { , x } for x ∈ R . Then f ′ = f , so everyorbit of f ′ is bounded from below, but 2 ∈ ∂f (0).9 xample 4. To prove that ( viii ) does not imply ( vii ) consider f : R → R given by f (cid:18) x x (cid:19) = (cid:18) e x + e x − x (cid:19) for x = ( x , x ) ∈ R . Then f ′ (cid:18) x x (cid:19) = (cid:18) (cid:19) (cid:18) x x (cid:19) for all x = ( x , x ) ∈ R , so that O ( − u ; f ′ ) is unbounded from below for u ≫
0. But clearly everyorbit of f is bounded from below.We use the following notion of stability which is weaker than ordinaryLyapunov stability according to Proposition 3.2. Definition 3.3.
Let f : D → D be a convex monotone map and v ∈ D be a fixed point of f . We say that v is a tangentially stable , or, t-stable ,fixed point if f ′ v has all its orbits bounded from above. Similarly, we call aperiodic point ξ ∈ D with period p tangentially stable if it is a t-stable fixedpoint of f p .We note that if v is a t-stable periodic point of f with period p , then f m ( v ) is also t-stable for each 0 < m < p . Indeed, as( f p ) ′ v = ( f p − m ) ′ f m ( v ) ◦ ( f m ) ′ v and ( f p ) ′ f m ( v ) = ( f m ) ′ v ◦ ( f p − m ) ′ f m ( v ) , we get that(( f p ) ′ f m ( v ) ) k = (( f m ) ′ v ◦ ( f p − m ) ′ f m ( v ) ) k = ( f m ) ′ v ◦ (( f p ) ′ v ) k − ◦ ( f p − m ) ′ f m ( v ) . Thus, every point in the orbit of a t-stable periodic point is also t-stable.We denote by E ( f ) the set of all fixed points of f and we let E t ( f ) = { v ∈ E ( f ) : v is t-stable } . The subdifferential of a t-stable fixed point consists of nonnegative stablematrices by Proposition 3.2. In the next section we collect some resultsconcerning stable matrices that will be useful in the analysis.
To an n × n nonnegative matrix P = ( p ij ) we associate a directed graph G ( P ) on n nodes, in the usual way, by letting an arrow go from node i to10 if p ij >
0. We say that a node i has access to a node j if there is a(directed) path in G ( P ) from i to j . Using the notion of access one definesan equivalence relation ∼ on { , . . . , n } by i ∼ j if i has access to j andvice versa. The equivalence classes are called classes of P . A nonnegativematrix P is called irreducible if it has only one class. Otherwise it is said tobe reducible .The spectral radius of P is given by ρ ( P ) = max {| λ | : λ eigenvalue of P } .If P is a stable nonnegative matrix, we call a class C of P critical if ρ ( P CC ) =1. Recall that in Perron-Frobenius theory a class C of P is called basic if ρ ( P CC ) = ρ ( P ). Thus, all critical classes of a stable nonnegative matrix arebasic by Proposition 3.1. The following proposition is a direct consequenceof [27, Theorem 3] (see also [29, Corollary 3.4]) and Proposition 3.1. Proposition 4.1. If P is a nonnegative stable matrix and C and C ′ aretwo distinct critical classes of P , then no i ∈ C has access to any j ∈ C ′ . Moreover, we have the following general fact concerning nonnegativematrices (cf. [11, Chapter XIII § Proposition 4.2. If P is a nonnegative matrix, then ρ ( P ′ ) ≤ ρ ( P ) for allprincipal submatrices P ′ of P . If P is reducible, then the equality holds forat least one principal submatrix P ′ of P with P ′ = P . Using these proposition we now prove the following useful normal formfor stable nonnegative matrices.
Proposition 4.3. If P is a nonnegative stable n × n matrix, then there exista permutation matrix Π and a unique partition of { , . . . , n } into disjointsets U , C , D , and I such that(i) Π T P Π = P UU P UC P UD P UI P CC P CD
00 0 P DD
00 0 P ID P II , (9) where C is the disjoint union of the critical classes C , . . . , C r of P ,(ii) P CC is block-diagonal, with blocks P C i C i for ≤ i ≤ r ,(iii) every i ∈ U has access to some j ∈ C , and for every i ∈ D there exists j ∈ C that has access to i ,(iv) I = { , . . . , n } \ ( U ∪ C ∪ D ) . CU D
Figure 1: Partition associated with a stable matrix
Moreover, in that case, we that ρ ( P UU ) < , ρ ( P DD ) < , and ρ ( P II ) < .Proof. Let C be the union of the critical classes C , . . . , C r of P . Then P CC is block diagonal by Proposition 4.1. Let U be the set of nodes of G ( P ) notin C that have access to some i ∈ C . In addition, let D be the set of nodes i in G ( P ) that are not in C , but from which there exists j ∈ C such that j has access to i . We remark that U ∩ D = ∅ . Indeed, if i ∈ U ∩ D , thenthere exist j , j ∈ C such that i has access to j and j has access to i ,By Proposition 4.1, j and j are both in a single class, say C m . But thisimplies that i ∈ C m , which is a contradiction. In fact, the same argumentshows that P DU = 0, P CU = 0, and P DC = 0.Now let I = { , . . . , n } \ ( U ∪ C ∪ D ). By definition no node in I hasaccess to a node C , nor can it be accessed by a node in C . Hence P IC = 0and P CI = 0. Furthermore, the definition of U and D implies that P IU = 0and P DI = 0. Thus, the sets U , C , D , and I , communicate as in Figure1 and hence there exists a permutation matrix Π such that Π T P Π satisfies(9).To prove the final assertion, we remark that if ρ ( P UU ) = 1, then thereexists a critical class U ∗ ⊆ U such that ρ ( P U ∗ U ∗ ) = ρ ( P UU ) = 1 and P U ∗ U ∗ is irreducible by Proposition 4.2, which is a contradiction. In exactly thesame way it can be shown that ρ ( P DD ) < ρ ( P II ) < U , C , D , and I in G ( P ) are respectively called up-stream nodes , critical nodes , downstream nodes , and independent nodes . Byusing the normal form and the Perron-Frobenius theorem we now prove thefollowing assertion. Proposition 4.4. If P is nonnegative stable n × n matrix and P z ≤ z , then(i) P CC z C = z C ,(ii) z D = 0 and ( P z ) C ∪ D = z C ∪ D , iii) z I ≥ ,(iv) if, in addition, z S ≥ for some S ⊆ { , . . . , n } that contains at leastone element of each critical class of P , then z ≥ .Proof. As P z ≤ z , it follows from Proposition 4.3 that P DD z D ≤ z D . Since ρ ( P DD ) <
1, we get that ( P DD ) k z D → k → ∞ , and hence z D ≥
0. Thisimplies that z C ≥ P CC z C + P CD z D ≥ P CC z C . (10)Let C , . . . , C r be the critical classes of P . As P C i C i is nonnegative andirreducible, it follows from the Perron-Frobenius theorem [11, Theorem ??]that there exists for each 1 ≤ i ≤ r a positive m i ∈ R C i such that m i P C i C i = m i . Put m = ( m , . . . , m r ) ∈ R C and remark that, as P CC is block diagonalthat mP CC = m . Multiplication by m from the left in (10) gives mz C ≥ mP CC z C + mP CD z D ≥ mP CC z C = mz C . As m is positive, we deduce that P CD z D = 0 and P CC z C = z C , (11)which proves ( i ).Recall that z D ≥
0. To show ( ii ) we assume by way of contradictionthat z j > j ∈ D . By definition there exists a path ( i , . . . , i q ) in G ( P ) with i = i ∈ C , i , . . . , i q ∈ D , and j = i q . Since P z ≤ z we get that( P CD z D ) i = P iD z D ≥ P i i z i ≥ P i i P i i z i ≥ . . . ≥ (cid:0) q − Y k =1 P i k i k − (cid:1) z i q > , which contradicts (11) and hence z D = 0. By Proposition 4.3( i ) we also findthat ( P z ) C ∪ D = z C ∪ D .To prove that z I ≥
0, we remark that z I ≥ P ID z D + P II z I = P II z I by( ii ). As ρ ( P II ) <
1, we get that z I ≥ P kII z I → k → ∞ , so that z I ≥ S ⊆ { , . . . , n } contains at least one element in eachcritical class of P and z S ≫
0. Remark that z U ≥ P UU z U + P UC z C + P UI z I ≥ P UU z U + P UC z C , since z I ≥
0. As ρ ( P UU ) <
1, we get that( I − P UU ) − = P k ≥ P kUU is nonnegative and z U ≥ ( I − P UU ) − P UC z C . (12)As each P C i C i is irreducible and ρ ( P C i C i ) = 1, it follows from the Perron-Frobenius theorem that z C i is a multiple of the unique positive eigenvectorof P C i C i . By assumption z C i has at least one positive coordinate, and hence z C ≥
0. It now follows from (12) that z U ≥ z ≥ z ∈ R n and S ⊆ { , . . . , n } we simply say that z = 0 on S , or, z is zero on S , if z S = 0. Similar terminology will be used for z S ≥ z S ≫
0. Given a collection P of nonnegative n × n matrices we define G ( P ) = [ P ∈P G ( P ) , (13)and we observe that the following lemma holds. Lemma 4.5. If P is a convex set of nonnegative n × n matrices, then thereexists M ∈ P such that G ( M ) = G ( P ) .Proof. Since the number of edges in G ( P ) is finite there exist F ⊆ P finitesuch that G ( F ) = G ( P ). Define M = |F | − P Q ∈F Q and note that M ∈ P ,as P is convex. Moreover, G ( M ) = G ( F ) = G ( P ).For a stable nonnegative matrix P , we let N c ( P ) denote the set of criticalnodes of G ( P ) and we let G c ( P ) denote the restriction of G ( P ) to N c ( P ).For a collection of stable nonnegative n × n matrices, P , we define N c ( P ) = [ P ∈P N c ( P ) and G c ( P ) = [ P ∈P G c ( P ) . (14)Using these concepts we can now present the main theorem of this section. Theorem 4.6. If P is a convex rectangular set of stable nonnegative n × n matrices, then there exists M ∈ P such that G c ( M ) = G c ( P ) .Proof. The assertion is trivial if G c ( P ) is empty. Let F be a finite set ofmatrices in P such that G c ( F ) = G c ( P ). For each k ∈ N c ( P ) we let Q k = { P k : P ∈ F and k ∈ N c ( P ) } . For k N c ( P ) we pick an arbitrary P ∈ P and put Q k = { P k } . Subse-quently we define an n × n nonnegative matrix M by M k = |Q k | − X q ∈Q k q for 1 ≤ k ≤ n. As P is convex and rectangular, M ∈ P and hence G c ( M ) ⊆ G c ( P ).We claim that G c ( P ) ⊆ G ( M ) by construction. Indeed, if ( i, j ) is an arcin G c ( P ), then there exists P ∈ F such that ( i, j ) is an arc in G c ( P ). Thisimplies that i ∈ N c ( P ) ⊆ N c ( P ) and P i ∈ Q i . As P ij >
0, we get that M ij = |Q i | − X q ∈Q i q j ≥ p ij |Q i | > . G c ( M ) ⊆ G c ( P ) ⊆ G ( M ), so that G c ( P ) | N c ( M ) = G c ( M )(Here G c ( P ) | N c ( M ) denotes the restriction of the graph G c ( P ) to the nodesin N c ( M ).) As N c ( M ) ⊆ N c ( P ), it remains to prove that N c ( P ) ⊆ N c ( M )to establish the equality G c ( M ) = G c ( P ). To show the inclusion we use thefollowing claim. Claim. If C is the set of nodes of a strongly connected component of G c ( P ),then ρ ( M CC ) = 1.If we assume the claim for the moment and take i ∈ N c ( P ), then thereexists a strongly connected component C in G c ( P ) such that i ∈ C . Clearlythere exists a class C ∗ of M such that C ⊆ C ∗ and hence 1 = ρ ( M CC ) ≤ ρ ( M C ∗ C ∗ ) by Proposition 4.2 and the claim. This implies that C ∗ is a criticalclass of M and therefore i ∈ N c ( M ).To prove the claim we consider a nonlinear map g : R C + → R C + given by, g k ( y ) = sup q ∈Q k q C y for k ∈ C and y ∈ R C + . We begin by constructing an eigenvector u ≫ g . As g is monotone,positively homogeneous, and continuous, we can use the Brouwer fixed pointtheorem to find u ∈ R C + , with u = 0, and λ ≥ g ( u ) = λu (see[3, pp.152–154] or [17, p.201]). Since F is finite, Q k is finite, and hence thesup is attained for u and k ∈ C , say by q k ∈ Q k . Now let Q be the n × n nonnegative matrix with Q k = q k for all k ∈ C and Q k is some element in Q k for all k C . As P is rectangular, Q ∈ P . Moreover, Q CC u = g ( u ) = λu and ρ ( Q CC ) ≤ ρ ( Q ) ≤
1, as Q is stable. Thus, we find that λ ≤ i , i ) is an arrow in G c ( P ), then there exists q ∈ Q i with q i >
0. This implies that if x ∈ R C + and x i >
0, then g i ( x ) ≥ q i x i > . (15)Since C is a strongly connected component of G c ( P ), there exists a pathfrom any i to any j in C . Recall that u ∈ R C + and u = 0. Hence there exists j ∈ C such that u j >
0. Now let ( i , i , . . . , i r ) be a path in G c ( P ) | C from i = i to j = i r . By (15), g i r − ( u ) ≥ q i r u i r >
0, and g i r − ( u ) ≥ q i r − g i r ( u ) ≥ q i r − q i r u i r >
0. By repeating the argument we get that u i = g ri ( u ) = g ri ( u ) ≥ (cid:0) r Y k =1 q i k (cid:1) u i r > . u i > i ∈ C .Let k ∈ C and q ∈ Q k . Then there exists P ∈ F such that q = P k and k ∈ N c ( P ). Moreover, there exists a critical class C ′ of P with k ∈ C ′ ⊆ C and ρ ( P C ′ C ′ ) = 1. We note that P C ′ C ′ u C ′ ≤ P C ′ C u ≤ g ( u ) C ′ = λu C ′ ≤ u C ′ , (16)as P l ∈ Q l for all l ∈ C ′ . Since u is positive on C ′ , it follows from Proposition4.4( i ) that P C ′ C ′ u C ′ = u C ′ , so that λ = 1 and P C ′ C u = u C ′ by (16). There-fore, if k ∈ C and q ∈ Q k , then q C u = P kC u = u k , so that M kC u = u k forall k ∈ C . From this we conclude that M CC u = u and hence ρ ( M CC ) = 1,which proves the claim. By using the results from the previous section we can now start analysingthe t-stable fixed points of monotone convex maps. To begin, we have thefollowing lemma.
Lemma 5.1. If f : D → D is a convex monotone map and v and w aret-stable fixed points of f , then G c ( ∂f ( v )) = G c ( ∂f ( w )) .Proof. From Propositions 2.1, 2.2, and 3.2 it follows that ∂f ( v ) and ∂f ( w )are convex rectangular sets of stable nonnegative n × n matrices. By Theo-rem 4.6 there exists M ∈ ∂f ( v ) such that G c ( M ) = G c ( ∂f ( v )). Moreover, w − v = f ( w ) − f ( v ) ≥ M ( w − v ) , so that w − v = M ( w − v ) on C ∪ D , by Proposition 4.4. (Here C and D are the critical nodes and the downstream nodes of M .) This implies that f i ( v ) − f i ( w ) = ( v − w ) i = M i ( v − w ) for all i ∈ C ∪ D. From this equality we deduce that f i ( x ) − f i ( w ) = f i ( x ) − f i ( v ) + f i ( v ) − f i ( w ) ≥ M i ( x − v ) + M i ( v − w )= M i ( x − w )for all x ∈ D and i ∈ C ∪ D . Thus, M i ∈ ∂f i ( w ) for all i ∈ C ∪ D . Now let P ∈ ∂f ( w ) and define Q ∈ P n,n by Q i = (cid:26) M i if i ∈ C ∪ DP i otherwise.16s ∂f ( w ) is rectangular, Q ∈ ∂f ( w ). Clearly G c ( f ( w )) ⊇ G c ( Q ) ⊇ G c ( M ) = G c ( ∂f ( v )). By interchanging the roles of v and w we obtain the desiredequality.By Lemma 5.1 we can define for a convex monotone map f : D → D witha t-stable fixed point v ∈ D , the set of critical nodes of f and the criticalgraph of f respectively by, N c ( f ) = N c ( ∂f ( v )) and G c ( f ) = G c ( ∂f ( v )) . Lemma 5.2.
Let f : D → D be a convex monotone map and let v ∈ D bea t-stable fixed point of f . Let S ⊆ { , . . . , n } be a set that contains at leastone node in each connected component of G c ( ∂f ( v )) . If w ∈ D is a fixedpoint of f and v ≤ w on S , then v ≤ w .Proof. By Theorem 4.6 there exists M ∈ ∂f ( v ) such that G c ( M ) = G c ( ∂f ( v )).Then w − v = f ( w ) − f ( v ) ≥ M ( w − v ) and w − v ≥ S , so that w − v ≥ iv ).From the previous lemma we immediately deduce the following theoremfor t-stable fixed points. Theorem 5.3.
Let f : D → D be a convex monotone map and let v, w ∈ D be t-stable fixed points of f . If v = w on a set S ⊆ { , . . . , n } that has atleast one node in each strongly connected component of G c ( f ) , then v = w .In particular, the t-stable fixed point is unique, if N c ( f ) is empty. To analyse the geometry of the t-stable fixed point set E t ( f ) and t-stable periodic points, we need some preliminary results concerning convexmonotone positively homogeneous maps. If h : R n → R n is a monotone convex positively homogeneous map, then 0is a fixed point and we can associate to h a graph G ( h ) by G ( h ) = G ( ∂h (0)) . If, in addition, 0 is t-stable, then we define A ( h ) = { i : there exists a path in G ( h ) from i to some j ∈ N c ( h ) } and we put B ( h ) = { , . . . , n } \ A ( h ). Convex monotone positively homo-geneous maps, which have 0 as a t-stable fixed point, have the followingproperties. 17 emma 6.1. Let h : R n → R n be a convex monotone positively homogeneousmap, with as a t-stable fixed point. Write C = N c ( h ) , A = A ( h ) , and B = B ( h ) , and identify each x ∈ R n with ( x A , x B ) ∈ R A × R B . Then C ⊆ A and the map h can be rewritten in the form h ( x A , x B ) = ( h A ( x A , x B ) , h B ( x B )) , where h A : R A × R B → R A and h B : R B → R B are convex monotone posi-tively homogeneous maps. Moreover, if h A : R A → R A is given by, h A ( y ) = h A ( y, for all y ∈ R A , then for each y ∈ R A + with y C ≫ and each i ∈ A , there exists k ≥ suchthat ( h A ) ki ( y ) > .Proof. Since h is convex and positively hommogeneous, h ′ = h and h ( x ) =sup P ∈ ∂h (0) P x for all x ∈ R n . Since B = { , . . . , n } \ A , we can write h inthe form h ( x A , x B ) = ( h A ( x A , x B ) , h B ( x A , x B )) . We note that P BA = 0 for all P ∈ ∂h (0). Indeed, otherwise there exists j ∈ B that has access to some node i ∈ A in G ( P ). But this implies thatthere exists a path from j to a node in C in G ( h ), as G ( P ) ⊆ G ( h ), whichcontradicts j ∈ B . Since ∂h (0) is rectangular, h B ( x A , x B ) = sup P ∈ ∂h (0) P BA x A + P BB x B = sup P ∈ ∂h (0) P BB x B . Thus, h B ( x A , x B ) is of the form h B ( x B ), and therefore h can be rewrittenas h ( x A , x B ) = ( h A ( x A , x B ) , h B ( x B )) , for all ( x A , x B ) ∈ R A × R B .To prove the last assertion we let y ∈ R A + be such that y C ≫ i ∈ A .By Lemma 4.5 there exists P ∈ ∂h (0) such that G ( P ) = G ( h ). We have that h A ( z ) = h A ( z, ≥ P AA z for all z ∈ R A . Hence ( h A ) k ( y ) ≥ ( P AA ) k y for all k ≥
1. Since there exists a path from i ∈ A to some node j ∈ C , say withlength m ≥
1, we get that ( h A )( y ) i ≥ ( P AA ) mij y j > Proposition 6.2. If h : R n → R n is a convex monotone positively homoge-neous map, with as a t-stable fixed point, and h has a fixed point v suchthat v ≫ on A ( h ) , then v = 0 on B ( h ) . roof. By Theorem 4.6 there exists M ∈ ∂h (0) such that G c ( M ) = G c ( h ).Let C , . . . , C r denote the critical classes of M , so C = C ∪ . . . ∪ C r . Bythe Perron-Frobenius theorem [11] there exists for each 1 ≤ i ≤ r a positiveeigenvector u i ∈ R C i such that M C i C i u i = u i . Define u ∈ R n by u C i = u i for 1 ≤ i ≤ r and u j = 0 if j C . Clearly u ≥ u ≫ C , and M u ≥ u .This implies that h ( u ) ≥ M u ≥ u ≥ . (17)As 0 is t-stable and h = h ′ , we know that O ( u, h ) is bounded. Moreover,it follows from (17) that ( h k ( u )) k is increasing and hence v = lim k →∞ h k ( u )exists. Obviously v is a fixed point of h and v ≥ u , so that v ≫ C .Remark that u B = 0, because B ∩ C = ∅ . Therefore v B = 0 by Lemma 6.1and hence v A is a fixed point of h A . By the second part of Lemma 6.1 weobtain that v A ≫
0, as v C ≫ h : R n → R n the spectral radius isdefined by τ ( h ) = sup { λ ≥ h ( x ) = λx for some x ∈ R n + \ { }} . (18) Proposition 6.3.
Let h : R n → R n be a convex monotone positively ho-mogeneous map,with as a t-stable fixed point. If h B : R B → R B is as inLemma 6.1, then τ ( h B ) < .Proof. Assume by way of contradiction that τ ( h B ) = r ≥
1. Then thereexists v ≥ v = 0 such that h B ( v ) = rv ≥ v . But h B ( v ) =sup P ∈ ∂h (0) P BB v and ∂h (0) is a rectangular compact set of stable nonnega-tive matrices. Hence h B ( v ) = Q BB v = rv for some Q ∈ ∂h (0). This impliesthat a class K of Q such that K ⊆ B and ρ ( Q KK ) = r ≥
1. As Q is stable, r = 1, and hence K ⊆ N c ( Q ) ⊆ N c ( h ) ⊆ A , which is a contradiction.It is shown by Nussbaum [21, Theorem 3.1] that τ ( h ) = τ ′ ( h ), where τ ′ ( h ) is the so called Collatz-Wielandt spectral radius of a monotone posi-tively homogeneous map h : R n → R n , which is given by τ ′ ( h ) = inf { µ > h ( x ) ≤ µx for some x ≫ } . (19)Thus, Proposition 6.3 has the following consequence. Corollary 6.4. If h : R n → R n is a convex monotone positively homoge-neous map, with as a t-stable fixed point, and h B is as in Lemma 6.1, thenthere exist < λ < and w ≫ such that h B ( w ) ≤ λw .
19e conclude this section by showing that a monotone convex positivelyhomogeneous map with 0 as a t-stable fixed point is non-expansive withrespect to a polyhedral norm. Recall that a norm on R n is called polyhedral if its unit ball is a polyhedron. Theorem 6.5.
Let h : R n → R n be a monotone convex positively homoge-neous map with as a t-stable fixed point, then there exist v ≫ and α > such that h is non-expansive with respect to the polyhedral norm, k x k v = max i ∈ A ( h ) | x i /v i | + α max i ∈ B ( h ) | x i /v i | for x ∈ R n , (20) where A ( h ) and B ( h ) are as in Lemma 6.1.Proof. We use the same notation as in Lemma 6.1. By Proposition 6.2 h has an eigenvector u ∈ R n such that u ≫ A = A ( h ) and u = 0 on B = B ( h ). Moreover, by Corollary 6.4 there also exists 0 < λ < w ∈ R B such that w ≫ h B ( w ) ≤ λw . Let v ∈ R A × R B be defined by v = u on A and v = w on B . Further let W be the diagonal matrix with v as its diagonal and define g : R n → R n by g ( x ) = ( W − ◦ h ◦ W )( x ) for all x ∈ R n . It follows from Lemma 6.1 that we can write g in the form g ( x ) = ( g A ( x A , x B ) , g B ( x B )) , where g A : R A × R B → R B and g B : R B → R B are convex monotone positivehomogeneous maps. Moreover, g A ( µ ,
0) = µ and g B ( µ ) ≤ λµ for all µ ≥
0, where denotes the vector with all coordinates unity.We write k · k ∞ to denote the sup-norm, so k x k ∞ = max i | x i | . Remarkthat g ( x ) B − g ( y ) B ≤ g ( x − y ) B = g B (( x − y ) B ) ≤ λ k ( x − y ) B k ∞ , as g is convex, monotone and positively homogeneous. By interchanging theroles of x and y we deduce that k g ( x ) B − g ( y ) B k ∞ ≤ λ k x B − y B k ∞ for all x, y ∈ R n . (21)Subsequently we remark that there exists C > g A (0 , x B ) ≤ C k x B k ∞ , as g A is continuously and positively homogeneous. (Recall that g ( x ) = sup P ∈ ∂g (0) P x for each x ∈ R n .) Thus, for each x ∈ R n , g ( x ) A = g A ( x A , x B ) ≤ g A ( x A ,
0) + g B (0 , x B ) ≤ k x A k ∞ + C k x B k ∞ . (22)20s g is convex, g ( x ) ≤ g ( x − y ) + g ( y ) , (23)so that g ( x ) − g ( y ) ≤ g ( x − y ) for all x, y ∈ R n . As g (0) = 0, we have that − g ( y ) ≤ g ( − y ), and hence it follows from (22) that k g ( x ) A k ∞ ≤ k x A k ∞ + C k x B k ∞ (24)for all x ∈ R n . It also follows from (23) that k g ( x ) A − g ( y ) A k ∞ ≤ max {k g ( x − y ) A k ∞ , k g ( y − x ) A k ∞ } . (25)Now let α > C/ (1 − λ ) and define k · k ′ on R n by k x k ′ = k x A k ∞ + α k x B k ∞ for all x ∈ R A × R B . It now follows from (21), (24) and (25) that k g ( x ) − g ( y ) k ′ = k g ( x ) A − g ( y ) A k ∞ + α k g ( x ) B − g ( y ) B k ∞ ≤ k ( x − y ) A k ∞ + C k ( x − y ) B k ∞ + αλ k ( x − y ) B k ∞ ≤ k ( x − y ) A k ∞ + α k ( x − y ) B k ∞ = k x − y k ′ . Finally, we recall that g ◦ W − = W − ◦ g , so that h is non-expansive withrespect to k W − ( · ) k ′ = k · k v and we are done. Throughout this section we assume, in addition to Hypothesis 2.3, that thedomain
D ⊆ R n satisfies the following property. Hypothesis 7.1.
The domain
D ⊆ R n is a downward set, i.e., if x ∈ D and y ≤ x , then y ∈ D . Given a convex monotone map f : D → D we define E + ( f ) = { z ∈D : f ( z ) ≤ z } . As f is convex, E + ( f ) is convex. There exists a naturalprojection from E + ( f ) onto E ( f ) if f has a t-stable fixed point (cf. [1, Lemma3.3]). Lemma 7.2. If f : D → D is a convex monotone map with a t-stable fixedpoint, then f ω ( z ) = lim k →∞ f k ( z ) (26) exists and f ω ( z ) = z on N c ( f ) for each z ∈ E + ( f ) . In addition, the map f ω : E + ( f ) → E ( f ) is a surjective convex monotone projection, i.e., ( f ω ) = f ω . roof. Since f : D → D has a t-stable fixed point, all orbits of f are boundedfrom below by Proposition 3.2. Therefore, f ω ( z ) = lim k →∞ f k ( z ) exists forall z ∈ E + ( f ), as ( f k ( z )) k is a decreasing sequence and D is downward. Bycontinuity of f , f ω ( z ) is a fixed point of f .Let v be a t-stable fixed point of f and z ∈ E + ( f ). By Theorem 4.6there exists Q ∈ ∂f ( v ) such that G c ( Q ) = G c ( ∂f ( v )) = G c ( f ). We also havethat z − v ≥ f ( z ) − v = f ( z ) − f ( v ) ≥ Q ( z − v ) . (27)From Proposition 4.4 it follows that z − v = Q ( z − v ) on N c ( f ) = N c ( Q ) andhence z = f ( z ) on N c ( f ). Replacing z by f k ( z ) in the previous argumentgives f k +1 ( z ) = f k ( z ) = . . . = z on N c ( f ) for all k ≥
1. Thus, f ω ( z ) =lim k →∞ f k ( z ) = z on N c ( f ). Clearly, f ω ( x ) = x if x ∈ E ( f ), so that f ω : E + ( f ) → E ( f ) is onto and ( f ω ) = f ω . Moreover, as f ω is the pointwiselimit of ( f k ) k , f ω is a convex monotone map.The fixed point set E ( f ) can be naturally equipped with a binary oper-ation ∧ f that turns ( E ( f ) , ∧ f ) into an inf-semilattice, if f : D → D has at-stable fixed point. The relation ∧ f is E ( f ) is defined by x ∧ f y = lim k →∞ f k ( x ∧ y ) . (28)We note that if x, y ∈ E ( f ), then f ( x ∧ y ) ≤ f ( x ) = x and f ( x ∧ y ) ≤ f ( y ) = y , so that f ( x ∧ y ) ≤ x ∧ y . As f : D → D has all its orbits bounded frombelow and D is downward, the limit (28) exists. To prove that ( E ( f ) , ∧ f ),is an inf-semilattice one has to show that ∧ f is associative, symmetric, andreflexive, which is a simple exercise. It also follows from Lemma 7.2 thatif we put C = N c ( f ) and define r C : E ( f ) → R C by r C ( x ) = x C for all x ∈ E ( f ), then r C ( E ( f )) is an inf-semilattice in R C , where ∧ is the infimumoperation induced by the partial ordering ≤ on R C . Indeed, if x, y ∈ E ( f )and v ∈ D is a t-stable fixed point of f : D → D , then there exists M ∈ ∂f ( v )such that G c ( M ) = G c ( f ). As f ( x ∧ y ) ≤ f ( x ) = x and f ( x ∧ y ) ≤ f ( y ) = y , f ( x ∧ y ) ≤ x ∧ y , so that f k ( x ∧ y ) ≤ f k − ( x ∧ y ) for all k ≥
1. This impliesthat f k − ( x ∧ y ) − v ≥ f k ( x ∧ y ) − f ( v ) ≥ M ( f k − ( x ∧ y ) − v ) . By Proposition 4.4 we get that f k − ( x ∧ y ) − v = f k ( x ∧ y ) − f ( v ) = f k ( x ∧ y ) − v C for all k ≥
1. Hence r C ( x ∧ f y ) = ( x ∧ f y ) C = lim k →∞ f k ( x ∧ y ) C = ( x ∧ y ) C = r C ( x ) ∧ r C ( y ) , so ( r C ( E ( f )) , ∧ ) is an inf-semilattice in R C . The difference between ( E ( f ) , ∧ f )and ( r C ( E ( f )) , ∧ ) is illustrated by the following simple example. Consider P = / /
20 1 00 0 1 . So, P is a projection. Clearly, E ( P ) = span { (1 / , , , (1 / , , } , but E ( P ) is not an inf-semilattice with respect to ∧ , as (1 / , ,
0) = (1 / , , ∧ (1 / , ,
1) is for instance not in E ( P ). In this case N c ( P ) = { , } andspan { (1 , , (0 , } is an inf-semilattice with respect to ∧ .Let us now analyse the t-stable fixed point set in more detail. We shallprove the following theorem. Theorem 7.3. If f : D → D is a convex monotone map with a t-stable fixedpoint, then ( E t ( f ) , ∧ f ) is an inf-semilattice and ( r C ( E t ( f )) , ∧ ) is a convexinf-semilattice in R C , where C = N c ( f ) . But first we give two preliminary lemmas.
Lemma 7.4.
Let f : D → D be a convex monotone map with a t-stable fixedpoint. Let z ∈ E + ( f ) and w = f ω ( z ) be given by (26). Write S = { i : w i
0) for all y ∈ R S and denote v = ( z − w ) S . Then v ≫ v ≥ h ( z − w ) S ≥ h S ( v ) . This implies that τ ( h S ) ≤
1, as τ ( h S ) = τ ′ ( h S ) by [21, Theorem 3.1]. (Here τ ( · ) and τ ′ ( · ) are as in (18) and (19), respectively.Assume by way of contradiction that τ ( h S ) = 1. Then there exists u ∈ R S + such that u = 0 and h S ( u ) = u . Then η = ( u, ∈ R S × R E satisfies h ( η ) = η by (29). Since ∂f ( w ) is a compact rectangular set ofnonnegative matrices and h ( η ) = sup P ∈ ∂f ( w ) P η , there exists P ∈ ∂f ( w )such that η = h ( η ) = P η . Let S ′ = { i : η i = 0 } and remark that S ′ ⊆ S and S ′ is a union of classes of P . To proceed we need the notion of a final class.A class of a nonnegative matrix is called final if it has no access to any otherclass. It is known (see [5, Theorem 3.10]) that a nonnegative matrix M hasa positive eigenvector if, and only if, each final class of M is basic. Clearly P S ′ S ′ has a final class, say F . As P S ′ S ′ η S ′ = η S ′ and η S ′ ≫
0, we find that F is a basic class of P S ′ S ′ , and hence ρ ( P F F ) = ρ ( P S ′ S ′ ) = 1. By (30) wehave that( z − w ) F ≥ ( f ( z ) − w ) F ≥ h ( z − w ) F ≥ ( P ( z − w )) F ≥ P F F ( z − w ) F . (31)By the Perron-Frobenius theorem there exists m ≫ R F such that mP F F = m . This implies that m ( z − w ) F ≥ mP F F ( z − w ) F = m ( z − w ) F ,and hence ( z − w ) F = P F F ( z − w ) F . Thus, f ( z ) F = z F by (31). Similarlywe deduce that f k ( z ) F = z F for all k ≥
1. Indeed, z − w ≥ f k ( z ) − w ≥ ( f k ) ′ w ( z − w ) = ( f ′ w ) k ( z − w ) = h k ( z − w )and h k ( z − w ) F ≥ ( P k ) F F ( z − w ) F ≥ ( P F F ) k ( z − w ) F . Recall that w F = lim k →∞ f k ( z ) F and therefore w F = z F . But thisimplies that F ⊆ E , which contradicts the fact that F ⊆ S ′ ⊆ S . Thus, weconclude that τ ( h S ) < Lemma 7.5.
Let h , h E , h S , S and E be as in Lemma 7.4. If h E : R E → R E has all its orbits bounded from above, then h has all its orbits bounded fromabove. roof. Since h : R n → R n is monotone, it suffices to prove that O ( x ; h ) isbounded from above for all x ∈ R n + . As h can be written in the form (29),we know that { h k ( x ) E : k ≥ } = { h kE ( x E ) : k ≥ } is bounded from above.It therefore remains to be shown that { h k ( x ) S : k ≥ } is bounded fromabove. Since τ ′ ( h S ) = τ ( h S ) <
1, there exist u ∈ R S + and 0 < α < u ≫ h S ( u ) ≤ αu . For y ∈ R S we define a norm by k y k u = max i ∈ S | y i /u i | . For each y ∈ R S + we have that y ≤ k y k u u , so that k h S ( y ) k u ≤ k h S ( k y k u u ) k u = k h S ( u ) k u k y k u ≤ α k y k u , as h S is positively homogeneous and monotone. Since { h k ( x ) E : k ≥ } isbounded from above, there exists v ≫ R E such that h k ( x ) E ≤ v for all k ≥
0. This implies that h (0 , h k ( x ) E ) S ≤ h (0 , v ) S ≤ γu for some γ >
0. Now using the fact that h is positively homogeneous andconvex, we get that0 ≤ h k +1 ( x ) S ≤ h ( h k ( x ) S , S + h (0 , h k ( x ) E ) S ≤ h S ( h k ( x ) S ) + γu, so that k h k +1 ( x ) S k u ≤ k h S ( h k ( x ) S ) k u + γ ≤ α k h k ( x ) S k u + γ. By induction we obtain k h k +1 ( x ) S k u ≤ α k k x S k u + γ − α , which shows that { h k ( x ) S : k ≥ } is bounded from above.Let us now prove Theorem 7.3 Proof of Theorem 7.3.
To prove that ( E t ( f ) , ∧ f ) is an inf-semilattice, it suf-fices to show that if x, y ∈ E t ( f ), then x ∧ f y ∈ E t ( f ), as ( E ( f ) , ∧ f ) isan inf-semilattice. So, suppose that x, y ∈ E t ( f ). Put z = x ∧ y and let w = x ∧ f y . We need to show that h = f ′ w : R n → R n has all its orbitsbounded from above by Proposition 3.2. By Lemma 7.4 we can write h inthe form of (29), since f ( z ) ≤ z . We also get that τ ( h S ) <
1. By Lemma 7.525t is sufficient to prove that h E : R E → R E has all its orbit bounded fromabove. For each i ∈ E we have that f i ( w ) = w i = x i ∧ y i = f i ( x ) ∧ f i ( y ) , as x, y, w ∈ E ( f ). Note that w ≤ x ∧ y implies that w + εu ≤ ( x ∧ y ) + εu = ( x + εu ) ∧ ( y + εu )for all u ∈ R n and ε >
0. Now let u ∈ R n and ε > x + εu, y + εu ∈ D . Then f i ( w + εu ) ≤ f i ( x + εu ) ∧ f i ( y + εu ) , since f i is monotone, and hence f i ( w + εu ) − f i ( w ) ≤ f i ( x + εu ) ∧ f i ( y + εu ) − f i ( x ) ∧ f i ( y ) ≤ max { f i ( x + εu ) − f i ( x ) , f i ( y + εu ) − f i ( y ) } for all i ∈ E . This implies that f ′ w ( u ) i = lim ε ↓ f i ( w + εu ) − f i ( w ) ε ≤ max { f ′ x ( u ) i , f ′ y ( u ) i } (32)for all i ∈ E .Applying the same argument for the map f k and using the fact that( f k ) ′ v = ( f ′ v ) k for all v ∈ E ( f ), we obtain that( f ′ w ) k ( u ) i ≤ max { ( f ′ x ) k ( u ) i , ( f ′ y ) k ( u ) i } for all u ∈ R n , i ∈ E and k ≥
1. But x and y are t-stable and therefore( h E ) k ( s ) = ( h k (0 , s )) E = ( f ′ w ) k (0 , s ) E is bounded from above as k → ∞ for all s ∈ R E . Thus, we conclude that w is a t-stable fixed point of f .Recall that ( r C ( E ( f )) , ∧ ) is an inf-semilattice in R C , where C = N c ( f ),and r C ( x ∧ f y ) = r C ( x ) ∧ r C ( y ) for all x, y ∈ E ( f ). This implies that( r C ( E t ( f )) , ∧ ) is a inf-semilattice in R C . To show that r C ( E t ( f )) is convexwe apply the same technique as before.Let x, y ∈ E t ( f ) and 0 < λ <
1. Put z = λx + (1 − λ ) y . Since f is convex, f ( z ) ≤ λf ( x ) + (1 − λ ) f ( y ) = λx + (1 − λ ) y = z. w = f ω ( z ) = lim k →∞ f k ( z ) exists and is a fixed point of f with w C = z C = λx C + (1 − λ ) y C . We need to show that w is t-stable. Let h = f ′ w and recall that is suffices to show that h E all its orbit bounded fromabove by Lemma 7.5. Let i ∈ E and note that as x, y, w ∈ E ( f ), f ki ( w ) = w i = λx i + (1 − λ ) y i = λf ki ( x ) + (1 − λ ) f ki ( y ) . Clearly w ≤ λx + (1 − λ ) y , so that w + εu ≤ λ ( x + εu ) + (1 − λ )( y + εu )for all u ∈ R n and ε >
0. Now fix u ∈ R n . Then for all ε > f ki ( w + εu ) ≤ f ki ( λ ( x + εu )+(1 − λ )( y + εu )) ≤ λf ki ( x + εu )+(1 − λ ) f ki ( y + εu )for all k ≥
1. Thus, f ki ( w + εu ) − f ki ( w ) ≤ λ ( f ki ( x + εu ) − f ki ( x )) + (1 − λ )( f ki ( y + εu ) − f ki ( y ))and hence ( f k ) ′ w ( u ) i ≤ λ ( f k ) ′ x ( u ) i + (1 − λ )( f k ) ′ y ( u ) i ≤ λ ( f ′ x ) k ( u ) i + (1 − λ )( f ′ y ) k ( u ) i for all k ≥ i ∈ E . As x, y ∈ E t ( f ), the right hand side is boundedfrom above as k → ∞ . Therefore (( f ′ w ) k (0 , s ) E ) k = (( h E ) k ( s )) k is boundedfrom above for all s ∈ R E , which shows that w is t-stable.We note that if E t ( f ) is compact, then it is a connected set. To showthis it suffices to prove that r − C is continuous, as r C ( E t ( f )) is convex. Notethat r C is one-to-one on E t ( f ) by Theorem 5.3. So, let y k → y in r C ( E t ( f )), r C ( x k ) = y k for all k , and r C ( x ) = y . If ( x k i ) i is a subsequence of ( x k ) k and x k i → z , then z ∈ E t ( f ) by compactness. Moreover, r C ( z ) = y , whichimplies that z C = x C . Thus, by Theorem 5.3, z = x , and hence r − C iscontinuous.Another consequence of Theorem 7.3 is the following. Recall that O ( ξ ) = { ξ, f ( ξ ) , . . . , f p − ( ξ ) } ⊆ D is a Lyapunov stable periodic orbit of f if for allneighbourhoods U i of f i ( ξ ), i = 0 , . . . , p −
1, there exist neighbourhoods V i of f i ( ξ ), i = 0 , . . . , p −
1, such that f kp + i ( y ) ∈ U i for all y ∈ V i for all k ≥ Corollary 7.6. If f : D → D is a convex monotone map with a t-stableperiodic point ξ ∈ D , then f has a t-stable fixed point. Moreover, if ξ has aLyapunov stable orbit, then f has a Lyapunov stable fixed point. roof. Let ξ ∈ D be a t-stable periodic point of f with period p . Put g = f p and note that f k ( ξ ) is a t-stable fixed point of g for all 0 ≤ k < p . Let z = ξ ∧ f ( ξ ) ∧ · · · ∧ f p − ( ξ ). Clearly g ( z ) ≤ z and hence u := g ω ( z ) existsand is a t-stable fixed point of g by Theorem 7.3. As f k ( z ) ≤ z for all0 ≤ k < p , we have that f k ( u ) = f k ( g ω ( z )) = g ω ( f k ( z )) ≤ u for all 0 ≤ k < p . In particular, f ( u ) ≤ u and u = g ( u ) = f p ( u ) ≤ f ( u ) ≤ u, so that f ( u ) = u . Moreover, as ( f ′ u ) p = g ′ u and f ′ u is continuous, we concludethat u is a t-stable fixed point of f .Now assume that ξ has a Lyapunov stable orbit. For each i = 0 , . . . , p − V i of f i ( ξ ) such that the orbit of each y ∈ V i is bounded from above. Let W i = { x ∈ D : x ≤ y for some y ∈ V i } and put W = ∪ p − i =0 W i . Note that, as f i ( ξ ) ∈ V i , u ∈ W and W is a neighbourhoodof u . As f is monotone and the orbit of each y ∈ V i , i = 0 , . . . , p −
1, isbounded from above under f , the orbit of each w ∈ W is bounded fromabove, and hence u is a Lyapunov stable fixed point.We also have the following result. Lemma 7.7.
Suppose f : D → D is a convex monotone map and v and w are fixed points with w ≤ v . If w ≪ v or v Lyapunov stable, then w isLyapunov stable.Proof. Suppose that w ≪ v and let V = { x ∈ D : x ≪ z be an openneighbourhood of w , where w ≪ z ≪ v . For each x ∈ V we have that x ≪ v , so that f k ( x ) ≤ f k ( v ) = v for all k ≥
0. Thus, the orbit of x isbounded from above and hence w is Lyapunov stable by Proposition 3.2.Now assume that w ≤ v and v is Lyapunov stable. Then there exists aneighbourhood U of v such that the orbit of each y ∈ U is bounded fromabove. Let W = { x ∈ D : x ≤ y for some y ∈ U } . Note that W is aneighbourhood of w , since v ∈ U and w ≤ v . As f is monotone, the orbitof each x ∈ W is bounded from above under f , and hence w is Lyapunovstable.We conclude this section by showing that every t-stable fixed point of aconvex monotone piece-wise affine map is Lyapunov stable. Recall that amap f : R n → R n is piece-wise affine if R n can be partitioned into polyhedrasuch that the restriction of f to each polyhedron is an affine map.28 orollary 7.8. If f : R n → R n is a convex monotone piece-wise affine map,then every t-stable fixed point of f is Lyapunov stable.Proof. Since f is piece-wise affine, we can find by [1, Lemma 6.4] a neighbor-hood W of 0 such that f ( v + x ) = f ( v ) + f ′ v ( x ) for all x ∈ W . By Theorem6.5 there exists a norm under which f ′ v is non-expansive. Since f ′ v (0) = 0, wecan take any open ball, B , around 0 for this norm, and get f ′ v ( B ) ⊆ B . Bytaking B of sufficiently small radius, we can guarantee that B ⊆ W . Since f ( v ) = v , we find for all x ∈ B that f ( v + x ) = v + f ′ v ( x ) ∈ v + B , whichshows that f ( v + B ) ⊆ v + B . Since this inclusion holds for all balls B ofsufficiently small radius, v is a Lyapunov stable fixed point of f . For a directed graph G and integer k ≥ G k be the directed graphthat has the same nodes as G and it has an arrow from node i to node j if,and only if, there exists a directed path of length k in G from i to j . Thereexists the following relation between G c ( f k ) and ( G c ( f )) k . Theorem 8.1. If f : D → D is a convex monotone map with a t-stable fixedpoint, then G c ( f k ) = ( G c ( f )) k for all k ≥ . To prove this theorem we reduce it to a special case, which was analysedin [1]. Recall that g : R n → R n is called additively homogeneous if g ( x + λ ) = g ( x ) + λ for all x ∈ R n and λ ∈ R . (Here is the vector in R n withall coordinates unity.) The map g is said to be additively subhomogeneous if g ( x + λ ) ≤ g ( x ) + λ for all x ∈ R n and λ ≥
0. The following theoremfor convex monotone additively homogeneous maps is proved in [1, Theorem4.1].
Theorem 8.2 ([1]) . If g : R n → R n is a convex monotone additively homo-geneous map with a fixed point, then G c ( g k ) = ( G c ( g )) k for all k ≥ . We note that every fixed point of a monotone subhomogeneous map g : R n → R n is stable, because g is non-expansive with respect to the sup-norm [8]. Using a standard ”cemetery state” argument, see [1, Section 1.4],we derive the following consequence of Theorem 8.2. Corollary 8.3. If g : R n → R n is a convex monotone additively subhomo-geneous map with a fixed point, then G c ( g k ) = ( G c ( g )) k for all k ≥ . roof. Define h : R n +1 → R n +1 by h ( x, x n +1 ) = (cid:18) x n +1 + g ( x − x n +1 ) x n +1 (cid:19) for all ( x, x n +1 ) ∈ R n +1 . It is easy to verify that h is a convex monotoneadditively homogeneous map. Let v ∈ R n be a fixed point of g and remarkthat w = ( v,
0) is a fixed point of h . Due to the triangular structure of h ,we see that ∂g ( v ) = ∂h ( w ) JJ , where J = { , . . . , n } and G c ( h ) is the unionof G c ( g ) with the loop { ( n + 1) , ( n + 1) } .The same is true for the critical graph of g k and h k , as h k ( x, x n +1 ) = (cid:18) x n +1 + g k ( x − x n +1 ) x n +1 (cid:19) for all ( x, x n +1 ) ∈ R n +1 . It follows from Theorem 8.2 that G c ( h k ) = ( G c ( h )) k for all k ≥
1. By considering the subgraph on the nodes { , . . . , n } we findthat G c ( g k ) = ( G c ( g )) k for all k ≥ Proposition 8.4. If f : D → D is a convex monotone map with a t-stablefixed point v ∈ D , then G c (( f ′ v ) k ) = ( G c ( f ′ v )) k for all k ≥ .Proof. Recall that f ′ v : R n → R n is a convex monotone positively homoge-neous map. Write h = f ′ v and let A , B , C , h A , h B and h A be as in Lemma6.1. Similar notation will be used for h k , so ( h k ) A , ( h k ) B and ( h A ) k . Bydefinition of A we know that G c ( h A ) = G c ( h ). As v is a t-stable fixed pointof f , 0 is t-stable for h . It follows from Lemma 6.1 that we can write h inthe form h ( x A , x B ) = ( h A ( x A , x B ) , h B ( x B )) , where h A : R A × R B → R A and h B : R B → R B are convex monotone posi-tively homogeneous maps. Due to the this form we have that ( h k ) B = ( h B ) k and ( h k ) A = ( h A ) k , where h A : R A → R A is given by h A ( x A ) = h A ( x A , G c (( h A ) k ) = G c ( h k ).By Proposition 6.2 there exists v A ≫ R A such that h A ( v A ) = v A .Let W be the | A | × | A | diagonal matrix with W ii = ( v A ) i for all i ∈ A . Nowdefine g : R A → R A by g ( x ) = ( W − ◦ h A ◦ W )( x ) for all x ∈ R A . G c (( h A ) k ) = G c ( g k ) for all k ≥
1. Moreover, g (0) = 0 and g ( ) = .so that g ( x + λ ) ≤ g ( x ) + λ for all x ∈ R A and λ ≥
0, as g is convex and positively homogeneous. ByCorollary 8.3 we get that G c ( g k ) = ( G c ( g )) k for all k ≥
1. From this itfollows that G c ( h k ) = G c (( h A ) k ) = G c ( g k ) = ( G c ( g )) k = ( G c ( h A )) k = ( G c ( h )) k , which completes the proof.By using Proposition 8.4 it is now easy to prove Theorem 8.1 Proof of Theorem 8.1.
Let f : D → D be a convex monotone map with at-stable fixed point v ∈ D . Then 0 is a t-stable fixed point of f ′ v . For each k ≥ ∂f k ( v ) = ∂ ( f k ) ′ v (0), so that G c ( f k ) = G c ( ∂f k ( v )) = G c ( ∂ ( f k ) ′ v (0)) = G c (( f k ) ′ v ) . Thus, it follows from Proposition 8.4 that G c ( f k ) = G c (( f k ) ′ v ) = G c (( f ′ v ) k ) = ( G c ( f ′ v )) k = ( G c ( f )) k for each k ≥
1, and we are done.To analyse the periods of t-stable periodic points we need to recall thenotion of cyclicity of a graph. The cyclicity of a strongly connected directedgraph G , denoted c ( G ), is the greatest common divisor of the lengths of itscircuits. The cyclicity of a general directed graph G is given by, c ( G ) = lcm { c ( G i ) : G i is a strongly connected component of G} . The cyclicity of a nonnegative stable matrix P is defined by c ( P ) = c ( G c ( P )).Note that c ( P ) is the order of a permutation on n letters, where n is the sizeof the matrix. The following consequence of the Perron-Frobenius theoremconcerning the cyclicity of stable nonnegative matrices can be found in [20,Theorem 9.1]. Theorem 8.5 ([20]) . If P is stable nonnegative matrix, then the period ofeach periodic point of P divides c ( P ) . For a convex monotone map f : D → D with a t-stable fixed point, wedefine the cyclicity of f by c ( f ) = c ( G c ( f )).31 heorem 8.6. If D ⊆ R n is downward and f : D → D is a convex monotonemap with a t-stable fixed point, then the period of each t-stable periodic pointof f divides c ( f ) . In particular, the period of each t-stable periodic point isthe order of a permutation on n letters.Proof. Let v ∈ D be a t-stable fixed point of f and let ξ ∈ D be a t-stableperiodic point of f with period p . Put g = f c ( f ) and note that c ( g ) = 1.Indeed, by Theorem 8.1 we get that G c ( g ) = G c ( f c ( f ) ) = ( G c ( f )) c ( f ) = ∪ si =1 G c ( f ) i , where G , . . . , G s are the disjoint strongly connected componentsof G c ( f ). Let c i = c ( G i ) for 1 ≤ i ≤ s and note that c i divides c ( f ). Itwell-known that if G i is a strongly connected graph with cyclicity c i , then c ( G kc i i ) = 1 for all k ≥ c ( G c ( f ) i ) = 1 for all 1 ≤ i ≤ s . Thus, c ( g ) = c ( ∪ si =1 G c ( f ) i ) = lcm { c ( G c ( f ) i ) : 1 ≤ i ≤ s } = 1 . By Theorem 4.6 there exists M ∈ ∂g ( v ) such that G c ( M ) = G c ( g ), so c ( M ) = c ( g ) = 1. Now let C , D , U , and I as in Proposition 4.3. As g ( v ) = v and g ( x ) − g ( v ) ≥ M ( x − v ) for all x ∈ D , we get that g k ( x ) ≥ M k ( x − v ) + v for all x ∈ D and k ≥ . (33)In particular, ξ = g p ( ξ ) ≥ M p ( ξ − v ) + v, (34)so that ξ − v ≥ M p ( ξ − v ). It follows that ξ − v = M p ( ξ − v ) on C ∪ D and ξ − v = 0 on D from Proposition 4.4. This implies that g p ( ξ ) ≥ ξ on C ∪ D .Put F = C ∪ D and G = U ∪ I . By definition we have that M F G = 0,and so M kF G = 0 for all k ≥
1. From this we deduce that( ξ − v ) F = ( M p ( ξ − v )) F = ( M F F ) p ( ξ − v ) F = M pF F ( ξ − v ) F . The matrix M is stable and has cyclicity one. Therefore the matrix M F F is also stable and has cyclicity one. This implies that any periodic point of M F F must have period one by Theorem 8.5. Thus, we find that ( ξ − v ) F = M F F ( ξ − v ) F , so that ( ξ − v ) F = M kF F ( ξ − v ) F for all k ≥
1. Since M F G = 0we deduce that ( ξ − v ) = M k ( ξ − v ) on F. From (33) it follows that g k ( ξ ) ≥ ξ on F . Let z = ξ ∧ g ( ξ ) ∧ . . . ∧ g p − ( ξ ).Clearly z ≤ ξ and z = ξ on F . As g ( z ) ≤ z , it follows from Lemma 7.2 that g ω ( z ) = lim k →∞ g k ( z ) exists and g ω ( z ) = z on C . Thus, g ω ( z ) = ξ on C .32ote that g ω ( z ) and ξ are fixed points of g p , and ξ is a t-stable fixed pointof g p = f pc ( f ) . Therefore it follows from Lemma 5.2 that g ω ( z ) ≥ ξ . As g ω ( z ) ≤ z ≤ ξ , we conclude that g ω ( z ) = z = ξ . Hence f c ( f ) ( ξ ) = g ( ξ ) = ξ from which we conclude that the period of ξ divides c ( f ).Remark that if f is strongly monotone convex map with a t-stable fixedpoint v ∈ D , then every P ∈ ∂f ( v ) is positive by Proposition 2.2. Hence c ( f ) = 1 in that case, and therefore f has no t-stable periodic points exceptits t-stable fixed points. We also like to point out that in Theorem 8.6 thet-stability assumptions are essential. Indeed, consider A = cos α sin α − sin α cos α
00 0 b and P = − α − α , then B = P AP − ∼ αb b − α ( b −
2) 1 b − α ( b − α ( b − b when α > b >
2, the matrix B is nonnegativefor all sufficiently small α >
0. Now if α = 2 π/p , then A has a periodic pointof period p and hence B has one too. This shows that a monotone convexmap may have (unstable) periodic orbits with arbitrary large periods. Wealso remark that if α > π , then B has anunstable bounded orbit that does not converge to a periodic orbit. In this final section we give a condition under which every orbit of a convexmonotone map f : D → D , where D = R n , converges to a Lyapunov stableperiodic orbit. To present it we need the notion of the recession map ˆ f of f , which can be defined byˆ f ( x ) = lim λ →∞ λ f ( λx ) for all x ∈ R n . Since f : R n → R n is convex, ˆ f ( x ) ∈ ( R ∪ {∞} ) n exists for each x ∈ R n andis equal to ˆ f ( x ) = sup y ∈ R n f ( y + x ) − f ( y ) (35)(see [25, Theorem 8.2]). As ˆ f is the point-wise limit of a convex monotonemap, ˆ f is also convex and monotone. The next theorem shows that if ˆ f has33ll its orbits bounded from above, then f is non-expansive with respect toa polyhedral norm. Theorem 9.1. If f : R n → R n is a convex monotone map and the recessionmap ˆ f has all its orbits bounded from above, then f is non-expansive withrespect to the norm k · k v given in Theorem 6.5.Proof. We remark that ˆ f is a convex, monotone, and positively homogeneousmap, which has 0 as a t-stable fixed point, because ( ˆ f ) ′ = ˆ f and ˆ f has allits orbits bounded from above. Moreover, − ˆ f ( − x ) ≤ f ( y + x ) − f ( y ) ≤ ˆ f ( x ) (36)for all x ∈ R n by (35).Let k · k v be the polyhedral norm from Theorem 6.5 and remark that ˆ f is non-expansive with respect to k · k v . Clearly k u k v ≤ k w k v if u, w ∈ R n are such that 0 ≤ | u | ≤ | w | , where | z | = ( | z | . . . . , | z n | ). Therefore it followsfrom (36) that k f ( y + x ) − f ( y ) k v ≤ max {k ˆ f ( − x ) k v , k ˆ f ( x ) k v } ≤ k x k v for all x, y ∈ R n . Thus, f is also non-expansive with respect to k · k v .This theorem has the following consequence. Corollary 9.2. If f : R n → R n is a convex monotone map and f is non-expansive with respect to some norm on R n , then f is non-expansive withrespect to a polyhedral norm.Proof. We note that ˆ f ( x ) = lim λ →∞ f ( λx ) /λ for all x ∈ R n . As f is non-expansive with respect to some norm, ˆ f will be non-expansive with respectto the same norm. This implies that ˆ f has all its orbits bounded from above,since ˆ f (0) = 0. From Theorem 9.1 we conclude that f is non-expansive withrespect to a polyhedral norm.It is proved in [22] that if a map f : R n → R n is non-expansive withrespect to a polyhedral norm, that every bounded orbit of f converges toa periodic orbit. Moreover, if the unit ball of the polyhedral norm has N facets, then the period of each periodic point of a non-expansive map doesnot exceed max k k (cid:0) ⌊ N/ ⌋ k (cid:1) , see [18]. By using these results, the followingglobal convergence theorem can be proved.34 heorem 9.3. If f : R n → R n is a convex monotone map, with a fixedpoint, and the recession map ˆ f has all its orbit bounded from above, thenevery orbit of f converges to a Lyapunov stable periodic orbit of f whoseperiod divides c ( f ) .Proof. As ˆ f has all its orbits bounded from above, we know by Theorem9.1 that f is non-expansive with respect to a polyhedral norm, so that allperiodic orbits of f are Lyapunov stable. Since f has a fixed point, it followsfrom [22] that every orbit of f converges to a periodic orbit. Proposition3.2 implies that every periodic point of f is t-stable, and hence the periodof each periodic point divides c ( f ) by Theorem 8.6, which completes theproof. References [1] M. Akian and S. Gaubert. Spectral theorem for convex monotone homogeneous maps,and ergodic control.
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