An abstract version of Lyapunov exponents is defined for positive homogeneous maps on Homogeneous Lattices and a sufficient condition is given for the asymptotic stability of the map. | Turk J Math 24 (2000) , 277 – 281. ¨ ITAK ˙ c TUB A Remark on the Asymptotic Properties of Positive Homogeneous Maps on Homogeneous Lattices Alp Eden Abstract An abstract version of Lyapunov exponents is defined for positive homogeneous maps on Homogeneous Lattices and a sufficient conditon is given for the asymptotic stability of the map. Key Words: Lyapunov Exponents, asymptotic stability 1. Introduction We will start with a typical example. Let X be a compact Hausdorff space, and let a : X → 0 on X, then the operator T is linear and positive. For any continuous, positive function f : X → 0, α(f ∨ g) = αf ∨ αg and α(f ∧ g) = αf ∧ αg. We will call the structure (Λ, ∨, ∧, ·) a Homogeneous Lattice. We will assume that the lattice Λ has two distinguished elements : the zero 0 and the unit e, where e ∨ 0 = e, e ∧ 0 = 0. Definition . The Positive Cone of the lattice Λ consists of the elements f such that f ∨ 0 = 0 and will be denoted by Λ+ . Definition . An order can be defined by f ≤ g ⇔ f ∨ g = g and f ∧ g = f . It follows that e ≥ 0 and the order structure is compatible with the scalar multiplication. 278 EDEN Clearly, f ∈ Λ+ ⇔ f ≥ 0 Example . Furnish C(X; 0; (iii) kT f k ≤ |T | kf k for f ∈ Λ+ ; (iv) |ST | ≤ |S| |T | . Proof. (i) follows from the positivity of T combined with kT ek ≤ |T | . (ii) follows from (i) and the homogeneity of the valuation k k . By the homogeneity of T, T 0 = 0, hence for (iii) we can assume that kf k 6= 0. Then ≤ e implies that T kf1k f ≤ |T | . Combining with T kf1k f = kf1k T (f ) = 1 kf k f 279 EDEN 1 kf k kT (f )k , (iii) follows. 2 (iv) follows from (i) and (iii) with f = Se. Definition . A sequence of real numbers {an } is said to be subadditive if an+m ≤ an + am . We state the following well-known fact about sub-additive sequences af real numbers ( a proof can be found . in Walters [4] , Theorem ) Lemma . If {an } is a subadditive sequence then lim an n→∞ .
