On Growth Rates of Subadditive Functions for Semiflows

Transcription

1 journal of differenial equaions 48, (998) aricle no. DE98347 On Growh Raes of Subaddiive Funcions for Semiflows Sebasian J. Schreiber Deparmen of Mahemaics, Wesern Washingon Universiy, Bellingham, Washingon sschreibcc.wwu.edu Received Augus 29, 997; revised March 24, 998 Le,: X_T + X be a semiflow on a compac meric space X. A funcion F: X_T + X is subaddiive wih respec o, if F(x, +s)f(x, )+F(,(x, ), s). We define he maximal growh rae of F o be sup sup () F(x, ). This growh rae is shown o equal he maximal growh rae of he subaddiive funcion resriced o he minimal cener of aracion of he semiflow. Applicaions o Birkhoff sums, characerisic exponens of linear skew-produc semiflows on Banach bundles, and average Lyapunov funcions are developed. In paricular, a relaionship beween he dynamical specrum and he measurable specrum of a linear skew-produc flow esablished by R. A. Johnson, K. J. Palmer, and G. R. Sell (SIAM J. Mah. Anal. 8, 987, 33) is exended o semiflows in an infinie dimensional seing. 998 Academic Press. INTRODUCTION Consider a coninuous-ime or discree-ime semiflow, on a compac meric space X. By his we mean a coninuous map,: X_T + X,,(x, )=, x ha saisfies, 0 x=x and, s+ x=, s, x for s, # T + where T + equals eiher Z + in he discree-ime case or R + in he coninuous-ime case. An imporan class of coninuous funcions F: X_T + R associaed wih, are hose ha saisfy a subaddiiviy condiion: F(x, +s)f(x, )+F(, x, s). () These funcions naurally arise in various seings including he sudy of Birkhoff sums [0], average Lyapunov funcions [2, 3], and characerisic Copyrigh 998 by Academic Press All righs of reproducion in any form reserved. 334

2 GROWTH RATES OF SUBADDITIVE FUNCTIONS 335 exponens for smooh dynamical sysems [3, 4], homeomorphisms of meric spaces [7], and linear skew-produc semiflows [5]. In each of hese seings, i is he asympoic behavior of hese subaddiive funcions as ha is of ineres. In his ground breaking work, Kingman [8] provided he firs sysemaic sudy of he long-erm behavior of subaddiive funcions from an ergodic poin of view. Kingman's subaddiive ergodic heorem [8] assures ha () F(x, ) has a well-defined i almos surely for any,-invarian measure. The purpose of his paper is o provide uniform upper bounds for he iing values of () F(x, ) in erms of hese well-defined is. The paper is organized as follows. In Secion 2, we inroduce he main definiions: he maximal growh rae of a subaddiive funcion and he growh rae of a subbaddiive funcion wih respec o an ergodic measure. The main heorem assers ha he maximal growh rae equals he supremum of he growh raes wih respec o ergodic measures. In Secion 3, we prove he main resul. In Secion 4, we derive hree applicaions of he main resul. Firs, we show ha he maximal (respecively minimal) growh rae of he Birkhoff sums of a coninuous funcion equals he supremum (respecively infimum) of he average of his funcion wih respec o any ergodic measure. Second, we consider he dynamical specrum and he measurable specrum of a skew-produc semiflow on Banach bundles. Sacker and Sell [6] defined he dynamical specrum 7 dyn for a finie-dimensional linear skew-produc flow? over a suiable base space X o be he se of values * # R where he shifed semiflow? * fails o have an exponenial dichoomy. This definiion was exended o an infinie dimensional seing by Magalha~ es [9]. Alernaively, Johnson e al. [5] defined he measurable specrum 7 meas o be he closure of he characerisic exponens of? as deermined by he muliplicaive ergodic heorem [5, 3, 4]. The exisence of he measurable specrum in he infinie dimensional seing was proven by Ruelle [5] and Man~ e []. In he spiri of Johnson e al. [5] we prove ha 7 dyn 7 meas 7 dyn hereby exending heir resul o he infinie dimensional seing. As our final applicaion, he main resul is applied o he sudy of average Lyapunov funcions [2, 3] ha are used o prove ha cerain posiively invarian ses are repelling. They arise ofen in biological applicaions [4] and in hese cases i is useful o know on wha se i is necessary o check wheher a candidae funcion is in fac an average Lyapunov funcion. We show ha i is sufficien o check on he minimal cener of aracion of he semiflow (a subse of he Birkhoff cener of he semiflow).

3 336 SEBASTIAN J. SCHREIBER 2. DEFINITIONS AND STATEMENT OF MAIN RESULT Given a semiflow, we resric our aenion o coninuous funcions F: X_T + R, F (x)=f(x, ) ha are subaddiive wih respec o, (i.e., saisfy ()). To sudy he measure-heoreic growh raes of hese funcions, le M inv (,) denoe he space of Borel probabiliy measures ha are,-invarian and le M erg (,)M inv (,) denoe hose invarian measures for which, is ergodic. Given + # M erg (,), Kingman's subaddiive ergodic heorem [8] assers ha here exiss a Borel se UX such ha +(U)= and F (x)= inf >0 F d+ X for all x # U. Hence, i makes sense o define he growh rae of F wih respec o + o be GR(F, +)= inf >0 F d+. X In general, however, he growh rae of F is no well defined a every poin of X. Therefore, a bes we can hope o find a uniform upper bound for he growh rae of a subaddiive funcion. Wih his purpose in mind, we define he maximal growh rae of F o be GR + (F)=sup sup F (x). Our main resul relaes hese measure-heoreic and dynamical definiions. Theorem. Le F: X_T + R be a coninuous subaddiive funcion wih respec o he semiflow,. Then GR + (F)=sup[GR(F, +) :+ # M erg (,)] = inf >0 sup F (x). Theorem shows for wha invarian subse KX i is sufficien o evaluae he growh rae of F. This se is called he minimal cener of aracion (see [0] or [2]) of,, he unique compac posiively invarian se MC(,) which saisfies wo condiions:

4 GROWTH RATES OF SUBADDITIVE FUNCTIONS 337 () If U is any neighborhood of MC(,), le U be he characerisic funcion for U (i.e., U (x)= if x # U else U (x)=0). Then for all if T + =R + or 0 i=0 U (, s x) ds= & : U (, i x) ds= if T + =Z +. (2) If KX is any oher compac posiively invarian se saisfying condiion (), hen MC(,)K. The Birkhoff ergodic heorem implies (see, e.g., Exercises I.8.3 and II..5 in [0]) ha MC(,)=. supp(+), where he union is aken over + # M erg (,) and where supp(+) denoes he suppor of +. Consequenly, Theorem implies GR + (F)=GR + (F MC(,)). + I is worh noing ha by he Poincare recurrence heorem MC(,) is conained in he Birkhoff cener of, (i.e., he closure of he recurren poins). This inclusion can be proper. For insance, Nemyskii and Sepanov [2] provide an example of a flow on a wo orus whose Birkhoff cener is he enire orus bu whose minimal cener of aracion is a single poin. Theorem can also be used o find a uniform lower bound on he growh rae of a superaddiive funcion F wih respec o,: a coninuous funcion F: X_T + R ha saisfies F(x, +s)f(x, )+F(, x, s). In his case, &F(x, ) is subaddiive. Hence, applying Theorem o &F, we ge ha inf inf F (x)=inf[gr(f, +) :+ # M erg (,)]=sup >0 inf F (x). We shall denoe hese equivalen quaniies GR & (F), he minimal growh rae of he superaddiive funcion F.

5 338 SEBASTIAN J. SCHREIBER 3. PROOF OF THEOREM To avoid confusion, hroughou his secion we le denoe an elemen of R + and n denoe an elemen of Z +. We begin by assuming ha T + =Z + and by proving GR + (F)sup[GR(F, +) :+ # M erg (,)] inf n sup n F n(x). (2) Kingman's subaddiive ergodic heorem implies he firs inequaliy in (2). To prove he second inequaliy in (2), choose =>0. We will show ha here exiss a + # M erg (,) such ha GR(F, +)+2= inf n sup n F n(x). (3) To prove (3), for every n # Z + choose y n # X such ha n F n( y n )+= inf n sup n F n(x). (4) Define a sequence of Borel probabiliy measures ' n on X by ' n = n& n : i=0 $,i y n, where $ x is he Dirac measure concenraed a he poin x. Compacness of X implies here exiss a subsequence of measures ' nk ha converges o a measure & in he weak* opology. For noaional convenience, we wrie & k =' nk and x k = y nk. To show ha & is,-invarian, le f: X R be a coninuous funcion. Weak* convergence of he & k implies X f(, x) d&(x)= k : n k n k & i=0 = : k n k n k & i=0 = : k n k n k & i=0 = X f(x) d&(x). f(, i+ x k ) f(, i x k )+ ( f(, nk x k )&f(x k )) k n k f(, i x k ) Since f was an arbirary coninuous funcion, & is,-invarian.

6 GROWTH RATES OF SUBADDITIVE FUNCTIONS 339 Nex we prove a lemma based on esimaes found in Kaznelson and Weiss' proof of he subaddiive ergodic heorem [6]. Lemma. Then Le n k be a sricly increasing sequence of posiive inegers. sup k : mn k n k & i=0 for any and 0{m # Z +. F m (, i x) sup k F nk (x) n k Proof. Assume n k >m. For each i beween and m here exiss a unique choice of inegers c(i, k)0 and 0r(i, k)m such ha n k =i+c(i, k) m+r(i, k). By subaddiiviy, F nk (x)f i (x)+f c(i, k) m (, i x)+f r(i, k) (, i+c(i, k) m x) c(i, k)& F i (x)+ : F m (, i+ jm x)+f r(i, k) (, i+c(i, k) m x). j=0 Summing boh sides over i from o m, we ge m mf nk (x) : i= m = : i= m F i (x)+ : i= n k &m F i (x)+ : i= c(i, k)& : j=0 m F m (, i+ jm x)+ : m F m (, i x)+ : i= i= F r(i, k) (, i+c(i, k) m x), F r(i, k) (, i+c(i, k) m x) where he second line follows from he definiion of c(i, k) and r(i, k). Dividing boh sides by mn k and rearranging erms, we ge F nk (x) n k n k m\ : m i= n k & & : i=n k &m+ n k & F i (x)+ : i=0 m F m (, i x)+ : F m (, i x)&f m (x) i= F r(i, k) (, i+c(i, k) m x) +. As 0r(i, k)m and 0n k &c(m, k) m2m, coninuiy of F and compacness of X imply ha he i k n k m\ : m i= n k & F i (x)&f m (x)& : m + : F r(i, k) (, i+c(i, k) m x) + i= i=n k &m+ F m (, i x)

7 340 SEBASTIAN J. SCHREIBER exiss and equals zero. Thus, aking he sup on boh sides of (5) complees he proof of he lemma. K Reurning o he proof of Theorem, recall ha x k = y nk. Lemma and (4) imply ha inf m m F m d&= inf X m sup k inf n k : mn k n k & i=0 F nk (x k ) n k sup n F n(x)&=. F m (, i x k ) The ergodic decomposiion heorem (see [0, Chap. II, Theorem 6.4], implies X F m d&= X\ X F m d& x+ d&, where & x are Borel probabiliy measures for which, is ergodic. I follows ha here exiss an ergodic measure +=& x for some such ha (3) holds. Taking he i as = 0 complees he proof of he second inequaliy in (2). To complee he proof in he discree case, we need he following wellknown lemma (see, for example, [, p. 28]). Lemma 2. If a n is a sequence of real numbers such ha a n+m a n +a m for all n, m # Z +, hen n n a n= inf n n a n. Similarly if a: R + R is a coninuous funcion such ha a(+s) a()+a(s) for all s, # R +, hen a = inf >0 a.

8 GROWTH RATES OF SUBADDITIVE FUNCTIONS 34 Subaddiiviy of F implies ha he sequence a n =sup F n (x) saisfies he condiions of Lemma 2. Therefore inf n sup n F n(x)= n n sup F n (x)gr + (F n ) which complees he proof of Theorem in he discree case. Now consider a coninuous-ime semiflow,(, x) and a subaddiive funcion F(x, ) where T + =R +. Define, (x, n)=,(x, n) o be he ime-one map for,. Wih, we associae he subaddiive funcion F (x, n)=f(x, n). The proof of Theorem in he coninuous- ime case follows from he discree-ime case and he nex lemma. Lemma 3. GR + (F )=GR + (F) (6) inf sup n n F n (x)= inf sup >0 F (x) (7) sup[gr(f, +) :+ # M erg (, )]=sup[gr(f, +) :+ # M erg (,)]. (8) Proof. Given # R le [] denoe is ineger par. Coninuiy of F and compacness of X implies here exiss K>0 such ha F(x, ) K for all and 0. Given >, subaddiiviy of F implies ha F(x, )&F(x, [])F(, [] x, &[])K. (9) Since ([])=, (9) implies sup F(x, ) sup F(x, [])= sup n n F (x, n). Alernaively, since Z + /R +, he opposie inequaliy holds and we have sup F(x, )= sup n F(x, n). (0) n Equaion (0) implies (6). To prove (7), se a =sup F (x). Lemma 2 implies ha n n a n= inf n n a n, a = inf >0 a. Since Z + /R +, i follows ha n (n) a n = () a complees he proof of (7). which

9 342 SEBASTIAN J. SCHREIBER To prove (8), noice ha he characerizaion of MC(, ) and MC(,) as he closure of he suppors of he ergodic measures implies ha sup[gr(f, +) :+ # M erg (, )]=GR + (F MC(, )) sup[gr(f, +) :+ # M erg (,)]=GR + (F MC(,)). Therefore by (0) i is sufficien o show ha MC(, )=MC(,). Since any ergodic measure + for, is an ergodic measure for,, i follows ha MC(,)MC(, ). The inclusion in he opposie direcion follows from he definiion of he minimal cener of aracion. K 4. APPLICATIONS 4.. Birkhoff Sums An immediae applicaion of Theorem is o Birkhoff sums (see Exercise I.8.5 in [0]). We sae he resul in he case when T + =R +. An analogous saemen holds for he discree-ime case. Corollary. Le, be a coninuous-ime semiflow on a compac meric space X. If f: X R is a coninuous funcion hen sup inf sup f(, s x) ds=sup { fd+: + # M erg (,) = 0 X inf f(, s x) ds=inf { fd+: + # M erg (,) 0 =. X Proof. Define F(x, )= f(, 0 sx) ds. F is coninuous and addiive (i.e., superaddiive and subaddiive). Theorem implies ha GR + (F)= sup + GR(F, +) and GR & (F)=inf + GR(F, +). The proof of he corollary is compleed by observing ha he Birkhoff ergodic heorem implies ha GR(F, +)= X fd Specra for Linear Skew-Produc Semiflows on Banach Bundles In his secion we assume ha,: X_R X is a coninuous-ime flow on a compac meric space X. Following he work of Sacker and Sell [6], Johnson e al. [5], and Magalha~ es [9], we sudy he specral properies of linear-skew produc semiflows on Banach vecor bundles over,. ABanach vecor bundle E over X is a riad (E, p, &}&) where E is a opological space, p: E X a coninuous map (called he canonical projecion), &}&: E R a coninuous funcion and for every he se E(x)=p & (x) is endowed wih a vecor space srucure such ha &}& x : E(x) R is a Banach norm

10 GROWTH RATES OF SUBADDITIVE FUNCTIONS 343 on E(x) and he opology induced by his norm coincides wih he relaive opology of E(x). E(x) is called he fibre over x. Poins in E can be represened as ordered pairs (x, v) where and v # E(x). A semiflow?: E_R + E is said o be a linear skew-produc semiflow on E if?(x, v, )=(,(x, ), 8(x, ) v), where 8(x, ) is a bounded linear map ha sends he fiber E(x) o he fiber E(, x). Since? is only defined for 0, i is useful o idenify he poins (x, v) ine hrough which here is a backward coninuaion of?. To his end, we define he se B=[(x, v)# E : here is exacly one coninuous funcion (u, w): (&, 0] E such ha u(0)=x, w(0)=v and?(u(s), w(s), )=(u(+s), w(+s)) for all s0 and all #[0,&s]]. For any poin (x, v)# B and 0 we se 8(x, ) v equal o (u(), w()) where (u, w) is he unique backward coninuaion of (x, v). We define he sable se of? by and he unsable se of? by S=[(x, v)#e : 8(x, ) v =0] U=[(x, v)#b : 8(x, ) v =0]. & Noice ha he se S is posiively invarian under? (i.e.,? SS for all 0) and he se U is invarian under? (i.e., UB and? UU for all # R). I is easy o check ha S and U are vecor sub-bundles of E. The linear skew-produc flow? is said o have an exponenial dichoomy provided ha here exiss a coninuous family of linear projecors P(x) of he fibers E(x) and consans K, :>0 such ha (i) N=[(x, v)#e : P(x) v=0]b. (ii) &8(x, ) P(x)&Ke &: for all 0 and. (iii) &8(x, )(I&P(x))&Ke : for all 0 and. Noice ha (i) implies ha (iii) makes sense. Whenever? admis an exponenial dichoomy, he unsable se U equals N and he sable se S equals [(x, P(x) v) :(x, v)#e]. Consequenly, E=SU where denoes a Whiney sum.

11 344 SEBASTIAN J. SCHREIBER Given * # R, define he linear skew-produc semiflow? * by? * (x, v, )=(, x, e &* 8(x, ) v) for 0 and (x, v)#e. The resolven \(E,?) of? is defined o be he se \(E,?)=[* # R + :? * admis an exponenial dichoomy]. Given * # \(E,?), le U * and S * denoe he unsable and sable ses of? *. The dynamical specrum of? is defined o be he se 7 dyn (E,?)=R"\(E,?). Magalha~ es [9, Theorem 2.] provided he following characerizaion of he dynamical specrum. Theorem 2 (Magalha~ es, 987). Le?=(,, 8) be a linear skew-produc semiflow over a compac conneced meric space X. Assume ha 8(x, ) is a compac linear operaor for all 0 and. Then he dynamical specrum 7 dyn (E,?) is closed, bounded above, and equals he union of closed inervals. These inervals are called he specral inervals and in his seing an inerval [a, b] is allowed o degenerae o a poin when a=b. Associaed wih each specral inerval here is a specral bundle V of E which saisfies he following properies: () If +, * # \(E,?) and (+, *) & 7 dyn (E,?)=[a, b] hen he specral bundle V associaed wih [a, b] has finie dimension, saisfies V=U + & S * and is invarian under?. (2) If * # \(E,?) and (&, *) & 7 dyn (E,?)=(&, b], hen he specral bundle V associaed wih (&, b] saisfies V=S * and is posiively invarian under?. (3) If * # \(E,?) hen he number of specral inervals included in (*, ) is finie. Remarks. Magalha~ es original saemen of he heorem assumed hax is a compac conneced smooh Banach manifold. I is easily seen ha his proof holds for compac conneced meric spaces. To define he measurable counerpar o 7 dyn (E,?), we firs sae a heorem of Man~ e [] which is he infinie dimensional counerpar o Oseledec's muliplicaive ergodic heorem [3]. Ruelle [5] proved a similar heorem for Hilber space vecor bundles. Theorem 3 (Man~ e, 983). Le?=(,, 8) be a linear skew-produc semiflow over a compac meric space X. Assume ha 8(x, ) is compac and

12 GROWTH RATES OF SUBADDITIVE FUNCTIONS 345 injecive for all 0 and. Then here is a Borel se such ha +()= for all + # M inv (,) and such ha every x # saisfies one of he following hree condiions () () ln&8(x, )&=&. (2) There exiss a k(x)#z + and a spliing E(x)=E (x) }}} E k(x) (x)f (x) and numbers * (x)>}}}>* k(x) (x) such ha (a) E i (x) is finie dimensional for all ik(x). (b) \ () ln&8(x, ) v&=* i (x) for all 0{v # E i (x) and ik(x). (c) () ln&8(x, ) F (x)&=&. (3) There exis subspaces E i (x), F i (x), i=, 2,..., and real numbers * (x)>* 2 (x)> }}} such ha: (a) E i (x) is finie dimensional for all i. (b) n * n (x)=&. (c) E (x) }}}E i (x)f i (x)=e(x) for all i. (d) For all i and 0{v # E i (x), (e) For all i, \ ln &8(x, ) v&=* i(x). ln &8(x, ) F i (x)&=* i+ (x). Following Johnson e al. [5] we define he measurable specrum of a linear skew-produc semiflow? by 7 meas (E,?)=[* i (x) :x #, ik(x)], where he * i (x) are he characerisic exponens as defined in Man~ e's heorem and where we se k(x)=0 or when x # corresponds o a poin in case () or (3) of Theorem 3. Using Theorem in conjuncion wih he resuls of Magalha~ es and Man~ e, we ge he following resul.

13 346 SEBASTIAN J. SCHREIBER Theorem 4. Le?=(,, 8) be a linear skew-produc semiflow over a compac conneced meric space X. Assume ha 8(, x) is a compac and injecive operaor for all and 0. Then 7 dyn (E,?)7 meas (E,?)7 dyn (E,?), where 7 dyn (E,?) denoes he boundary of 7 dyn (E,?). Proof. We firs show ha 7 meas (E,?)7 dyn (E,?). To his end, le * # R and ( y, w)#e be such ha ln 8( y, ) w =*. () We wan o show ha * # 7 dyn (E,?). Arguing negaively, suppose ha * # \(E,?). Then? * admis an exponenial dichoomy E=S * U * and here exis K, ;>0 such ha 8(x, ) v s Ke (*&;) v s for all, v s # S * (x), 0 (2) 8(x, &) v u Ke &(;+*) v u for all, v u # U * (x), 0. (3) In his case, we can wrie w=w s +w u for some w s # S * ( y) and w u # U * ( y). Relaions () and (2) imply ha w u canno equal zero. Relaion (3) implies ha for 0 Hence, we ge ha for all 0 w u = 8(, y,&) 8( y, ) w u Ke &(;+*) 8( y, ) w u. 8( y, ) w u K e(*+;) w u which conradics (). Nex we prove he inclusion, 7 dyn (E,?)7 meas (E,?).

14 GROWTH RATES OF SUBADDITIVE FUNCTIONS 347 To his end, le I be a specral inerval of he form [a, b] or(&, b] and le V be is associaed specral bundle as given by Theorem 2. Define F(x, )=ln &8(x, ) V(x)& for all and 0. F is coninuous and subaddiive wih respec o,. For any + # M erg (,), GR(F, +) is he maximal Lyapunov exponen of? V wih respec o + (see, e.g., [] or [5]). In paricular here exiss ( y, w)#v such ha ln 8( y, ) w =GR(F, +). Our argumens in he previous paragraph applied o? V imply ha GR(F, +)#7 dyn (V,?)=I. On he oher hand, Theorem implies ha GR + (F)=sup[GR(F, +) :+ # M erg (,)]. Hence GR + (F)#I. We claim ha GR + (F)=b. Arguing negaively assume ha GR + (F)<b hen Theorem implies ha here exiss =>0 such ha inf >0 sup F(x, )<b&=. Therefore here exiss T>0 such ha e &bt &8(x, T ) V(x)&e &=T for all. Submuliplicaiviy of linear operaors wih respec o he operaor norm implies ha for all n # Z +,, Le e &bnt &8(x, nt ) V(x)&e &=nt. (4) K= sup e T &8(x, ) V(x)&., 0T Given any 0, here is a unique nonnegaive ineger n and real 0rT such ha =nt+r. This observaion, inequaliy (4) and our choice of K imply e &b &8(x, ) V(x)&Ke &=. This inequaliy implies ha? b V admis an exponenial dichoomy wih projecors P(x) equal o he ideniy map. Hence b # \(V,?) conradicing

15 348 SEBASTIAN J. SCHREIBER our choice of b. Therefore GR + (F)=b and by Theorem, b # 7 meas (V,?) 7 meas (E,?). When he specral inerval I is of he form [a, b], we claim ha a # 7 meas (E,?). To prove his claim, we define he superaddiive funcion G(x, )=&ln &8(x, &) V(x)& for and 0. In his case, for any + # M erg (+), GR(G, +) is he minimal Lyapunov exponen of? V. Mimicking our previous argumens for b, i follows ha a=gr & (G)=inf[GR(G, +) :+ # M erg (,)]. Hence a # 7 meas (E,?). K Remarks. Johnson e al. [5, Theorem 2.3] proved Theorem 4 for linear skew-produc flows on finie-dimensional spaces. A relaed heorem was proven in he discree case by he auhor [7] Average Lyapunov Funcions Consider a semiflow,: Y_T + Y on a locally compac meric space Y. Assume X is a compac subse of Y wih empy inerior such ha X and Y"X are posiively invarian. Moivaed by applicaions, various mehods have been developed o deermine wheher X is a uniform repellor, i.e., here exiss '>0 such ha for all y # Y"X, inf d(, y, X )>' (see, for example, [4]). One of hese mehods uses wha is commonly referred o as an average Lyapunov funcion [2, 3]: Given UY an open neighborhood of X and a coninuous funcion P: U R +, define F: X_T + R by F(x, )=ln P(, y) inf y x, y # U"X P( y). (5) P is called an average Lyapunov funcion provided ha P & (0)=X and for all. sup F(x, )>0 >0 Remarks. Recall P is Lyapunov funcion if P(, y)>p( y) for all y # U"X and >0. No all Lyapunov funcions are average Lyapunov funcions. For example, consider x* =x(&x 2 ) wih P(x)=x. However, he advanage of an average Lyapunov funcion is ha i gives a condiion ha only needs be checked a X. Theorem (Huson, 984). Le Y, X, and, be as defined above. If here exiss an average Lyapunov funcion for X, hen X is uniformly repelling.

16 GROWTH RATES OF SUBADDITIVE FUNCTIONS 349 As F is superaddiive, Theorem immediaely implies he following corollary. Corollary 2. Le U, X, Y, and, be as defined above. Le P: U R + be a coninuous funcion such ha P & (0)=X. If F(x, ) as defined in (5) is coninuous and inf inf { F(x, ) :x # M(, X ) = >0, hen X is uniformly repelling. Remarks. Huson [2, Corollary 2.3] proved ha i suffices o check ha inf () F(x, )>0 for x # L + (, X )= (x) where (x) denoes he -i se of he poin x. Since L + (, X ) conains he Birkhoff cener of, X, Corollary 3 improves his resul. Under addiional assumpions (i.e.,, is a dissipaive Lipschiz flow on Y=R n and + X=Rn + ), Huson [3, Theorem 5.2] proved ha if X is uniformly repelling hen here exiss an average Lyapunov funcion P such ha F as defined by (5) is coninuous. Hence Corollary 3 can be inerpreed as saying ha behavior of, near M(, X ) deermines wheher X is a uniform repellor. ACKNOWLEDGMENTS The auhor hanks Keih Burns for poining ou an error in an earlier incarnaion of his paper, an anonymous reviewer for suggesing he applicaion of he main heorem o specral properiies of linear skew-produc flows, and Edoh Amiran for helpful commens. REFERENCES. R. Bowen, ``Equilibrium Saes and he Ergodic Theory of Anosov Diffeomorphisms,'' Lecure Noes in Mahemaics, Vol. 470, Springer-Verlag, Berlin, V. Huson, A heorem on average Liapunov funcions, Monash. Mah. 98 (984), V. Huson, The sabiliy under perurbaions of repulsive ses, J. Differenial Equaions 76 (988), V. Huson and K. Schmi, Permanence and he dynamics of biological sysems, Mah. Biosci. (992), R. A. Johnson, K. J. Palmer, and G. R. Sell, Ergodic properies of linear dynamical sysems, SIAM J. Mah. Anal. 8 (987), Y. Kaznelson and B. Weiss, A simple proof of some ergodic heorems, Israel J. Mah. 42 (982), Y. Kifer, Characerisic exponens of dynamical sysems in meric spaces, Ergodic Theory Dynamical Sysems 3 (983), 927.

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