P-ARCS IN STRONGLY MONOTONE DISCRETE-TIME DYNAMICAL …

Differential and Integral Equations, Volume 7, Number 6, November 1994, pp. 1473-1494.

P-ARCS IN STRONGLY MONOTONE DISCRETE-TIME DYNAMICAL SYSTEMS

JANUSZ MIERCZYNSKI*

Institute of Mathematics, Technical University ofWroclaw Wybrzeze Wyspianskiego 27, PL-50-370 Wroclaw, Poland

Dedicated to the memory of Professor Peter Hess (1941-1992)

Abstract. Strongly monotone discrete-time (semi)dynamical systems are generated by second order parabolic partial differential equations periodic in time, complemented with boundary conditions admitting the strong maximum principle. It has been recently proved by P. Hess and P. Poliicik that, under some additional conditions, the set of periods of periodic points which are not linearly unstable is bounded from above. In the present paper we use the above theorem, together with results by P. Takac, to state and prove several propositions about classification of points and genericity of some properties. A main tool in the proofs is the concept of nondegenerate p-are, that is, a simply ordered invariant set diffeomorphic to [0, 1]. The paper is divided into four sections. Sections 1 and 2 contain preliminary results. In Section 3 basic properties of p-ares are discussed. In Section 4 we investigate generic behavior, that is, long-time behavior of points belonging to some open dense subset of the phase space.

The theory of strongly monotone dynamical systems with continuous time, initiated independently by M.W. Hirsch ([21]) and H. Matano ([29], [30]), has undergone a mighty development since then, whether from the theoretical point of view (see e.g. [44], [45]) or from the point of view of applications (see e.g. [10], or [16]). In the discrete time case however, only quite recently a general abstract theory was developed by P. Takac in [46] and [47] (but see the earlier papers [20], [1], [3], and for the finite-dimensional case [43]). At the same time, new results have been obtained by P. Hess, P. Polacik and I. Terescak: ([38] and [19]).

In the present paper we generalize and improve on some results from [ 47] as well as from [38], assuming that the strongly monotone mapping considered is of class C 1. We introduce the concept of nondegenerate p-ares (see Section 3), which are defined to be simply ordered invariant sets diffeomorphic to [0, 1]. (In a sense they can be considered "duals" of Takac's d-hypersurfaces, see [ 47]). If the equilibria contained in a nondegenerate p-are satisfy some spectral property then the p-are is an attractive normally hyperbolic C1 one-dimensional manifold with boundary, which allows us to use such tools as forward invariant laminations (foliations). In

Received August 1993. *Research supported by the KBN grant PB 666/2/91. AMS Subject Classifications: 58F39, 58F12, 35K55, 35B40, 34C05.

1473

1476 JANUSZ MIERCZYNSKI

monotone mapping):

. C_DQ) There. exist No E N and an open dense Y c X such that for each y E y its 1Im1t set w (y) IS a cycle of least period at most No .

. From (DQ) we derive that there is N EN (e.g. N = No!) such that for each y E y the set (J) (y; SN) is a singleton (that is, an sN -equilibrium).

Hypothesis (DQ) is quite frequently met in the applications: (1) If Sis the time-one mapping of a strongly monotone semifiow, the fact that

for points in some open and dense set their limit sets are singletons is well known for various classes of such semifiows, see e.g. [25], or [35].

(2) For S being an injective strongly monotone mapping of class ci+a, a > O, it was proved by P. Hess and P. Poh'icik in [19], Corollary 4, that (DQ) is often satisfied (they proved in fact more: the set of points whose limit sets are not linearly unstable cycles is open dense).

For S : X --* X a strongly monotone mapping we introduce some concepts, following [47]. A point x E X is called lower (resp. upper) approximable if there exists a sequence {xk}, xk < x (resp. xk > x ), lim xk = x such that w (x) as well as

k--+oo all w (xk) are compact non empty and, moreover, the union

00

w (x) U U w (xk) k=l

is precompact in X. For x E X lower (resp.) upper approximable, the lower w-limit set (lower limit set, for short) of x E X is defined as

00 00

w_(x) := nclx(Uwcxz)), k=l l=k

where xk is as in the definition of lower approximable. The upper w-limit set w+ (x) of an upper approximable x EX is defined analogously (see [47], pp. 117-119). It should be remarked that the definition of lower (upper) w-limit set does not depend on the choice of the sequence {xk} (see Cor. 3.2 in [47]). We say that a lower (resp. upper) approximable point x E X (or its orbit) is lower w-stable (resp. upper w-stable) if w_ (x) = w (x) (resp. w+ (x) = w (x )). The set of lower (upper) w-stable points is denoted by X E s_ (x E S+). The points belonging to sl/2 := s_ u s+ are called W-Semistable, whereas those belonging to S := S_ n S+ are called order (i)-Stable. A }JOint X E s_ is called lower asymptotically (i)-Stable (written X E A_) if there is y E X, y < x, with the property that w (y) = w (x). The points upper asymptotically w-stable are defined analogously. Further, we define the set A of asymptotically w-stable points as A:= A_ n A+, and the set A 1; 2 of asymptotically w-semistable points as A 1; 2 := A_ U~. Further, we refer to the lower (resp. upper) approximable points belonging to U_ := X \ S_ (resp. U+ := X \ S+) as lower w-unstable (resp. upper w-unstable). Finally, the sets U := U_ U U+ and U2 := U_ n U+ are called the sets of w-unstable and w-biunstable points, respectively.

STRONGLY MONOTONE DYNAMICAL SYSTEMS 1477

An orbit 0 (x) is eventually increasing (resp. eventually decreasing) if there is n EN such that sn+!x « snx (resp. sn+lx » snx). The set of points having eventually increasing (resp. eventually decreasing) orbits is denoted by M+ (resp. M_). The orbits of points in M := M_ U M+ are called eventually monotone. (Notice that the above concepts differ from those in [47].) Sometimes we will say that points rather than their orbits are eventually increasing; etc.

Finally, N is defined to be the set of those convergent ~ for which there is a simply ordered L 3 ~' L c X\ M, homeomorphic to (0, 1).

It should be noted that most of our results are formulated only for the « relation (e.g. A_, S_ etc.). Obviously their counterparts for» are true.

Acknowledgment. The author thanks Morris W. Hirsch and the anonymous referee for their valuable remarks. The first version of the paper has been circulated as a preprint entitled Generic asymptotic behavior in strongly monotone discrete-time

dynamical systems.

1. Preliminary facts about strongly monotone systems. In the present section we state some rather well-known results which will be frequently used in the sequel. Our first result is a form of the Kre1n-Rutman theorem.

Proposition 1.1. Assume that S : X ~ X is a compact C 1 strongly monotone mapping. Let x E E. Then

(i) The spectral radius p (x) of D S (x) is positive and is a simple eigenvalue; (ii) The one-dimensional eigenspace Vr (x) of V belonging to p (x) is contained

in v_;: u - v_;: u {O}; (iii) The one-codimensional invariant subspace V2 (x) complementary to V1 (x)

intersects v+ only at {0}; (iv) The spectral radius r(x) of DS(x)lv2 (x) is less than p(x); (v) The mappings x f-7- Vr (x), x f-7- V2(x), x f-7- p(x), x f-7- r(x), where x E E,

are continuous.

Proof. As S is compact, DS(x) is a completely continuous linear operator (see e.g. Prop. 8.2 in [14]). Strong positivity of DS(x) allows us to employ the K.re1nRutman theorem (see e.g. Thm. 19.3 in [14]), which gives immediately parts (i) through (iv). The continuous dependence of p(x) and V1 (x) on x is a standard result (see [23]). To see the rest it suffices to notice that V2 (x) is the nullspace of the corresponding eigenfunctional for the linear operator adjoint to DS(x) (the continuous dependence of V2(x) on xis equivalent to the continuous dependence of those normalized functionals). 0

For x E Ewe denote by v 1 (x) the unique unit vector contained in V1 (x) n v_;:. Further, by the attraction set of x E E we understand the set K (x) := {y E X : Sn y ~ X as n ~ 00}.

Proposition 1.2 (Principle of Linearized Stability). Assume that S : X ~ X is a C 1 strongly monotone mapping. Then for x E Es there are a forward invariant


Proposition 2.3. Assume X E E and Sy » y for each y E wE-(x). Then X E

A_ n (Es U En), and K(x) n M+ is nonempty.

Proof. Fixay E wE-(x). ByProposition1.5,cu(y) =zforsomez E A_n(EsUEn). We have z E cl wE-(x), so z =X. This means that y E K(x). Evidently y EM+, so K (x) n M+ is non empty. D

Proposition 2.4. Assume X E E and Sy « y for each y E wE-(x). Then X E

u_ n (Eu u En), and there exists z E A+ n (Es u En), z «X, such that cu (~) = z for each~ E wE-(X).

Proof. Fix a y E WE-(x). By Proposition 1.5, z := cu(y) E ~ n (Es U En) and z « y « X. Take any ~ E wE-(x) n [[y, x]]. We have cu(~) ~ cu(y) = z. Again by Proposition 1.5, cu(~) E E and cu(O « ~ « x. It is impossible that all sn ~ » y' for then cu ( ~) would belong to wE- (x) n [y' X]] n E = 0. Denote by no the greatest positive integer such that sno~ ~ y. So, sno+l~ is in the simply ordered set scwE-(x) n [y, x]) and Sy :::::; sno+l~ « y. Consequently cu(~) « y, hence cu ( ~) :::::; z. Therefore cu ( ~) = z for any ~ E wE- (x). In particular, there is a sequence {yk} C wE-(x), Yk / x ask ---7- oo, such that cu(yk) = z « y. This means that x E U_. The fact that x E Eu U En is a consequence of the Principle of Linearized Stability (Proposition 1.2). D

Proposition 2.5. Assume X E E and there is a sequence {xk} c En wE-(x), Xk /X as k ---7- 00. Then X E s_ n En \ A and K (x) n c_ (x) = 0.

Proof. It is straightforward that x is lower approximable and i ¢. A_. Takac's Discontinuity Principle (Prop. 3.3 in [ 47]) implies that X E s_. To prove that X E En notice that, since W (x) is tangent at x to vr (x), the equality

II Sxk- Sx II //DS(x) Vr(x)/1 = k!~ 1/Sxk _ Sx/1 = 1

is valid. Now, suppose to the contrary that some y E K(x) n C_(x). From Proposition 1.3(v) we deduce that, as v1 (x) is in the interior of the cone V+, for sufficiently large n one of the inequalities sny « X or S11 y » X holds. Fix such ann. Let z E C(x) satisfy y « z. The inequality S11 y » x cannot hold, since then we would have X« S 11 y « snz with X and snz in C(x), which is impossible (C(x) is strongly unordered). Therefore y E K(x) n M+. The sequence {xk} is convergent to x in the original topology, so a fortiori in the order topology, hence there is ko such that xk E [[Sny, x]] for k0 ~ k. From this we get xk = cu(xk) = x, which is absurd. D

3. P-ares: basic properties. In the present section the standing assumption is: S: X ---7- X is a compact C1 strongly monotone mapping. By a p-are we understand a simply ordered invariant set J C X such that there

is an increasing C 1 diffeomorphism h from a compact interval I c lR onto J. (his increasing if ~1 < ~2 implies h~1 « h~2). A p-are J not being a singleton is called


nondegenerate. The image under h of the left (resp. right) endpoint of I is called the lower (resp. upper) endpoint of J and denoted by a_ J (resp. a+ J). For X, y in a p-are J, x « y, we write (x, y) 1 := J n [x, y]. Notice that (x, y) 1 is compact.

Lemma 3.1. Endpoints of a p-are J are equilibria.

Proof. For J a singleton there is nothing to prove. Assume then a_ J « a+ J, and suppose to the contrary that S(a_J) i= a_J. Since J is simply ordered we have S(a_J) » a_J. Sis strongly monotone, so SJ c (S(a_J), S(a+J)) 1 i= J, which contradicts invariance of J. The case S(a+J) -=f. a+J is considered in an analogous way. D

Lemma 3.2. For a p-are J the mapping SIJ is a homeomorphism.

Proof. The continuous mapping Sl J is surjective by invariance and injective by strong monotonicity. The continuity of (SI 1 )-1 follows by compactness. D

The tangent bundle of a p-are J is denoted by 'IT' J. By 'IT' x J we understand the set of vectors in V such that { x} x 'IT' x J forms the fiber of 'IT' J over x E J.

Lemma 3.3. Let J be a nondegenerate p-are. Then 'IT'x J C V+ U - V+ U {0} for eachxEJ.

Proof. Since the diffeomorphism h : I -7- J is increasing, we have

dh (5) = lim h(5 + t) - h(5) E V \ {0} dt t-+0 t + ,

for each 5 E I. The strong monotonicity of S and invariance of J yield the desired result. D

Proposition 3.4. Let J be a p-are. Then S!J is a C 1 diffeomorphism of J, C 1 conjugate to an orientation-preserving diffeomorphism of I.

Proof. If J is a singleton there is nothing to prove. Assume then J to be nondegenerate. The mapping F : I -7- I defined as F := h - 1 o (S 11 ) o h is an orientation-preserving homeomorphism of I. Further, from the strong positivity of DS (Assumption (Al)) we obtain by simple calculation that F'(5) > 0 for each 5 E J. Consequently, F is a diffeomorphism of I. Now we need to notice that (Si 1 )- 1 = h o F-1 o h-1 is a diffeomorphism of J. D

Proposition 3.5. Let J be a nondegenerate p-are. Then (i) The bundle J XV decomposes invariantly into the direct sum of two sub bundles,

JxV='IT'lEBlU. (ii) For each x E J, Dx n V = {0}, where U = U {x} x Ux (set-theoretically).

xEJ (iii) There are constants c > 0 and f-L, 0 < f-L < 1, such that

forallx E J, v E 'IT'xl \ {0}, wE Ux.


(vi) The set U Cx is a bounded neighborhood of J in (C(iLJ) U C+(iLJ)) n xeJ

ceca+ I) u c_(a+J)); (vii) For X E J n (Es u En, Cx c C(x); (viii) Each Cx is strongly unordered.

Sketch of Proof. We will prove the existence of a lamination by a Graph Transform argument.

Throughouttheproof,for~ E Jlet17 := (SIJ)- 1 ~. Foro> OwriteCJ(B2(8)) := {g E C 1(B2(8)) : g(O) = 0 and Dg(O) = 0}. Further, let 9(8) := {G : J --+ CJ(Bz(8)) : G is continuous}. 9(8) endowed with the uniform norm is a Banach space. Given G E 9(8) and~ E J, put Gs := G(~) (E CJ(B2(8)). For~ E J and

Gs E CJ(Bz(8)), set £Gs := {~ + Psw + Gs(w)Vt : w E Bz(8)}. £Gs is a C 1

embedded one-codimensional submanifold tangent at~ to 1Us. Finally, given 8 > 0 and .B > 0, we write 9(8; ,B) := {G E 9(8) : IIGsiii :::; .B for each~ E J}. 9(8; ,B) is a bounded closed convex subset of 9(8).

By the Implicit Function Theorem (see e.g. p. 17 in [24]) the preimage s-1 (£Gs) contains a C 1 embedded one-codimensional submanifold, tangent at 17 to 1U17 • The tangent space 1UlJ at 17 is transverse to vr. Consequently, there are 81 and HlJ E

CJ (Bz(8r)) such that £H7J C s-1(£Gs). Moreover, HlJ depends continuously on~· Define~ by ~Gs := HlJ. ~will be our graph transform. (Notice that for the time being the definition of ~ is not complete, since we do not know yet what is its target space.)

By Proposition 3.7, there are d > 0 and v, 0 < v < 1, such that IIDSn(1J)huy II :::; dvn for each 17 E J and each n E N. Further, from the continuity of 1] 1-+ 1U1J, compactness of J and the fact that all1UlJ are transverse to v1 it follows that there is K > 0 such that II IT huq II < K for each 17 E J (recall that IT is the projection of V on Vz along Vr). Thus we have

Replacing s by sNI for Nt sufficiently large, we get

II (ITo DS(17))v II < vII vii for each 17 E J and each v E 1UlJ.

Using the continuity agaitl, we prove that there exists K > 0 such that for each 1] E J and each v E V with l!lP'1(ry)vl! :::; Kl!lP'2(1])VII (recall that lP'1, lP'z are the bundle projections corresponding to the decomposition J x V = 'II' J E9 1U), the inequality

II (ITo DS(17))v II < v llv II

holds. This implies that there are 80 > 0 and .Bo > 0 with the following property: For each 8, ,8, 0 < 8 < 80 , 0 < .B < f3o, each H E 9(8; {3), and each 1] E J, the projection of S(£H1J) along vr on~+ V2 lies within distance v8 of~·


For a one-codimensional subspace W transverse to v 1, denote by R w the bounded linear functional in V2* such that W = { w + Rww · v1 : w E V2}. Write dist(W1, W2 )

:= II Rw1 - Rw2 [[. The sub spaces U1J depend continuously on 'fJ E J and are transverse to v1, hence the operators Rv~ are well defined and depend continuously on 'fJ E J, and the set {Rv~ : 'fJ E J} is compact.

The exponential separation (Proposition 3.5(iii)) implies that, after possibly replacing S by SN1 , there is L > 0 such that

dist((DS(~))- 1 W1, (DS(~))- 1 W2) < vdist(W1, W2)

for any 'fJ E J and any one-codimensional subspaces W1, W2, transverse to v1 and to the vector tangent at 'fJ to J, with dist(W1, U1J) < L, dist(W2, U1J) < L.

From the above it follows that there are 8' > 0 and /3' > . 0 such that 18 is a contraction mapping from 9(8', f3') into itself. By the Contraction Mapping Theorem there exists precisely one fixed point of 18, which corresponds to a lamination, forward invariant under S, satisfying (i) through (iii), as well as (v).

In fact, we have proved the existence of a lamination forward invariant under some iterate of S. From part (v) it follows in a standard way that, after possibly taking smaller leaves, the lamination obtained is forward invariant under S. (Here is the place where we assume that the diameters 8(x) can depend on x.)

Parts (iv) and (vi) follow from (v) in the standard way, cf..the proof ofThm. 4(v)(vi) on p. 1181 in [15]. Part (vii) is a consequence of part (vi) and Proposition 1.3.

It remains to prove (viii). Suppose to the contrary that there are Z1 < z2 contained insome£x. We can assumez1 « Z2 (if not, replace them by Sz1 andSz2respectively). By the continuity of the embeddings F, we can finds such that z1 « s « z2 and s ELy for some y E J, y « x. As Z1, Z2 E Lx, from the first inequality in part (vi) we deduce that w(z1) = w(z2) = w(x) =: X· If x = x then by part (viii) we see z1, z2 E C(x), which contradicts Proposition 1.3(ii). So assume x « x, say. In particular, we have x tf. C(x), y tf. C(x). From Proposition 1.3(v) we deduce that the direction of snx- sny converges to the direction of V1 (x) as n -+ 00. As snx- sny » 0 for each n EN, we have that

snx -sny IISnx- snyll -+ v1 Cx) as n-+ oo.

On the other hand, from the second inequality in part (vi) it follows that

II snz1- S"y snx- S"y II ------ -+ 0 asn-+ oo. IISnz1- S"yll IISnx- snyll

S"z1 S"y But the above gives that IJS"z1::::s"yiJ E - V+ converges to v1 (x) E v_;:, a contradic-tion. D

For a p-are J satisfying the assumptions of Proposition 3.8, a forward invariant set U Lx will be called an admissible neighborhood of J in (C((LJ) U C+(a_J)) n

xel ccca+J) u c_ca+J)).

The next result is a generalization of Proposition 1.4(b) (cf. Thm. 5A.3 in [22]).


Proposition 3.9. Assume that w (~) is compact nonempty for each~ E X. Then for any X E & \ A_ the set w€- (x) is locally unique.

Proof. For x E Eu this is Proposition 1.4(b). If x E &8 then from Proposition 1.2 it follows that x E A. So assume x E &0 • From Theorem 2.1 we infer that there is ~ E w€-(x) such that S~:::; ~· As

and Slw,-cx) is a homeomorphism onto its image, it follows that for each y E w€-(x) there exists a (unique) sequence {y_n} c w€-(x), y0 = y, such that Sy_n = Y-n+l for each n EN. ForE> 0 put

It is obvious that {U1;d form a neighborhood base of x in the original topology. Since Proposition 1.1 (iv) implies r (x) < 1, we can find E > 0 so small that

for each Yl, Yz E UE, (3.2)

where r' := (r(x)+ 1)/2 < 1 and P2 (x) denotes the projection on V2 (x) along v1 (x). Now, let wE- (x) be the portion of another center manifold. From the facts proved earlierweconcludethatforeachy E wE-(x)nU€, whereE > Oischosensothat(3.2) holds, there is a sequence {y_n} C w€-(x) n U€, Yo= y, such that Sy_n = Y-n+l for eachn EN. Butfrom(3.2)itfollowsthatineachleaf{y+w: wE Vz(x), llwll < E}, y E wE- (x) fixed, there can be only one ~ for which the above sequence exists, that is, ~ = y. In particular, w€- (x) is unique. D

Theorem 3.10. Assume that each w(~) is nonempty compact. Then for a lower unstable equilibrium x there exists a nondegenerate p-are J with a+ J = x, such that w(~) = a_J for each~ E [a_l, x]] n X. Further, J c J' for any nondegenerate p-are J' With a+J' =X.

Proof. Fix a y E w€-(x). By Theorem 2.1 we have Sy « y, and by Proposition 1.5, w (y) = z with z « Sy « y « x. Proposition 2.4 asserts that z = w (5) for ~ E w€-(x) andz E ~·Fix as E w:(z). By Theorem 2.1, Ss « s, consequently [[z, sJJ is forward invariant. Now, from Proposition 1.3(ii), (v) we deduce that w€-(x) c K (z) n M_. Put (Sy, y) := [Sy, y] n w€-(x). The compact set (Sy, y) is contained in K (z) n M_, so there is n0 such that sno (Sy, y) is contained in [[z, ?;]]. Because of forward invariance of [[z, sJJ, the union

00

B := U S"(Sy, y) c [[z, s]]. n=no

For each n, the set sn(Sy, y) is diffeomorphic to [0, 1], and also each finite union ll2

U sn (Sy, y) is diffeomorphic to [0, 1], where the diffeomorphisms can be chosen n=n1


to be increasing. Consequently B can be proved to be diffeomorphic to (0, 1] in an increasing way. We claim that the closure of B in V equals B U {z}. Let {~k} c B

ll]

converge to some ~ E V. If there is n 1 such that all ~k E U sn ( Sy, y) then n=no

ll]

~ E U sn(Sy, y). So assume there is a subsequence (denoted also by {~d) such n=no

that {~d c snk(Sy, y) and nk -+ 00 ask-+ 00. Then ~k E [Snk+ 1y, snky], and, consequently, ~ = z. Proposition 1.3(v) implies that

1. ~- Z ( ) yo s~ II~- zll = v1 z E +· sEE

As a result, the set {z} U B U W;(x) U {x} is diffeomorphic to [0, 1] in an increasing way, so it is a p-are with lower endpoint z and upper endpoint x.

Now, let J' be any nondegenerate p-are with a+J' = X. By Lemma 3.6, J' is tangent atx to wE-(x). Proceeding as above we prove that points from J' sufficiently close to x hav!; z as their cu-limit,..Point, so z E J'. Put J := J' n [z, x]. Let l; be the least point in J such that [l;, x] n J = [l;, x] n J. By Proposition 3.9 wE-(x) is unique, hence l; « x. We claim that l; = z. Suppose to the contrary that z « l;. Then there is a sequence Sk / l; ask-+ oo, with Sk E l\ J. The mappings Sly and SJ 1 are homeomoryhisms, so there is a sequence ~k / ~ask -+ oo, with s~k = sk> s~ = c;, and ~k E J \ J. Since l; = S~ « ~, it follows that l; « ~k for sufficiently great k, which contradicts the choice of l;. Therefore l; = z. D

Remark. For earlier results see e.g. [28].

Our next result is in a sense an analog of Proposition 1.7.

Proposition 3.11. Assume that cu(~) is compact nonempty for each~ E X. Let x E

En n U_. Then K(x) n C_(x) = 0.

Proof. Suppose to the contrary that there are y E K(x) and z E C(x) such that y « z. By Proposition 1.3, y E M+, so there is n0 such that sny « x for n :::::; n0 .

Now Theorem 3.10 asserts that there is l; E E, l; « x, such that cu(~) = l; for any ~ < X, ~ sufficiently close to X. In particular, cu csn y) = ~ for sufficiently great n, a contradiction. D

The next result refines, for the case that all limit sets are compact nonempty, a classification provided in Theorem 2.1 (compare also [47]).

Theorem 3.12. Assume that cu (~) is compact nonempty for each~ E X. Let y E X be convergent. Then one ofthefollowing possibilities occurs:

(i) y E An M+ and cu(y) E A_; (ii) y E A_ n C(cu(y)) and cu(y) E A_; (iii) y E An M_ and w(y) E A+· (iv) y E ~ n C(cv(y)) and cv(y) E A+;


(v) y E s_ \A_ andy E C(w(y)) and w(y) E s_ \A_;

(vi) y E s+ \~andy E C(w(y)) and w(y) E s+ \A+; (vii) y E U_ and w(y) E U_. !fin addition w(y) E [n then y E C(w(y)); (viii) y E U+ and w(y) E U+- /fin addition w(y) E [n then y E C(w(y)).

Proof. Put x : = w (y). First, assume y E A_. From Proposition 1. 7 we infer that x E Es U &n. We have either y E M+, or y EM_, or else y E C(x). Whenever y EM+, Proposition 1.3(v) yields that there is n0 EN such that

Sny « sn+!y « sn+2 y «X for n ~no.

Because of continuity of sn+I there are y1 « y « y2 such that w(ry) = x for all 1} E [yl' Y2] n X, that is, y E A. If X E £s then by Proposition 1.2 X E A. If, on the other hand, x E &n, then from Proposition 3.5 we get x E S_. At any rate, Theorem 2.1(A) gives x E A_. The casey EM_ is treated in an analogous way. Now, let y E C(x). As y E A_, it follows that there is z E X, z « x such that W(1]) =X for an., E [[z, y] n X. From Proposition 1.3(iii) and (v) we deduce that [[z, y]] n X c K (x) n M+. Reasoning as above we obtain x E A_.

Now, assume that y ¢. A_. We claim xis not asymptotically lower stable. Suppose to the contrary x E A_. The casey E M is excluded, since then y would be in A. So assume y E C (x) (notice that x E A_ implies x E £8 U &n)· By Theorem 2.1 there is ~ E W;(x) such that sn~ /' x as n ~ oo. As~« x, one can find n0 EN such that sno y » ~. Foro > 0 define Lo := {y - ev1 : 0 < e ::::; o}. By strong monotonicity, sno IL, is a homeomorphism onto its image (for sufficiently small o). It follows that we can find 8 > 0 so small that ~ « sno L8 « y. Strong monotonicity yields that w(S) = w(u) = w(y) = x for all u E L 8, therefore y E A_, a contradiction. By Theorem 2.1, either (A), X E u_ n (£u u &n), or (B), there is a sequence {xz} c [ n wE-(X), Xz /'X as f ~ 00, andx E S_ \A.

Assume (A). By Theorem 3.10 there exists an equilibrium z E A+, z « x, such that w(17) = z for any 17 E [z, x]] n X. Take a sequence Yk /' y ask~ oo. Because of strong monotonicity, z ::::; w (yk) ::::; x, at least for sufficiently large k. Suppose that for some k the set w (yk) contains x. In fact we have then w (yk) = x, as the compact limit set is strongly unordered (Prop. 2.2 in [47]). This means that y E A_, from which it follows that Yk+I E M+ and x E A_, a contradiction. Applying Theorem 3.10 we obtain w(yk) = z « x for sufficiently large k. We have thus that y is lower approximable and in u_.

Assume (B). From Proposition 2.5 we deduce that y E C(x) U C+(x). But if y were in C+(x) then by Proposition 1.3(v) it would be in M_, hence in A, a contradiction. Soy E C(x). As r(x) < 1 and r(~) continuously depends on~ E £, it follows that there is lo such that r(~) < 1 for each~ E [ n wE-(x) n [xzo, x]. Put J := cwE-(x) u {x}) n [xzo, x]. Let u be an admissible neighborhood of J and let no be so large that sno y E U. Take a sequence Yk /' y as k ~ oo, such that sno Yk E U. (The existence of such a sequence implies that y is lower approximable.) Recall that the neighborhood U is laminated by the leaves £z depending continuously on z E J. For each k E N let xk E J be the unique point such that Yk E Lxk. We have xk /' x as


k--+ oo. From this it readily follows that y E S_. The fact that y ¢.A_ is obtained as above.

We have thus proved parts (v) and (vii). Parts (vi) and (viii) are proved in an analogous way. D

As a by-product of the proof of the previous result we obtain the following.

Corollary. Assume that w(~) is compact nonempty for each ~ E X. Then each convergent point is lower and upper approximable.

Proposition 3.13. Assume that w (~) is compact nonempty for each~ E X. Then a point y belongs to N if and only if there is a nondegenerate p-are J 3 w (y) such that J c E and a_l « w(y) « a+J.

Proof. First, assume y EN. Put x := w(y). Let L be as in the definition of N. We claim y ¢. U. Suppose to the contrary that (say) y E U_. Copying the proof of part (vii) of Theorem 3.12 we get that for points rJ E L, rJ < y, sufficiently close toy, one has w(rJ) = z E En~ and rJ E M_, a contradiction. Now we claim L c S \A. Suppose to the contrary that there is z E L n A_ (say). As z ¢. M, from Theorem 3.12(ii) it follows that w(z) EA._, hence w(z) E Es U En (by Theorem 2.1). Further, there is s E L, s < y, such that [s, z] n X c K(w(z)), in particular 0 =1 ([s, z] \ {z}) n L c K(w(z)). Proposition 1.2 yields ([s, z] \ {z}) n L c M+, a contradiction. From Theorems 3.12(v) and 2.1 it follows that x E S n En\ A 112.

By a (now) standard method, there is a p-are J 3 X, a_ I « X « a+J, such that r(~) < 1 for all r E J n E. We claim that (after possibly taking a smaller J with the same properties) J C E. Suppose to the contrary that there is a sequence ~k ? x as k --+ oo, ~k E J \ E (say). As SIJ is conjugate to an orientation preserving homeomorphism of [0, 1], there exist (say) a sequence {xk} c J n E, xk? x, and a sequence {rk} of positive numbers, lim rk = 0, such that rJ « S17 for each 17 « xk>

k--+oo II rJ - Xk II < rk. By Theorem 2.1, each Xk is in A._, which is impossible.

Conversely, assume x belongs to some p-are J c E with a_ J « x « a+ J. Take an admissible neighborhood fJ of J, and let N E N be such that sN y E U. As y is in the open set (SN)-1 U, one can find E > 0 so small that L := {y + evo : 1e1 ::S E} C (SN)-1U. From Theorem 2.1 it follows that K(z) = C(z) for each z E J \{a_ I, a+J}, hence no~ E L can be in M. D

Corollary. Assume that w (~) is compact nonempty for each~ E X. Then the set N is order-open in X.

Proof. Let y E N. It is easily verified that N is open. Let Yr « y « y2 be such that the p-are J with lower endpoint w (y1) and upper endpoint w (y2 ) consists entirely of equilibria. Fix a z E UYr, Y2Jl Its limit set w (z) is nonempty compact and contained in [[w(yr), w(y2)]]. By a standard argument we prove that in fact w (z) is an equilibrium. D

Remark. A result analogous to the above Corollary was proved in [47], Thm. 5.7, under the assumption that X is order-connected. That assumption, however, is too strong in view of the applications in Section 4.


A p-are J is called maximal if there is no p-are J' =!= J, such that J c J'.

Proposition 3.14. Assume that w (5) is compact nonempty for each 5 E X. Let J be a maximal p-are. Then a_ J E A_, a+ J E A+·

Proof. Suppose to the contrary that (say) a_f ¢. A_. If a_J E U_, then by Theorem 3.10 we see that there is a nondegenerate p-are J 1 with a+J1 = a_J. So J is contained in J U J1 and J =/= J U 1). For a_J E S_ \ A_ we put f) := W;(a_J) n [X, a_f), Where X E C n wE-(a_J), X « a_J is SUfficiently close to a_ I (compare the proof of Proposition 3.9). The upper endpoint of J is treated analogously. 0

The assumption that each limit set is compact nonempty cannot in general guarantee the existence of maximal p-ares (as a counterexample take the identity mapping on JR.). In the last part of the present section we give conditions which imply the existence of maximal p-ares.

Theorem 3.15. Assume that w (5) is compact nonempty for each 5 E X and that the set£ is compact. Then each p-are J is contained in some maximal p-are I max·

Proof. Denote by J the family of all p-ares containing J, partially ordered by inclusion. Let Jt := {Ky : y E f} be a simply ordered subfamily indexed by a simply ordered index set r. We intend to prove that there exists a p-are K contained in any K' ~ U Ky ~ K. Because£ is compact, the set {vi(a_Ky) : y E f} c JR.

yEr

is bounded. Moreover, vi(B_Ky1) 2: vi(a_Ky2 ) whenever Y1 ::::; yz. Put u := inf{vi(a_Ky) : y E r}. Let {y(k)hEN be a sequence with lim vi(a_Ky(k)) = u.

k-+oo We extract a subsequence (denoted again by {y(k)}) such that a_Ky(k) ~ x E £as k --7 oo. Evidently the simply ordered set {y(k) :kEN} is cofinal with r. From this it follows that ifthe sequence {a_Ky(k)} stabilizes then so does {a_Ky }. In such a case there is Yo E f such that x = a_KY! = a_KY2 for Yo ::::; Yl ::::; Y2· Now assume X « a_Ky(k) for each k. From Theorem 2.1 it follows that X E s+ n Cn \A+· Further, the points a_Ky(k) sufficiently close to X belong to the unique w:(x). Suppose that for some other sequence {y'(k)} cofinal with r, lim a_Ky'(k) = x' =!= x. As

k-+oo vi(x') = vi(x), the set {x, x'} is strongly unordered, from which it follows that there are k1, k2 EN such that a_Ky(k1) and a_Ky'(kz) are strongly unordered, a contradiction. Therefore there is an index Yo E r with the property that a_Ky E wE+(x) for each y 2: y0 . Proceeding in an analogous way with upper endpoints, we obtain a desired K'. The statement of the theorem follows by Zorn's lemma. 0

Theorem 3.16. Assume that X is order-convex in V and that S : X --7 X is ordercompact. Then for any pair a, b of equilibria with a « b there exists a p-are J with a_J =a, a+J =b.

Proof. By order-convexity, [a, b] c X. Denote by J[a,bJ the family of all p-ares contained in [a, b]. Obviously {a}, {b} E J[a,bJ· A p-are J E J[a,bJ is called [a, b)maximal if there is no J' E J [a,bJ such that J' ~ J, J' =/= J. As S is order-compact,


the set En [a, b] is compact. Applying Theorem 3.15 with S replaced by Sl[a,bJ

we get the existence of an [a, b ]-maximal p-are I max· We claim that a_ I max = a and a+lmax = b. Suppose to the contrary that a_lmax » a. If a_lmax ¢. A_, then copying the proof of Proposition 3.14 we obtain a nondegenerate p-are J" with a+J" = a_ fmaX> a_ J" E (a, a_ fmaxJ. Then J" U fmax ~ fmax, a COntradiction. So a_lmax E A_. Let h be an [a, a_lmax]-maximal p-are containing a. We have a+ h « a_ I max, since otherwise the p-are h u I max would be a proper superset of lmax· Moreover, a+JI E A+· Denote by J't the family of all simply ordered subsets of E n [a+ h , a_ J max], partially ordered by inclusion. Proceeding again in the same spirit as in the proof of Theorem 3.15 we obtain the existence of a maximal Kmax E J't. Evidently a+h, a_lmax E Kmax· As a_lmax E A_, Theorem 3.10 implies that there is c E Kmax> c « a_lmax such that X ::; c for each X E Kmax \ {a_lmax}. In particular, c and a_lmax are the only equilibria in [c, a_lmax1· Applying Thm. 1.4 in [46] or Thm. 2.4 in [47] we obtain that Ci!(TJ) = a_lmax for all rJ E [[c, a_lmaxJ, hence c E U+. But this implies, via Theorem 3.10, that there is a p-are with lower endpoint c and upper endpoint a_Jmax, which contradicts [a, b]-maximality of lmax· We have thus proved that a_lmax =a. Analogously, we prove that a+lmax =b. D

Special cases of Theorem 3.16 were known earlier; see e.g. [42], [2], [13], [52].

4. Genericity of some properties. In the present section we assume that S : X ---+ X is a compact C1 strongly monotone mapping satisfying (DQ), such

that each limit set is compact nonempty. The letter Twill stand for sN with N EN such that w(y; sN) is a singleton for

each y E Y.

Theorem 4.1. The set M(T) U N(T) is dense in Y, hence in X.

Proof. Assume y E Y \ (M(T) U N(T)). Put x := w(y; T). First consider the casey E U_(T). By Theorem 3.12 applied toT, x E U_(T), too. Theorem 3.10 asserts there is z E E(T), z « x, such that w(~; T) = z for each~ E [z, x]] n X. TakeN E N so large that yN y » z. Let U C Y be a neighborhood of yN y such that U » z. Obviously the inequality Ci!(rJ; T) ::0: z holds for each fJ E U. In fact, for fJ E U, rJ « TNy, we have Ci!(rJ; T) = z, since Ci!(rJ; T) = x would imply x E A_(T). Therefore for each~ E (TN)- 1U, ~ < y, we have w(~; T) = z. Take E > 0 so small that L := {y - ev! : e E (0, E]} c (TN)-1 U. The simply ordered set Lis contained in K (z; T), so by Proposition 1.3 at most one point in L can be in C(z; T) and the remaining ones must be in M(T). We have thus proved that there are points in M (T) arbitrarily close to y. The case y E U+ (T) is considered in an analogous way. The case y E U (T) is thus done.

For y E A1;2(T) n C(x; T) the proof is similar. It remains to consider the casey E S(T) \ A 1; 2 (T). By Theorem 3.12(v)-(vi), y E C(x; T) and x E S(T) \ A 1; 2 (T). Proposition 3.9 asserts that the manifold WE (x; T) is locally unique. From Theorem 2.1 we derive that there are T -equilibria x1, x 2 E WE (x; T), x 1 « x « x 2.

Put J := [xi, x2] n WE(x). As y ¢. N(T), from Proposition 3.13 we deduce that there is a sequence {~k} c J \ E(T), ~k / x (say) as k ---+ oo. Consequently,


copying the corresponding part of the proof of Proposition 3.13 we get that there is a sequence xk ? x, xk E J n S(T) n .A._(T) (say). Let fJ be an admissible neighborhood of J and let N E N be so large that TN y E fJ. Take E > 0 so small that M := {y + ev1 : 1e1 :::; E} c (TN)- 1U. As the mapping TIM is continuous and strongly monotone, it is a homeomorphism onto its image. Consequently there is a sequence {yk} C M, Yk ? y as k ---7 oo, with Yk E £xk C C(xk). By Theorem 3.12(ii), Yk E .A._(T). By the above, each Yk can be approximated by points from M (T). 0

A natural question arises whether the set M(T) U N(T) is large in a "measuretheoretical" sense, too (provided, of course, that the ambient Banach space V is separable, cf. Section 7 in [21], or Corollary 5.6 in [47]). As the following simple example shows, generally (but not generically!) this is not the case.

Let k r+ uk be a bijection of the set of positive integers onto the set of all rationals contained in [0, 1]. Put

CXl 1 1 B := [0, 1] n U (uk- k+2 ' uk + k+2).

k=l 2 2

B is open dense in the relative topology of [0, 1], but the Lebesgue measure of B is :::; 1/2. ThesetA := [0, 1]\BisnowheredenseofpositiveLebesguemeasure. Takea C00 real function f on lR such that f (x) = 0 if and only if x E A, f (x) > 0 for x < 0, f (x) < 0 for x > 1. It is a standard fact that the ordinary differential equation i = f (x) generates a C00 semifiow on JR. DefineS to be the time-one map of this semifiow. It is evident that S : lR ---7 lR is a C1 (C00 in fact) strongly monotone mapping of the strongly ordered Banach space JR. Further, [0, 1] is the unique maximal p-are. The set M = lR \ A is open dense, but its complement A has positive Lebesgue measure. The set N is empty, as A contains no open intervals. Therefore lR \ (M UN) = A has positive Lebesgue measure. By Prop. 2(v) in [34] A is not a Gaussian null set.

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