# Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

Authors: Qiuju Xing; Yuming Shi
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.


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## Body

## 1. Introduction

Distribution of a set of maps with some dynamical properties in some continuous map space is a very interesting topic. In the 1960s, Smale   [1] studied density of hyperbolicity. Some scholars believed that hyperbolic systems are dense in spaces of all dimensions, but it was shown that the conjecture is false in the late 1960s for diffeomorphisms on manifolds of dimension ≥2. The problem whether hyperbolic systems are dense in the one-dimension case was studied by many scholars. It was solved in the C1 topology by Jakobson   [2], a partial solution was given in the C2 topology by Blokh and Misiurewicz   [3], and C2 density was finally proved by Shen   [4]. In 2007, Kozlovki et al. got the result in Ck topology; that is, hyperbolic (i.e., Axiom A) maps are dense in the space of Ck maps defined in a compact interval or circle, k=1,2,…,∞,ω   [5]. At the same time, some other scholars considered the distribution of hyperbolic diffeomorphisms in Diff(M), where M is a manifold. Just like the work of Smale, Palis [6, 7] gave the following conjecture: (1) any f∈Diff(M) can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting a homoclinic bifurcation (tangency or cycle), (2) any diffeomorphism can be Cr approximated by a Morse-Smale one or by one exhibiting transversal homoclinic orbit. Later, it was shown that the conjecture (1) holds for C1 diffeomorphisms of surfaces   [8]. And some good results have been obtained, such as any diffeomorphism can be C1 approximated by a Morse-Smale one or by one displaying a transversal homoclinic orbit   [9], any diffeomorphism can be C1 approximated by one that exhibits either a homoclinic tangency or a heterodimensional cycle or by one that is essentially hyperbolic   [10].In 1963, Smale gave the well-known Smale-Birkhoff homoclinic theorem for diffeomorphisms   [11], from which one can easily know that if a diffeomorphism F on a manifold M has a transversal homoclinic orbit, then there exists an integer k>0 such that Fk is chaotic in the sense of both Li-Yorke and Devaney. Later, in 1986, Hale and Lin introduced a generalized definition of transversal homoclinic orbit for continuous maps and got the generalized Smale-Birkhoff homoclinic theorem, that is, a transversal homoclinic orbit implies chaos in the sense of both Li-Yorke and Devaney for continuous maps in Banach spaces   [12]. In the meanwhile, some scholars studied the density of maps which are chaotic in the sense of Li-Yorke or Devaney. In particular, some results have been obtained in one-dimensional maps (cf. [13–15]).Since 2004, Shi, Chen, and Yu extended the result about turbulent maps for one-dimensional maps introduced by Block and Coppel in 1992   [16] to maps in metric spaces. This map is termed by a new terminology: coupled-expanding map. Under certain conditions, the authors showed that a strictly coupled-expanding map is chaotic [17, 18]. Applying this coupled-expansion theory, they extended the criterion of chaos induced by snap-back repellers for finite-dimensional maps, introduced by Marotto in 1978   [19], to maps in metric spaces [17, 20, 21]. Recently, we studied the distribution of chaotic maps in continuous map spaces, in which maps are defined in general Banach spaces and finite-dimensional normed spaces, and obtained that the following several types of chaotic maps are dense in some continuous map spaces: (1) maps that are chaotic in the sense of both Li-Yorke and Devaney; (2) maps that are strictly coupled-expanding; (3) maps that have nondegenerated and regular snap-back repeller; (4) maps that have nondegenerate and regular homoclinic orbit to a repeller   [22].In the present paper, we will construct a set of continuous chaotic maps with generalized transversal homoclinic orbits, and show that the set is dense in the continuous map space. The method used in the present paper is motivated by the idea in   [22].This paper is organized as follows. In Section2, we first introduce some notations and basic concepts including the Li-Yorke and Devaney chaos, hyperbolic fixed point, and transversal homoclinic orbit, and then give a useful lemma. In Section 3, we pay our attention to distribution of maps with generalized transversal homoclinic orbits in a continuous map space, in which every map is defined in a closed, bounded, and convex set or a closed bounded set in a general Banach space. Constructing a continuous map with a generalized transversal homoclinic orbit, we simultaneously show the density of maps which are chaotic in the sense of both Li-Yorke and Devaney in the map space.


## 2. Preliminaries

In this section, some notations and basic concepts are first introduced, including Li-Yorke and Devaney chaos, hyperbolic fixed point, and transversal homoclinic orbit. And then a useful lemma is given.First, we give two definitions of chaos which will be used in the paper.Definition 2.1 (see [23]).
Let(X,d) be a metric space, f:X→X a map, and S a set of X with at least two points. Then, S is called a scrambled set of f if, for any two distinct points x,y∈S,
(2.1)lim⁡inf⁡n→∞⁡d(fn(x),fn(y))=0,limsup⁡n→∞⁡d(fn(x),fn(y))>0.
The map f is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of f.Definition 2.2 (see [24]).
Let(X,d) be a metric space. A map f:V⊂X→V is said to be chaotic on V in the sense of Devaney if (i)
the periodic points off in V are dense in V;(ii)
f is topologically transitive in V;(iii)
f has sensitive dependence on initial conditions in V.Now, we give the definition of hyperbolic fixed point.Definition 2.3 (see [25]).
LetX be a Banach space, U⊂X be a set, and F:U⊂X→X be a map. Assume that z is a fixed point of F, F is continuously differentiable in some neighborhood of z, and denote DF(z) the Fréchet derivative of F at z. The fixed point z is called hyperbolic if σ(DF(z))∩{λ∈C:|λ|=1}=∅, where σ(A) denotes the spectrum of a linear operator A. The hyperbolic fixed point z is called a saddle point if σ(DF(z))∩{λ∈C:|λ|>1}≠∅ and σ(DF(z))∩{λ∈C:|λ|<1}≠∅.
IfF is invertible, then for any p0∈X, the set OF+(p0):={Fk(p0):k≥0} is said to be the forward orbit of F from p0, the set OF-(p0):={Fk(p0):k≤0} is said to be the backward orbit of F from p0. Since it is not required that F is invertible in this paper, a backward orbit of p0 is a set OF-(p0)={pj:j≤0} with pj+1=F(pj),j≤-1, which may not exist, or exist but may not be unique. A whole orbit of p0 is the union OF+(p0)∪OF-(p0), denoted by OF(p0) in the case that it has a backward orbit OF-(p0). The stable set Ws(z,F) and the unstable set Wu(z,F) of a hyperbolic fixed point z of F are defined by
(2.2)Ws(z,F):={p0∈X:theforwardorbit{pj}j=0+∞ofFfromp0suchthat,pj⟶zasj⟶+∞{pj}j=0+∞},Wu(z,F):={p0∈X:thereexistsabackwardorbit{p-j}j=0+∞ofFfromp0suchthatp-j⟶zasj⟶+∞{p-j}j=0+∞},
respectively. The local stable and unstable sets are defined by
(2.3)Wloc⁡s(z,U,F):=Ws(z,F)∩U,Wloc⁡u(z,U,F):=Wu(z,F)∩U,
respectively, where U is some neighborhood of z. If U=B(z,r) or B̅(z,r) for some r>0, then the corresponding local stable and unstable sets of F are denoted by Wlocs(z,r,F) and Wlocu(z,r,F), respectively. By the Stable Manifold Theorem   [26], if F is continuously differentiable in some neighborhood of a saddle point z, then there exists a neighborhood U of z such that the corresponding local stable and unstable set of z are submanifolds of X, respectively.
In the following, we first give the definition that two manifolds intersect transversally and then give the definition of transversal homoclinic orbit for continuous maps.Definition 2.4 (see [25]).
Two submanifoldsV and W in a manifold M are transverse (in M) provided for any point q∈V∩W, we have that TqV+TqW=TqM, where TqV and TqW denote the tangent spaces of V and W at q, respectively, and “+” means the sum of the two subspaces (this allows for the possibility that V∩W=∅).Remark 2.5.
IfM=Rn, then V and W in M are transverse (in M) provided for any point q∈V∩W, we have that TqV+TqW=Rn. Obviously, if dimTqV+dimTqW=n, then the sum of the two subspaces TqV and TqW is a direct one, denoted by ⊕.Definition 2.6 (see [12]).
LetX be a Banach space, F:X→X be a map, and z∈X be a saddle point of F.  (i)
An orbitOF(p0)={pj}j=-∞+∞ is said to be a homoclinic orbit (asymptotic) to z if p0≠z and limj→+∞pj=limj→-∞pj=z.(ii)
A homoclinic orbitOF(p0)={pj}j=-∞+∞ to z is said to be transversal if there exists an open neighborhood U of z such that p-i∈Wlocu(z,U,F) and pj∈Wlocs(z,U,F) for any sufficiently large integers i,j≥0, and Fi+j sends a disc in Wlocu(z,U,F) containing p-i diffeomorphically onto its image that is transversal to Wlocs(z,U,F) at pj.
The following lemma is taken from Theorems  3.1, and 5.2, Corollary  6.1, and the result in Section  7 of   [12].Lemma 2.7.
LetF:Z→Z be a map, where Z=X×Y, and X and Y are Banach spaces. (i)
LetA and B be linear continuous maps in X and Y, respectively, with the absolute values of the spectrum of A less than 1 and the absolute values of the spectrum of B larger than 1, and ∥A∥,∥B-1∥≤λ0 for some constant 0<λ0<1.(ii)
Assume thatU is an open neighborhood of 0 in Z and f1:U→X,f2:U→Y are Ck(k≥1) maps with fi(0)=0,Dfi(0)=0,i=1,2. Further, assume that Df1,Df2 are uniformly continuous in U, and satisfies that for some constants 0<θ<1-λ0 and γ>0, ∥Df1(x,y)∥,∥Df2(x,y)∥<θ for all (x,y)∈B̅(0,γ)⊂U.(iii)
LetF:U→Z be of the following form:
(2.4)F(x,y)=(Ax+f1(x,y),By+f2(x,y)),
and have the local stable and unstable manifolds Wlocs(0,U,F)={(x,y)∣(x,y)∈U,y=0} and Wlocu(0,U,F)={(x,y)∣(x,y)∈U,x=0}≠{0}.(iv)
Assume thatO(p0)={pi}i=-∞∞ is a homoclinic orbit of F with pi→0 as i→±∞, and there exists an integer N>0 such that p-N∈Wlocu(0,U,F), pN∈Wlocs(0,U,F), and
(iv1)
F2N sends a disc O1∩Wlocu(0,U,F) centered at p-N diffeomorphically onto O2=F2N(O1) containing pN;(iv2)
O2 intersects Wlocs(0,U,F) transversally at pN.Then O(p0) is a transversal homoclinic orbit of F. Furthermore, there exists an integer k>0 and a subset Λ in a neighborhood of O(p0) such that Fk on Λ is topologically conjugate to the full shift map on the doubly infinite sequence of two symbols. Consequently, F is chaotic in the sense of both Li-Yorke and Devaney, and its topological entropy h(F)≥log2/k.Note that it is not required thatF is a diffeomorphism, even F may not be continuous on the whole space Z in Lemma 2.7.


## 3. Distribution of Maps with Transversal Homoclinic Orbits

In this section, we first consider distribution of maps with transversal homoclinic orbits in a continuous self-map space, which consists of continuous maps that transform a closed, bounded, and convex set in a Banach space into itself. At the end of this section, we discuss distribution of chaotic maps in a continuous map space, in which a map may not transform its domain into itself.Without special illustration, we always assume that(X,∥·∥X) and (Y,∥·∥Y) are Banach spaces, and DX and DY are bounded, convex, and open sets in X and Y, respectively. It is evident that D=DX×DY is a bounded, convex, and open set in Z=X×Y, where the norm ∥·∥ on Z is defined by ∥(x,y)∥=max{∥x∥X,∥y∥Y}, for any (x,y)∈Z, where x∈X,y∈Y. Introduce the following map space:
(3.1)C0(D̅,D̅):={f:D̅⟶D̅iscontinuousandhasafixedpointinD̅}.
For any f∈C0(D̅,D̅), let
(3.2)‖f‖:=sup⁡{‖f(x)‖:x∈D̅},
and for any f,g∈C0(D̅,D̅), let
(3.3)d(f,g):=‖f-g‖.
Then (C0(D̅,D̅),d) is a metric space. It may not be complete because a limit of a sequence of maps in C0(D̅,D̅) is continuous and bounded, but may not have a fixed point in D̅. But in the special case that Z is finite-dimensional, (C0(D̅,D̅),d) is a complete metric space by the Schauder fixed point theorem.In this section, we first study distribution of maps with transversal homoclinic orbits inC0(D̅,D̅).For convenience, by(x,y)∈Z denote x∈X and y∈Y, by Fix(f) denote the set of all the fixed points of f. For (x1,y1),(x2,y2)∈Z with (x1,y1)≠(x2,y2), by l((x1,y1),(x2,y2)) denote the straight half-line connecting (x1,y1) and (x2,y2): (3.4)l((x1,y1),(x2,y2))≔{u=(x1,y1)+t((x2,y2)-(x1,y1)):t≥0}.Lemma 3.1 (see [22, Lemma  3.1]).
For every mapf∈C0(D̅,D̅) and any ε>0, there exists a map g∈C0(D̅,D̅) such that d(f,g)<ε, Fix(g)∩D≠∅, and g is continuously differentiable in some neighborhood of some point x*∈Fix(g)∩D.Lemma 3.2.
For every mapf∈C0(D̅,D̅) and every ε>0, there exists a map F∈C0(D̅,D̅) with d(f,F)<ε such that F has a transversal homoclinic orbit in D.Proof.
Fix anyf∈C0(D̅,D̅). By Lemma 3.1, it suffices to consider the case that f has a fixed point z=(z1,z2)∈D with z1∈X,z2∈Y, and is continuously differentiable in some neighborhood of z.
For anyε>0, there exists a positive constant r0<ε/4 with B̅(z,r0)⊂D such that
(3.5)‖f(x,y)-z‖<ε4,(x,y)∈B̅(z,r0).
Letr=r0/4, take two constants a,b with r<a<b<r0/3 and take two points p1=(x0,z2)∈B(z,r),p0=(z1,y0)∈B(z,b)∖B̅(z,a), where x0∈X,y0∈Y.
The rest of the proof is divided into three steps.
Step 1.
Construct a mapF that is locally controlled near z.
Define(3.6)F(x,y)={(λ(x-z1)+z1,μ(y-z2)+z2),(x,y)∈B̅(z,r),(x+(x0-z1),y-(y0-z2)),(x,y)∈B̅(z,b)∖B(z,a),
where x∈X, y∈Y, and λ and μ are real parameters and satisfy
(3.7)|λ|<1,br<|μ|<r0r.
Note that |μ|>1 since r<b.
For any(x,y)∈B(z,a)∖B̅(z,r), F(x,y) is defined as follows. Let (x1′,y1′) and (x2′,y2′) be the intersection points of the straight line l(z,(x,y)) with ∂B̅(z,r) and ∂B̅(z,a), respectively, (see Figure 1). Set
(3.8)F(x,y)=F(x1′,y1′)+t(x,y)(F(x2′,y2′)-F(x1′,y1′)),
where t(x,y)∈(0,1) is determined as follows;
(3.9)(x,y)=(x1′,y1′)+t(x,y)((x2′,y2′)-(x1′,y1′)).
It is noted that when (x,y) continuously varies in B(z,a)∖B̅(z,r), so do the intersection points (x1′,y1′) and (x2′,y2′). Consequently, t(x,y) and then F(x,y) are continuous in B(z,a)∖B̅(z,r).
Next, defineF(x,y)=f(x,y) for (x,y)∈D̅∖B(z,r0). Finally, for any (x,y)∈B(z,r0)∖B̅(z,b), suppose that (x1′′,y1′′) and (x2′′,y2′′) are the intersection points of the straight line l(z,(x,y)) with ∂B̅(z,b) and ∂B̅(z,r0), respectively. Define F(x,y) as that in (3.8), where t(x,y) is determined by (3.9) with (x1′,y1′) and (x2′,y2′) replaced by (x1′′,y1′′) and (x2′′,y2′′), respectively. Hence, F(x,y) is continuous in B(z,r0)∖B̅(z,b).
Obviously,z is a saddle fixed point of F, and
(3.10){(x,y):x=z1,‖y-z2‖Y<b}⊂Wloc⁡u(z,b,F),{(x,y):y=z2,‖x-z1‖X<r}⊂Wloc⁡s(z,r,F).Step 2.
F∈C0(D̅,D̅) and satisfies that d(f,F)<ε.
From the definition ofF, it is easy to know that F is continuous on D̅ and has a fixed point z∈D̅, that is, F∈C0(D̅,D̅).
Next, we will prove thatd(f,F)<ε. For (x,y)∈D̅∖B(z,r0), ∥F(x,y)-f(x,y)∥=0<ε. For (x,y)∈B̅(z,r), it follows from (3.5) and (3.6) that
(3.11)‖F(x,y)-f(x,y)‖≤‖F(x,y)-z‖+‖f(x,y)-z‖≤r0+ε4<ε.
For (x,y)∈B̅(z,a)∖B(z,r), it follows from (3.5), (3.6), and (3.8) that
(3.12)‖F(x,y)-f(x,y)‖=‖F(x1′,y1′)+t(x,y)(F(x2′,y2′)-F(x1′,y1′))-f(x,y)‖≤|1-t(x,y)|‖F(x1′,y1′)-z‖+|t(x,y)|‖F(x2′,y2′)-z‖+‖f(x,y)-z‖≤r0+2b+ε4<ε.
For (x,y)∈B̅(z,b)∖B(z,a), from (3.5) and (3.6), one has
(3.13)‖F(x,y)-f(x,y)‖≤‖F(x,y)-z‖+‖f(x,y)-z‖<2b+ε4<ε.
For (x,y)∈B̅(z,r0)∖B(z,b), from (3.5), (3.6), and (3.8), one has
(3.14)‖F(x,y)-f(x,y)‖=‖F(x1′′,y1′′)+t(x,y)(F(x2′′,y2′′)-F(x1′′,y1′′))-f(x,y)‖≤|1-t(x,y)|‖F(x1′′,y1′′)-z‖+|t(x,y)|‖F(x2′′,y2′′)-z‖+‖f(x,y)-z‖≤r0+ε4+ε4<ε.
Therefore, from the above discussion, d(f,F)=∥F-f∥<ε.Step 3.
F has a transversal homoclinic orbit in D.
It follows from (3.6) that F(p-1)=p0 and F(p0)=p1, where p-1=(z1,μ-1(y0-z2)+z2)∈B(z,r) by (3.7). So, by (3.10) one has that p-1∈Wlocu(z,r,F), p1∈Wlocs(z,r,F). Hence, O(p0)={pj}j=-∞∞ satisfies that limj→∞pj=limj→-∞pj=z. Thus, O(p0)⊂D and is a homoclinic orbit of F.
Set a positive constantδ satisfying
(3.15)δ<min⁡⁡{b-‖y0-z2‖Y,‖y0-z2‖Y-a,|μ|-1b,r-|μ|-1b}.
Then 0<δ<r by (3.7), and consequently it follows from (3.10) that the disc
(3.16)O0≔{(x,y):x=z1,‖y-y0‖Y≤δ}⊂(B(z,b)∖B̅(z,a))∩Wloc⁡u(z,b,F).
Further, set the discs
(3.17)O-1≔{(x,y):x=z1,‖y-(μ-1(y0-z2)+z2)‖Y≤|μ|-1δ},O1≔{(x,y):x=x0,‖y-z2‖Y≤δ}.
Then O-1⊂Wlocu(z,r,F), O0=F(O-1), and O1=F(O0) (see Figure 2).
It is evident thatO1 intersects Wlocs(z,r,F) transversally at point p1. In addition, by the definition of F in B̅(z,b), one can get that F2:O-1→O1 is a diffeomorphism. Therefore, O(p0) is a transversal homoclinic orbit asymptotic to z of F by Definition 2.6, where N=1.
The entire proof is complete.Figure 1Figure 2Theorem 3.3.
LetX and Y be Banach spaces, Z=X×Y, and D be a bounded, convex, and open set in Z. Then, for every map f∈C0(D̅,D̅) and for any ε>0, there exists a map F∈C0(D̅,D̅) satisfying (1)
d(f,F)<ε;(2)
F has a transversal homoclinic orbit in D;(3)
F is chaotic in the sense of both Li-Yorke and Devaney;(4)
the topological entropyh(F)>0.Proof.
LetF be defined as in Lemma 3.2. Then (1) and (2) hold by Lemma 3.2.
LetF, r, λ, μ, and U be specified in the proof of Lemma 3.2. Without loss of generality, suppose that the fixed point z of F is the origin.
SetU=B(0,r), A=λI,B=μI, f1|U̅=0,f2|U̅=0. So A and B satisfy assumption (i) in Lemma 2.7, where λ0=max{|λ|,|μ|-1}<1. Take θ=(1-λ0)/2. Then ∥Df1(x,y)∥=∥Df2(x,y)∥=0<θ<1-λ0 for (x,y)∈B̅(0,r). Further,
(3.18)Wloc⁡u(0,r,F)={(x,y):x=0,‖y‖Y<r},Wloc⁡s(0,r,F)={(x,y):y=0,‖x‖X<r}.
Hence, F satisfies assumptions (ii) and (iii) in Lemma 2.7 with γ=r.
By the discussions in Step3 in the proof of Lemma 3.2, F satisfies assumption (iv) in Lemma 2.7, where O(p0), O-1, O0, and O1 are the same as those in the proof of Lemma 3.2 and N=1. So, all the assumptions in Lemma 2.7 are satisfied. Consequently, (3) and (4) hold by Lemma 2.7. The proof is complete.When it is not required that a map transforms its domainD into itself, the convexity of domain D̅ can be removed and all the corresponding results to Lemmas 3.1 and 3.2 and Theorem 3.3 still hold. In detail, let S be a bounded open set in Z and
(3.19)C0(S̅,Z)∶={f:S̅⟶Ziscontinuousandbounded,andhasafixedpointinS̅}.
Then (C0(S̅,Z),d) is a metric space, where d is defined the same as that in (3.3). The results of Lemma 3.2 and Theorem 3.3 hold, where C0(D̅,D̅) is replaced by C0(S̅,Z). Their proofs are similar.Now, we only present the detailed result corresponding to Theorem3.3.Theorem 3.4.
LetX and Y be Banach spaces, Z=X×Y, and S be a bounded open set in Z. Then, for every map f∈C0(S̅,Z) and for any ε>0, there exists a map F∈C0(S̅,Z) satisfying the following(1)
d(f,F)<ε;(2)
F has a transversal homoclinic orbit in S;(3)
F is chaotic in the sense of both Li-Yorke and Devaney;(4)
the topological entropyh(F)>0.Remark 3.5.
A general Banach spaceZ may not be discomposed into a product of two Banach spaces with dimension greater than or equal to 1. However, it is true for Z=Rn with n≥2. So Theorem 3.3 holds for each n-dimensional space Rn with n≥2. In addition, if D is a bounded and convex set in Rn, every continuous map f:D̅→D̅ has a fixed point in D̅ by the Schauder fixed point theorem. In this case one has that
(3.20)C0(D̅,D̅)=C(D̅,D̅)={f:D̅⟶D̅iscontinuous}.Remark 3.6.
(1) As we all know, underC1 perturbation, the hyperbolicity of a map is preserved. But it is obvious that the conclusion does not hold in the C0 sense.
(2) In theC0 topology, Theorems 3.3 and 3.4 show the density of distributions of maps with transversal homoclinic orbits, and consequently in the sense of both Li-Yorke and Devaney. However, it is not true in the C1 topology. For example, consider the map
(3.21)f(x,y)=(x2,y2),(x,y)∈I2=[0,1]×[0,1].
Clearly, f∈C0(I2,I2) and z=0 is a globally asymptotically stable fixed point of f in I2. By Theorem 3.3, for each ε>0, there exists a map F∈C0(I2,I2) with ∥F-f∥<ε such that F is chaotic in the sense of both Li-Yorke and Devaney. But, in the C1 topology, for each positive constant ε<1/2 and for every map F∈C1(I2,I2) with
(3.22)‖F-f‖C1∶=max{‖F-f‖,‖F′-f′‖}<ε,F is globally asymptotically stable in I2, and so is not chaotic in any sense.



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