Nonlinearity EST. 1865 PAPER • OPEN ACCESS On backward attractors of interval maps" To cite this article: Jana Hantakova and Samuel Roth 2021 Nonlinearity 34 7415 View the article online for updates and enhancements. Y o u m a y a l s o like - Dynamics of skew products of interval maps L. S. Efremova - A non-autonomous scalar one-dimensional dissipative parabolic problem: the description of the dynamics Rita de Cassia D S Broche, Alexandre N Carvalho and Jose Valero - Pressure. Poincare series and box dimension of the boundary Godofredo lommi and Anibal Velozo This content was downloaded from IPaddress 178.72.247.114 on 30/11/2021 at14:59 OPEN ACCESS I O P Publishing I London Mathematical Society Nonlinearity Nonlinearity 34 (2021) 7415-7445 https://doi.org/10.1088/1361-6544/ac23b6 On backward attractors of interval maps* J a n a H a n t a k o v a a n d S a m u e l R o t h 1 1 Mathematical Institute of the Silesian University in Opava, Na Rybnicku 1, 74601, Opava, Czech Republic 2 Faculty of Applied Mathematics, A G H University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland E-mail: j ana.hantakova@ math, slu.cz Received 19 January 2021, revised 15 July 2021 Accepted for publication 3 September 2021 Published 20 September 2021 CrossMark Abstract Special a-limit sets (sa-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sa-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting centre that completely characterizes which interval maps have all sa-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic w-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sa-limit sets to the dynamics within them. For example, we show that the isolated points in a sa-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the sa-limit set is nowhere dense. Moreover, we show that sa-limit sets in the interval are always both Fa and G$. Finally, since sa-limit sets need not be closed, we propose a new notion of /3-limit sets to serve as backward attractors. The /3-limit set of x is the smallest closed set to *The research was supported by RVO, Czech Republic funding for IC47813059. The first author was supported by the Marie Sktodowska-Curie Grant Agreement No. 883748 from the European Union's Horizon 2020 research and innovation programme. "Author to whom any correspondence should be addressed. Recommend by Dr Lorenzo J Diaz. |/££v ^ I Original content from this work may be used under the terms of the Creative Commons K r ^ K H Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1361-6544/21/117415+31$33.00 ©2021 IOP Publishing Ltd & London Mathematical Society Printed in the UK 7415 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth which all backward orbit branches of x converge, and it coincides with the closure of the sa-limit set. A t the end of the paper we suggest several new problems about backward attractors. Keywords: interval map, transitivity, a-limit set, special a-limit set, /3-limit set, backward attractor Mathematics Subject Classification numbers: primary: 37E05, 37B20 secondary: 26A18. 1. I n t r o d u c t i o n Let a discrete dynamical system be defined as an ordered pair (X, f) where X is a compact metric space and / is a continuous map acting on X. To understand the dynamical properties of such a system it is necessary to analyse the behaviour of the trajectories of any point i £ l under the iteration of / . Limit sets of trajectories are a helpful tool for this purpose since they can be used to understand the long term behaviour of the dynamical system. The co-limit sets (ui(x) for short), i.e. the sets of limit points of forward trajectories, were deeply studied by many authors. For instance, one can ask for a criterion which determines whether a given closed invariant subset of X is an w-limit set of some point x G l The question is very hard in general, however the answer for w-limit sets of a continuous map acting on the compact interval was provided by Blokh et al in [5]. A closely related question is that of characterizing all those dynamical systems which may occur as restrictions of some system to one of its w-limit sets. These abstract w-limit sets were studied by Bowen [6] and Dowker and Frielander [12]. It was also proved that each w-limit set of a continuous map of the interval is contained in a maximal one by Sharkovsky [24]. Backward limit sets were introduced as a dual concept to w-limit sets in order to capture the 'source' of the trajectory of a point. For invertible systems they are defined quite simply: one just reverses the direction of time and considers the w-limit sets of the inverse system. These so-called a-limit sets are very important in the study of flows, where they are used to define unstable manifolds, homoclinic and heteroclinic trajectories, and the Morse decompositions at the heart of Conley index theory [7, 14]. But when we study continuous mappings / : X —> X (not necessarily invertible), a point x may have many preimages (or none at all), and we must clarify what kind of backward limit set we wish to speak of. Several definitions have been proposed including conical limit sets, a-limit sets, branch a-limit sets, special a-limit sets, and others [1,7, 11, 15, 16, 20]. One of the most classic applications of forward limit sets in dynamics is due to Birkhoff. There are many notions of recurrence in topological dynamics (such as periodicity, nonwandering behaviour, chain-recurrence, etc), but the term recurrent point has been reserved for those points x which belong to their own w-limit sets. Birkhoff showed that these points can be used to identify the Birkhoff centre (Birkhoff called it the 'set of central motions') of a topological dynamical system (X, / ) , which is obtained by restricting / to its non-wandering set, then restricting that system to its non-wandering set, and so on through transfinite induction (taking intersections at limit ordinals) until reaching some countable ordinal (the 'depth') at which the sequence stabilizes. Birkhoff's result is that the centre of the system obtained in this way is the same as the closure of the set of recurrent points [2]. In light of Birkhoff's work, one can ask the analogous question, what is the significance of a point belonging to its own backward limit setl If we consider homeomorphisms than Birkhoff's results already apply (just using the inverse map), so i f we wish to get something new we must consider general continuous mappings / : X —> X. In one-dimensional dynamics 7416 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth some very good answers to this question have appeared for two kinds of backward limit sets, namely, the a-limit set and the special a-limit set. Coven and Nitecki showed that a point x is non-wandering for a continuous interval map / : [0,1] —> [0,1] if and only if x belongs to its own a-limit set [9]. But there is a deeper result related to the attracting centre of an interval map / : [0,1] —> [0,1], defined by Xiong as the set of all points x such that x is in the w-limit set of some point x\, which itself is in the w-limit set of some point X2, and so on for some infinite sequence { x i } ^ of points in the interval [29]. Xiong showed that the attracting centre is a subset of the Birkhoff centre (they can coincide) and that if x\, X2 can be found as above, then x is already in the attracting centre (so the 'depth' here is at most 2) [29]. The connection to backward limit sets was made in 1992 by Hero, who defined special a-limit sets (sa-limit sets, for short) and showed that a point x belongs to the attracting centre of a continuous interval map if and only if it belongs to its own sa-limit set [15]. Partial generalizations of this result to graph maps and some dendrite maps have appeared since then [26-28]. We now recall Hero's definition of a special a-limit set. A backward orbit branch of a point x is any sequence {x,}^0 such that xn = x and /(x;+i) = x; for all i. The corresponding branch a-limit set is defined as the set of all limits of convergent subsequences JC,- . (analogously as w-limit sets are defined from forward trajectories). Then the special a-limit set of a point x, denoted sa(x), is defined as the union of all branch a-limit sets over all backward orbit branches o f x [15]. Several studies in recent years have focused on branch a-limit sets, and in light of the definition it is easy to deduce corresponding properties for sa-limit sets, for example: • Branch a-limit sets are always closed and strictly invariant, and therefore each sa-limit set is strictly invariant (i.e. f(sa(x)) = sa(x)) and contains the orbit closure of each of its points [18]. • Branch a-limit sets are always internally chain transitive [16], and therefore sa-limit sets are internally chain recurrent (but not necessarily chain transitive, see example 9 below). • A l l the recurrent points of a system are contained in the union of its sa-limit sets, since this property holds for the union of the branch a-limit sets [13]. • For interval maps / : [0,1] —> [0,1], each branch a-limit set is locally expanding and hence coincides with an w-limit set of the same map / [1]. It follows that each sa-limit set of an interval map / is a union of some of its w-limit sets. Notably lacking in the list above are purely topological properties. For example, it seems that Hero did not consider the basic question whether all sa-limit sets are closed. Outside the realm of one-dimensional dynamics the situation is even more complicated. It has been shown that sa-limit sets are always analytic, but not necessarily closed or even Borel [17, 18]. Therefore it seems prudent to study more closely the properties of sa-limit sets in onedimensional dynamics, and especially in the most important one-dimensional space where Hero's work began, the unit interval. Kolyada, Misiurewicz, and Snoha began this study as a systematic programme in [18]. They investigated special a-limit sets of interval maps and proved that for interval maps with a closed set of periodic points, every special a-limit set has to be closed. This result led to the following conjecture: C o n j e c t u r e 1 [18]. For all continuous maps of the unit interval all special a-limit sets are closed. We disprove the conjecture in theorem 45 by showing a counterexample of a continuous interval map with a special a-limit set which is not closed and give the properties of continuous interval maps that determine if all special a-limit sets are closed in theorem 41. In corollaries 7417 Nonlinearity 34 (2021) 7415 J Hantáková and S Roth 42-44 we identify three classes of continuous interval maps for which all sa-limit sets are closed, namely, piecewise monotone maps, zero entropy maps with a closed set of recurrent points and maps which are not L i - Y o r k e chaotic. On the other hand, we show that for all continuous maps of the unit interval all special a-limit sets are both Fa and G$ in theorem 40. We give further topological properties of special a-limit sets of interval maps. If sa(x) is not closed, then it is uncountable and nowhere dense by theorem 39. If sa(x) is closed, then it is the union of a nowhere dense set and finitely many (perhaps zero) closed intervals by theorem 24, and in section 4.2 we prove some amount of transitivity of / on those intervals. Since salimit sets need not be closed, we propose a new notion of /3-limit sets to serve as backward attractors in definition 49. The /3-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the sa-limit set. Kolyada et al also made the following conjecture. C o n j e c t u r e 2 [18]. The isolated points in a special a-limit set for a continuous interval map are always periodic. We verify this conjecture in theorem 21. We also show that a countable special a-limit set for an interval map is a union of periodic orbits. These results are opposite to the case of w-limit sets. The w-limit sets of a general dynamical system do not posses any periodic isolated points unless UJ(X) is a single periodic orbit [23]. The authors of [18] also investigated the properties of special a-limit sets of transitive interval maps and stated the following conjecture: C o n j e c t u r e 3 [18]. Let / : [0,1] -> [0,1] be a continuous map and x,y G [0,1]. • If x ^ y and sa(x) = sa(y) = [0,1], then / is transitive. • If sa(x) = [0,1] then either / is transitive or there is c G (0,1) such that f\[o,c] and /|[C ,i] are transitive. We show in theorem 26 that / is transitive if there are three distinct points x,y, z [0,1] and x,y G [0,1]. If Int(sa(x) n sa(y)) ^ 0 then sa(x) = sa(y). We correct this conjecture by showing that at most three distinct special a-limit sets of / can contain a given nonempty open set in corollary 29. One additional motivation for studying sa-limit sets is that they provide more information about a-limit sets (see section 2 for the definition), since there is always the containment sa(x) C a(x). For transitive interval maps this containment is in fact an equality sa(x) = a(x) for all x G [0,1] (this can be deduced from [18, proposition 3.10] or theorem 33). The question then arises whether this is the typical situation, or perhaps typically sa(x) = a(x) at least for 'most' points x. We show in section 4.5 that for the generic interval map / : [0,1] —> [0,1] (in the topology of uniform convergence) there is a whole interval of points x G [0,1] for which a(x, f) ^ sa(x, /). To summarize, the key properties of limit sets as they apply to continuous maps of the interval are as follows: 27. 7418 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth x is nonwandering x £ a(x), x is in the attracting centre X a continuous map. A sequence {xn}%L0 is called • The forward orbit of a point x i f f(x) = xn for all n > 0, • A preimage sequence of a point x i f fn {x„) = x for all n ^ 0, • A backward orbit branch of a point x if XQ = x and / ( x „ + i ) = x„ for all n ^ 0. A point y belongs to the co-limit set of a point x, denoted by ui(x), i f and only i f the forward orbit of x has a subsequence { x « , } £ n such that xni —> y. A point y belongs to the a-limit set of a point x, denoted by a(x), if and only i f some preimage sequence of x has a subsequence {xni such that xni —> y. A n d a point y belongs to the special a-limit set of a point x, also written as the sa-limit set and denoted sa(x), i f and only i f some backward orbit branch of x has a subsequence {x„t } ^ 0 such that xni —> y. If we wish to emphasize the map, we will write LO(X, / ) , a(x, f) and sa(x, /). To summarize, the to, a, and sa-limit sets of a point x are defined as follows. The set to(x) is the set of all accumulation points of its forward orbit and a(x) (resp. sa(x)) is the set of all accumulation points of all its preimage sequences (resp. of all its backward orbit branches). Let T: X —> X and F: Y —> Y be continuous maps of compact metric spaces. If there is a surjective map : X —> Y such that o T = F o then it is said that semiconjugates T to F and 4> is a semiconjugacy. Let / : X —> X be a continuous map. A set A C [0,1] is invariant if f(A) C A. The forward orbit of a point, regarded as a subset of X rather than a sequence, will be denoted by Orb(x) = {/"(x): n ^ 0}. The forward orbit of a set is Orb(A) = \J{f"(A): n > 0}. We call / transitive i f for any two nonempty open subsets U,V C X there is n ^ 0 such that f"(U) n V ^ 0. We call / topological^ mixing i f for any two nonempty open subsets U,V C X there is an integer N^0 such that fn (U) H V ^ 0 for all n > N. Now let / : [0,1] —7- [0,1] be an interval map. We say that a point x is periodic if f"(x) = x for some n ^ 1, x is recurrent if x 1. We write A ' ( / ) = {Jxe[0 {] ui(x) for the union of all w-limit sets of / and S A ( / ) = l j x g [ 0 ^ sa(x) for the union of all sa-limit sets. Following [29] we define the attracting centre of / as A (/) = UxeA1 */)6 "-^*)- The Birkhoff centre of / is the closure of the set of recurrent points R e c ( / ) and coincides with Per(/) [8]. 7419 Nonlinearity 34 (2021) 7415 J Hantáková and S Roth If the map / is clear from the context, we may drop it from the notation. The relation of these sets is given by the following summary theorem from the works of Hero and Xiong [15, 29]. T h e o r e m 5 [ 15, 29]. For any continuous interval mapf: [0,1] —> [0,1], we have Per c Rec c A 2 = S A c Ř e č c A 1 c fi. If K c [0,1] is a non-degenerate closed interval such that the sets K,f(K),... ,fk ~l (K) are pairwise disjoint and fk (K) = K, then we call the set M = Orb(K) a cycle of intervals and the period of this cycle is k. We may also call K an n-periodic interval. If f\u is transitive then we call M a transitive cycle f o r / 3. Maximal w-limit s e t s a n d their relation t o special a-limit sets A n important property of the w-limit sets of an interval map/is that each w-limit set is contained in a maximal one. These maximal w-limit sets come in three types: periodic orbits, basic sets, and solenoidal w-limit sets. A solenoidal co-lim it set is a maximal w-limit set which contains no periodic points. Any solenoidal w-limit set is uncountable and is contained in a nested sequence Orb(/o) D Orb(/i) D . . . of cycles of intervals with periods tending to infinity, also known as a generating sequence [4, assertion 4.2]. Here is the theorem relating sa-limit sets to solenoidal w-limit sets; the proof is given in section 3.1. T h e o r e m 6 (Solenoidal sets). Let Orb(/0 ) D Ovb(h) D ... be a nested sequence of cycles of intervals for the continuous interval m ap f with periods tending to infinity. Let Q = p| Orb(/„) andS=Qf] Rec(/). (a) Ifa(y) n Q ^ 0, then y G Q. (b) Ify G Q, then sa(y) D S and sa(y) DQ = S. A basic set is an w-limit set which is inf inite, maximal among w-limit sets, and contains some periodic point. If B is a basic set then with respect to inclusion there is a minimal cycle of intervals M which contains it, and B may be characterized as the set of those points i e M such that Orb(ř/) = M for every relative neighbourhood U of x in M, see [4]. Conversely, i f M is a cycle of intervals for / , then we will write B(M,f) = {x G M : for any relative neighbourhood U of x in M w e h a v e O r b ( ř / ) = M}, and if this set is infinite, then it is a basic set [4]. Here is the theorem relating sa-limit sets to basic sets; the proof is given in section 3.2. T h e o r e m 7 ( B a s i c sets). Let fbe a continuous int erval map and fix y G [0,1]. (a) Ifa(y) cont ains an infinit e subset of a basic set B = B(M, /), t hen y G M and sa(y) D B. (b) If sa(y) cont ains a preperiodic point x, t hen there is a basic set B = B(M,f) such t hat x G B C sa(y). The sharpness of the second claim of theorem 7 is illustrated in figure 1. The first map has two basic sets B( [0,1 ], / ) and B(M, / ) , where M is the invariant middle interval. It is easy to see that set B([0,1],/) is a Cantor set and it is contained in sa(l). But sa(l) does not contain the basic set B(M, f) althoug h it includes the left endpoint of M, which is preperiodic. The second map shows that we cannot weaken the assumption to a(y). The a-limit set of 1 includes the preperiodic endpoint of M but sa(l) does not contain any basic set. 7420 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth Figure 1. A map where the sa-limit set of 1 (respectively, the a-limit set of 1) contains a preperiodic point from a basic set B(M, / ) , but sa(l) ^>B(M, /). Figure 2. A map with an increasing nested sequence of sa-limit sets not contained in any maximal one. Periodic orbits may also be related to sa-limit sets. The following result is one of the main theorems in [ 18]. Moreover, it holds for all periodic orbits of an interval map, even those which are not maximal w-limit sets. T h e o r e m 8 [18, theorem 3.2]. Let P be a periodic orbit for the continuous interval map f. Ifaiy) n P ^ 0, then sa(y) D P. One additional observation is appropriate in this section. Unlike w-limit sets, the sa-limit sets of an interval map need not be contained in maximal ones. Example 9. Fix two sequences of real numbers 1 = a\ > b\ > a-i > b-i > ... both decreasing to 0 and consider the 'connect-the-dots' map / : [0,1] —> [0,1] where /(0) = 0, f(fi,) = a,, f{bi) = a i + 2 , ( i = l , 2 , . . . ) and / is linear on all the intervals b{\, [bi, a,-]. The graph of such a function / is shown in figure 2. The sa-limit sets of this map are sa(x) = {a[,... ,a„} for x 0} where is an element of the group of residues modulo m ; , for every j. Denote by r the translation in H(D) by the element (1,1,...). T h e o r e m 10 [4, theorem 3.1]. Let {/;}^L0 be generating sequence with periods D = {m^JL^for a solenoidal set Q = n^nOrb/,-. Then there exists a semiconjugacy 4>: 2 —> H(D) between /\Q and r with the following properties: (a) There exists a unique set SREC = Q H Rec(/) such that u>(x) = SREC for every x is a semiconjugacy we have f'([x, z]) c (j>~ (r'(r)) for all i ^ 0. But the intervals (f>~ ( r ' ( 0 ) a r e pairwise disjoint. This shows that [x, z] is a wandering interval. Claim, z G Int(Orb(/,)), for every j > 0. We will assume otherwise. Let K be the connected component of Orb(7iv), for some N ^ 0, where z is an endpoint of K. Let v be a point such that z G UJ(V). B y theorem 10 property (a), S = ui(z) C ui(v), we have n Orb(7]v) infinite and necessarily Orb(v) n Int(Orb(//v)) ^ 0. This implies f (v) G OrbC/v) for all sufficiently large fe. It follows that Orb(t>) accumulates on z from the interior of K and we can find k > 0 such that f (v) G (x, z). But [x, z] is a wandering interval, so Orb(t>) cannot accumulate on z which contradicts z G co(v). Let {y„}^Ln be a backward orbit branch of y with a subsequence { y « , } ^ 0 such that liirii-too y„. = z. Since z G Int(Orb/7 ) it follows from lemma 11 thaty„ G Orb(/7 ) for all j, n > 0. Therefore {y«}^L0 C Q. For every n ^ 1, denote 0(y„) = r„. Then by theorem 10 property (b) (f>~ (r„) are connected, pairwise disjoint sets, each containing an element s„ G S. Since y„ G 4>~l (r„), we have lim^oo s„t = z. But 5 is a closed set and z ^ S, which is impossible. Therefore sa(y) n (Q\5) = 0 and sa(y) n g C S. • Corollary 12. A sa-limit set contains at most one solenoidal set. Proof. Let Orb(7n) D Orb(/i) D ... and Orb(/( ') ) D Ovb(I[) D ... be nested sequences of cycles of intervals generating two solenoidal sets Q = f] Orb(/„) and Q' = p|„Orb(/^). If sa(y) n Q 7^ 0 and sa(y) ("I 2 ' 7^ 0 then, by theorem 6, y e g n g ' . Since two solenoidal sets g and g ' are either identical or disjoint we have Q = Q'. Then the only solenoidal set contained in sa(y) is S = Q n Rec(/). • 3.2. Basic sets This section is devoted to the proof of theorem 7. Let / be a continuous map acting on an interval /. We say that an endpoint y of / is accessible if there is x G Int(7) and n G N such that /"(x) = y. If y is not accessible, then it is called non-accessible. The following proposition is derived from [21, proposition 2.8]. P r o p o s i t i o n 13 [21]. Letf be a topologically mixing continuous map acting on an interval I. Let x G / and e > 0 be such that [x — e, x + e] C / and any endpoints of I in [x — e, x + e] are accessible. Then for every non degenerate interval U C /, there exists an integer N such thatf"(U) D [x - e, x + e],for all n > N. A n m-periodic transitive map of a cycle of intervals is a transitive map g : M —> M, where M C Mis afinite union of pairwise disjoint compact intervals I,g(J),... ,gm ~l (f), andgm (/) = /. 7423 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth We write End(M) for the endpoints of the connected components of M and refer to these points simply as endpoints of M. The set of exceptional points of g is defined oc E:=M\f)\Jg"(U), U n=\ where U ranges over all relatively open nonempty subsets of M. It is known that E is finite; it can contain some endpoints and at most one non-endpoint from each component of M. If gm |/ is topological!/ mixing, then E = \JfS0Ei, where E{ is the set of non-accessible endpoints of gm \gi(i) by proposition 13. If gm \i is transitive but not mixing, then by [21, proposition 2.16] there is an ra-periodic orbit en, c\,..., c m _ i of points such that c; i]a r e topologicals/ mixing interval maps, where gl (T) = [a; , b{\, for i = 0 , . . . , m — 1. Then E = \JfS0Ei, where E{ is the union of the sets of non-accessible endpoints of g2m \[ai c,] and g 2 m | B y [21, lemma 2.32], every point in E is periodic and therefore g(E) = E. B y the definition of non-accessible points, g~1 (E) ("IM = E. From [18, proposition 3.10] it follows that sa(x) D M for all x [0,1], then we set B(M,f) = {x M' and a monotone map M' such that (B) = M'. Moreover, for anyx G M1 , 1 < # { f ' ( i ) f l B } ^ 2 and lnt((j)~l (x)) (IB = 0, and so : (M,f) —> {M1 , g) the semiconjugacy to the m-periodic transitive map g given by theorem 14. Let E be the set of exceptional points of the map g acting on M'. Suppose y G M and (p(y) £E\J End(M'). Then sa(y) D B. Proof. Let x G B. There is e > 0 such that \(XyX+t) is not constant and 4>((x,x + e)) n End(M') = 0 or |(X_£jX) is not constant and \N is constant which is in a contradiction with x e B b y theorem 14. We can assume \(XyX+t) is not constant and ((x, x + e)) n End(M') = 0, and denote V = (x, x + e). Then U = (y) £ E U End(M'), there is <5 > 0 such that ((y) - S, (y) + 8) c M' and ((y) - 6, (y) + 5)C\E = $. The set E equals the union of non-accessible endpoints of a topological!/ mixing map gm (resp. g2m ) acting on the components of M', therefore we can apply the proposition 13 to the map gm (resp. g2m ) acting on the component / C M ' such that ((y) — S, (y) + S) C I. There is an N > 0 such that ^(10 D (4>(y) - S, (j>(y) + S). But (j>(fN (V)) = gN (U), so (j>(y) is in (f>(fN (V)). Since 7424 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth 6 is monotone this means either y G f{V) or 4>{y) is an endpoint of the interval {f (V)). But we have seen that it is not an endpoint. Therefore y G fN (V) and we can find yl G V and N[ = N such that fNl (yi) = y. Notice that 4>{yx) <£ E since gNl ((f>(yi)) = 4>(y) <£ E and g~Nl (E) DM' = E; and ^ End(M') since y t G V and 0(V) n End(M') = 0. B y the same procedure, we can find y 2 G (x, x + e/2) n M and N2 G N such that fN2 (yi) = yi • B y repeating this process we construct a sequence { j n } ^ ! converging to x which is a subsequence of a backward orbit branch of y. Since x 6 f i was arbitrary, this shows fi C sa(y). • C o r o l l a r y 16. For every basic set B, there is y G B SMC/Z ?/za? sa(y) D B. Proof. Let M, (f>, g, M', E be as in the previous proof. Since the map 4>\B is at most 2-to-l and £ is a finite set, there are uncountably many points y G B such that (y) £ E U End(M'). The result follows by lemma 15. • Before we proceed to the proof of theorem 7 we need to recall the definition of a prolongation set and its relation to basic sets. Let M be a cycle of intervals. Let the side T be either the left side T = L or the right side T = R of a point i £ M and WT(X) be a one-sided neighbourhood of x from the T-hand side, i.e. Wj{x) contains for some e > 0 the interval (x, x + e) (resp. (x — e, x)) when T = R (resp. T = L). We do not consider the side T = R (resp. T = L) when x is a right endpoint (resp. left endpoint) of a component of M. Now let PT M(X)= f| f){Jf(WT(x)nM), where the intersection is taken over the family of all one-sided neighbourhoods Wj(x) of x. We will write PT (x) instead of Pf0 ^(x). The following auxiliary lemmas 17-19 are taken from [4]. L e m m a 17 [4, lemma2.2]. Letx G [0,1]. Then PT (x) is a closed invariant set and only one of the following possibilities holds: • There exists a wandering interval WT(X) with pairwise disjoint forward images and PT (x) = co(x). • There exists a periodic point p such that PT (x) = Oib(p). • There exists a solenoidal set Q such that PT (x) = Q. • There exists a cycle of intervals M such that PT (x) = M. There is a close relation between prolongation sets and basic sets. If M is a cycle of intervals for / then we define E(M, f) = {x G M: there is a side T of x such that P ^ x ) = M } , and, for x G E(M,f), we call this side T a source side of x. B y [4, theorem 4.1] if there exists the basic set B = B(M, f) then E(M, f) = B. In particular, if E(M, / ) is infinite then E(M, f) = B(M, f) (see the discussion on page 48 in [4]). L e m m a 18 [4, lemma 4.5]. Let M be a cycle of intervals. If E(M,f) is a finite set then E(M,f) = Orb(p) where p is a periodic point. IfE(M,f) is infinite then E(M,f) = B(M,f). The next property of basic sets follows from step B 7 on page 47 in [4]: L e m m a 19 [4]. Let B(M,f) be a basic set. Then for any x G B(M,f) with a source side T and any one-sided neighbourhood from the T-hand side WV(x), we have WT(X) n B(M,f) ^ 0. 7425 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth Proof of t h e o r e m 7. (a) Let and g be the maps given in theorem 14 and E be the set of exceptional points of the map g acting on M'. Since \B is an at most 2-to-l map, (z) £ £ U E n d ( M ' ) and therefore z £ 4>~l (E U End(M')). The set 4>~l (EU End(M')) is a union of finitely many, possibly degenerate, closed intervals in M. Since z € (a(y) n B)\~l (E U End(M')), there is a pre-image / e M of y, y = fk (y'), for some k > 0, and simultaneously )/ <£ ~l (EU End(M')), which implies ^ £ U End(M'). Then y S M by the invariance of M . B y lemma 15 applied to y', sa(y') D B. But the containment sa(y) D sa(y') is clear from the definition of sa-limit sets, and so sa(y) D B. (b) Let {y,}^o be the backward orbit branch of y accumulating on x. Since x is not a periodic point, it is not contained in { y ; } ^ 0 more then one time and we can assume that { y , } ^ 0 accumulates on x from one side T. Consider the prolongation set PT (x). Clearly { y , } ^ 0 c PT (x). Since PT (x) is closed and invariant, x and Orb(x) belong to PT (x), we see that PT (x) contains both periodic and non-periodic points. B y lemma 17, there is only one possibility PT (x) = M, where M is a cycle of intervals. The other possibilities are ruled out—Orb(/?), where p is a periodic point and ui(x), where x is a preperiodic point, can not contain a non-periodic point; and a solenoidal set Q can not contain a periodic point. Since PT (x) = M contains { y ; } £ 0 m u s t contain a one-sided neighbourhood of x from the T-hand side and therefore PT M{x) = PT (x) = M. Let E(M, /) = {zeM : there is a side S of z such that Pff(z) = Af}. Since x e E(M,f) and x is not periodic, by lemma 18, E(M, f) = B(M, f) and T is a source side of x in B(M, / ) . Let and g be the maps given in theorem 14 and E be the set of exceptional points of the map g acting on M'. B y lemma 19, B(M,f) accumulates on x from the T-hand side and thus the map is not constant on some one-sided neighbourhood of x from the T-hand side Wj(x) (otherwise we can find a point y e B(M, f) n WT(X) such that y e Int(^_ 1 (^(x))) which is in the contradiction with the properties of from theorem 14). Since { y ; } £ 0 accumulates on x from the T-hand side and is not constant on WT(X), we can find j > 0 such that y ; € Wr(x) and <^(yp ^ E U End(M'). Then sa(y;-) D -S(M, / ) by lemma 15, and sa(y) D scuO^) since y^ is a preimage of y. We conclude that sa(y) D B(M, /). • We record here one corollary which we will need several times in the rest of the paper. Corollary 20. If sa(x) contains infinitely many points from a transitive cycle M, then x € M and sa(x) D M. Proof. In this case M is itself a basic set, so we may apply theorem 7. • 4. General p r o p e r t i e s of special a - l i m i t s e t s f o r interval m a p s 4.1. Isolated points are periodic Unless an w-limit set is a single periodic orbit, its isolated points are never periodic [23]. The opposite phenomenon holds for the sa-limit sets of an interval map. T h e o r e m 21. Isolated points in a sa-limit set for a continuous interval map are periodic. Proof. Let z € sa(y) such that z is neither periodic nor preperiodic. Then z is a point of an infinite maximal w-limit set, i.e. a basic set or a solenoidal set. This follows from Blokh's decomposition theorem, that A 1 (/) is the union of periodic orbits, solenoidal sets and basic sets, 7426 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth and from theorem 5, z [0,1]. Let M be the union of the non-degenerate components of sa(x). If M = 0 then sa(x) is nowhere dense. Otherwise M must be a finite or countable union of closed intervals, and since M contains the interior of the closure of the sa-limit set we know that s a ( x ) \ M is nowhere dense. Let K be any component of M. B y theorem 5 we have K c Per(/), and therefore periodic points are dense in K. Let n ^ 1 be minimal such that f"(K) n K =t 0. Since f"(K) is connected and K is a component of the invariant set sa(x), we know that f"(K) C K. Since periodic points are dense in K we must have f"(K) = K. Therefore Orb(K) is a cycle of intervals, and by lemma 11 we get x K is an interval map with a dense set of periodic points. There is a structure theorem 7427 Nonlinearity 34 (2021) 7415 J Hantäkovä and S Roth for interval maps with a dense set of periodic points [21, theorem 3.9]3 which tells us that K is a union of the transitive cycles and periodic orbits of g, B y lemma 23, sa(x, g) contains a dense subset of K. B y corollary 20 each transitive cycle L C K for g must contain x. Since transitive cycles have pairwise disjoint interiors, g has at most two transitive cycles. If their union is not K, then K must contain a non-degenerate interval of periodic points of g. But by an easy application of lemma 11, no sa-limit set can contain a dense subset of an interval of periodic points. Therefore K is the union of one or two transitive cycles for g. B y lemma 23, M is the union of one or two transitive cycles for / . Finally, if L is one of the (at most two) transitive cycles for / that compose M, then by corollary 20 we have sa(x) D L. Therefore M C sa(x). • Remark 25. If sa(x) contains a cycle of intervals M, then sa(x) is in fact a closed set, but we are not yet ready to prove this fact. See theorem 39 below. 4.3. Transitivity and points with sa(x) = [0,1] Let / : [0,1] —7- [0,1] be an interval map. We say that the point x has a full sa-limit set if sa(x) = [0,1]. Kolyada, Misiurewicz, and Snoha proved that when / is transitive, all points x [0,1] is transitive if sa(xi) = [0,1] for at least three distinct points x\, Xi, X3. Proof. Suppose that / is not transitive. We will prove that at most two points have a full sa-limit set. Suppose there is at least one point x with sa(x) = [0,1]. B y theorem 24 there are two transitive cycles L, L' for / such that [0,1] = L U L', and by corollary 20 every point with a full sa-limit set belongs to L n L'. We will show that the cardinality of L n L' is at most two. Let A\, A2,..., An be the components of L, numbered from left to right in [0,1]. Let a : { 1 , . . . , n} —> {1,..., n} be the cyclic permutation defined by /(A,-) = Aa&. If n ^ 3 then there must exist i such that \a(i) — a(i + 1)| > 2, so there is some Aj strictly between Aa&, Aa^+l). Let B be the component of L' between A, and A i + l , see figure 3. B y the intermediate value theorem, f(B) D Aj. This contradicts the invariance of L'. Therefore L has at most two components. For the same reason L' has at most two components. Moreover, L, L' cannot both have two components; otherwise the middle two of those four components have a point in common, but their images do not, again see figure 3. There are two cases remaining. L, L' can have one component each, and then t n L' has cardinality one. Otherwise, one of the cycles, say L, has two components, while L' has only one, and then t n L' has cardinality two. • In the course of the proof we have also shown the following result: Corollary 27. If a continuous map f: [0,1] —> [0,1] has one or two points with full sa-limit sets, but not more, then [0,1] is the union of two transitive cycles of intervals. 3 In fact, [21, theorem 3.9] tells us that all the transitive cycles for g have period at most 2, and the periodic orbits not contained in transitive cycles also have period at most 2. Some of this extra information can be shown quite easily; it comes up again in our proof of theorem 26. K = {L: L is a transitive cycle for g} J U Per(g) 7428 Nonlinearity 34 (2021) 7415 J Hantäkovä and S Roth B Ai+l common point I M O A4 A a(i+l) Figure 3. Diagrams for the proof of theorem 26. Figure 4. Maps for which sa(x) = [0,1] for only 1 or 2 points x. Corollary 27 corrects the second part of conjecture 3 (originally [18, conjecture 4.14]) to allow for two points with a full sa-limit set.4 Both possibilities from the corollary are shown in figure 4. One of the interval maps shown has exactly one point with a full sa-limit set, and the other has exactly two. 4.4. Special a-limit sets containing a common open set Now we study the sa-limit sets that contain a given transitive cycle of intervals. We get a sharpening of theorem 7 in the case when B = M, i.e. when our basic set is itself a transitive cycle of intervals. Let M c [0,1 ] be a transitive cycle of intervals for/. For the reader's convenience, we recall some definitions from section 3.2. We write End(M) for the endpoints of the connected components of M and refer to these points simply as endpoints of M. The main role in our analysis is played by the set of exceptional points of M oc E:=M\f]\Jf"(U), U n=\ where U ranges over all relatively open nonempty subsets of M. It is known that E is finite; it can contain some endpoints and at most one non-endpoint from each component of M. It is also known that E and M\E are both invariant under/, see [21]. Endpoints of M in E are called non-accessible endpoints, as explained in section 3.2. Recall from equation (1) that E= {xeM:sa(x)^M}. T h e o r e m 28. Let M be a transitive cycle of intervals for f: [0,1] —> [0,1] and let E be its set of exceptional points. 4 Incidentally, when there is exactly one point with a full sa-limit set, the conclusion of the conjecture holds as stated in [18]: there is c £ (0,1) such that such that/j[0-c] and/j[c ij are both transitive. 7429 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth (a) Each point x € M\ (E U End(M)) has the same sa-limit set. (b) At most three distinct sa-limit sets of f contain M. We will see in the course of the proof that if sa(x') D M is distinct from the sa-limit set described in part (a), then x' belongs to a periodic orbit contained in End(M). Since there are at most two such periodic orbits in End(M) we get part (b). Before giving the proof we discuss some consequences of this theorem. Corollary 29. At most three distinct special alpha-limit sets of a continuous map f can contain a given nonempty open set. Proof. Let U be a nonempty open set in [0,1]. If a sa-limit set of / contains U, then by theorem 24, U contains a whole subinterval from some transitive cycle M. Applying corollary 20, we see that any sa-limit set which contains U also contains M. B y theorem 28 there are at most three such sa-limit sets. • This corollary corrects conjecture 4 (originally [18, conjecture 4.10]), in which it was conjectured that two sa-limit sets which contain a common open set must be equal. For comparison, note that if two w-limit sets of an interval map contain a common open set, then they are in fact equal, since an w-limit set with nonempty interior is itself a transitive cycle [24]. We also remark that the number three in corollary 29 cannot be improved, as is shown by the following example. Example 30. Let / : [0,1] -> [0,1] be an interval map for which [0, | ] is a full twohorseshoe, [|, §] is a full three-horseshoe, and [|, 1] is a full two-horseshoe, as shown in figure 5. Then ^ belongs to only one of the three transitive invariant intervals for / and sa(|) = [|, | ] . But both | and | are accessible endpoints of adjacent transitive intervals and so sa(|) = [0, f] and sa(|) = [|, 1]. In what follows it is necessary to allow for a weaker notion of a cycle of intervals for / . A n interval is called non-degenerate if it contains more than one point. If U is a non-degenerate interval (not necessarily closed) such that U,f(U),.. .,fn ~(U) are pairwise disjoint nondegenerate intervals and f"(U) C U (not necessarily equal), then we will call Orb(t/) a weak cycle of intervals of period n. The next lemma records one of the standard ways to produce a weak cycle of intervals. Similar lemmas appear in [4] and several other papers, but since we were unable to find the exact statement we needed, we chose to give our own formulation here. 7430 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth L e m m a 31. If a subinterval U contains three distinct points from some orbit off then Ovb(U) = \J^LQf'(U) is a weak cycle of intervals for f. Proof. Let x, f(x), fm (x) be three distinct points in U, 0 < n < m. Clearly Orb(t/) is invariant. Since the intervals U, f'\U) both contain f'\x) we see that U U f'\U) is connected, i.e. it is an interval. Then also f"(U) U f2 "(U) is connected, and so on inductively. Therefore the set A = \J°IL0fjn (U) is connected. Then Ovb(U) = \J"Zof'(A) has at most n connected components. Let B D A be the component of Orb(L0 containing U, and let k ^ n be minimal such that B n fk (B) ^ 0. Then fk (B) is a connected subset of Orb(t/), so fk (B) C B. For 0 < i < j < k i f / ' ( B ) n f\B) ^ 0, then f+k ~j (B) n 7^ 0, so fi+k ~\B) f l B ^ t ) , contradicting the choice offe.This shows that B, / ( B ) , . . . , / 1 (B) are pairwise disjoint. It remains to show that they are all non-degenerate. Clearly all three points x , / " ( x ) , / m ( x ) are in B . From the disjointness of B, / ( B ) , . . . , fk ~l (B) it follows that n, m are multiples of k. A n d since x, f"(x), fm (x) are distinct we get n = jlk,m = j2k with 0 < jl < j2. If any / ' ( B ) is a singleton, i < fe, then so also is /*(B). Then using B D /^(B) D f2k (B) D . . . , we see that all the sets fjk (B), j ^ 1, are the same singleton. But in that case fhk (B) = {/"(x)}, fhk (B) = {/m (x)} are not the same singleton, which is a contradiction. • L e m m a 32. Let N be a weak cycle of intervals for f and M a transitive cycle of intervals. Let E be the set of exceptional points for M. IfN n M is nonempty, then the following hold: (a) Either N 3 M\E or else N O M is a periodic orbit contained in End(M), and (b) The period ofN is at most twice the period ofM. Proof. Let n, m be the periods of N, M, respectively. Let iV,-, M ; , be the components of N, M with the temporal ordering, so /(iV,) C iV,-+im o d « for all i < n and f(Mj) = Mj+l m o d m for all j < m. Suppose first that N DM is infinite. Then contains a non-degenerate interval U C M. We have D Oib(U) D M\E, where the first containment comes from the invariance of N and the second from the definition of the exceptional set E. Since E contains at most one non-endpoint of each component Mj, it follow that each component Mj meets at most two components iV,-. So in this case n ^ 2m. For the rest of the proof we suppose that N DM is finite. We no longer need transitivity and the sets N, M will play symmetric roles. Clearly each nonempty intersection Ni n Mj is at commonendpoints. This shows thatN DM C End(M). It also shows that each component JV; contains at most two points from M, and conversely each component Mj contains at most two points from N. Claim 1. Each component Ni contains the same number of points of M. Conversely, each component Mj contains the same number of points of N. Suppose first that some Ni contains two distinct points a, b from M. Then a, b belong to distinct components of M, and therefore f(d), f(b) also belong to distinct components of M. But they both belong to iV,-+i m o d»- Continuing in this way we see that each each component of Af contains two points from M. Now suppose instead that each component of contains at most one point from M. Surely some Ni contains at least one point a e M. Then Ni+i m o d „ contains the point f(a) € M. Continuing in this way we see that each component Ni contains exactly 1 point from M. Moreover, the whole argument still works if we reverse the roles of Af and M. This concludes the proof of claim 1. Claim 2. The intersection N C\M is a periodic orbit. Suppose first that each component of contains two points from M. We reuse an argument from the proof of theorem 26. Let A\,A2,... ,Am be the components of M in the spatial order, i.e. numbered from left to right in [0,1]. Let a : { 1 , . . . , m} —> { 1 , . . . , m} be the cyclic permutation defined by /(A,-) C Aa&. 7431 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth If m ^ 3 then there must exist i such that \a(i) — a(i + 1)| ^ 2 , so there is some Aj strictly between AC T (;),AC T (;+ i). Let Nk be the component of N which intersects both A, and A i + l . B y the intermediate value theorem, f(Nk) D Aj. This contradicts the fact t h a t M O N is finite. ThereforeM has only two components, and N has only one, i.e.N = Ni. The two endpoints of Ni belong to A\, A2, respectively, and are therefore interchanged by / . So in this case M flJV is a periodic orbit of period 2. The symmetric situation arises i f each component of M contains two points from N. Then N has two components and M has only 1, and again M ON is a periodic orbit of period 2. Now suppose that each Ni contains exactly one point from M, and each Mj contains exactly one point from N. Then m = n and we may assume the components are indexed such that Ni n Mj is nonempty i f and only if i = j. Let x; be the unique point of intersection of Ni and Mt. Since these intersection points are unique we get /(*,) = x,-+im o d « for all i. Thus N C\M is a periodic orbit of period m = n. • Proof Of t h e o r e m 28. We have already seen that the set of exceptional points in M may be characterized as E = {x y. For k > k', Vk is a weak cycle of intervals contained in Vki and with the same number of components. It follows that is the only component of Vk which meets V2. Therefore must contain x; as well. In particular, setting 8 = x, — y we find that V e > 0 , O r b ( ( y , y + e))D(y,y+<5). Now let A/ = Orb((y,y + (5)) = n e > o ° r b ( ( } ' , 3 ' + e))- It follows easily that N\{y} C saBa sin(y). For i f z € N, z 7^ y, then taking ei < min(i5, \z — y|) we find zi € Cy, y + ei) and « i ^ l such that f[ (zi) = z. Then taking e2 < zi — y we find z 2 € (y,y + £2) and « 2 ^ 1 such that f"2 (z2) = zi • Continuing inductively, we get a subsequence of a backward orbit branch of z which accumulates on y. 7432 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth Since (y, y + S) contains x;y for sufficiently large j, lemma 31 also implies that N is a weak cycle of intervals, and since it is forward invariant we have x G JV. This concludes step 1. Step 2: i f x, x' G M\E andy G sa(x)\M, then sa(x) C sa(x'). To prove this claim, fix an arbitrary point y G sa(x). We need to show that y [0,1] (in the topology of uniform convergence) there is a whole interval of points x G [0,1] for which a(x, / ) ^ sa(x, / ) . We show in theorem 33 that if there are no wandering intervals then a(x, / ) = sa(x, / ) for all x G [0,1]. T h e o r e m 33. Let f: [0,1] —> [0,1] be a continuous map. If£l(f) = [0,1] then a(x) = sa(x) for all x G [0,1]. Proof. B y [29, theorem 1] the set fi(/)\Rec(/) is at most countable and since R e c ( / ) is a closed set we have Per(/) = [0,1]. Thus / has a dense set of periodic points. B y [21, theorem 3.9] (see also the discussion on page 40 in [21]) every non-periodic point of / belongs to the interior of some transitive cycle of intervals L. Fix x G [0,1]. We will show that a(x) C sa(x). Let y G a(x). If y is periodic then by theorem 8 y G sa(x). Suppose that y 7433 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth is not periodic and let L be the transitive cycle of intervals for / with y y as j —> oo. The points from {x;}?^ are all distinct since x is not periodic and thus there are infinitely many preimages of x in L. In particular, one of the preimages is a non-exceptional point of L and y € L C sa(x) by equation (1). • Let C°([0,1]) be the complete metric space of all maps / : [0,1] —> [0,1] with the usual uniform metric d(f,g) = supx 6 [o,i]|/(x) — g(x)\. If some comeager subset of maps in C°([0,1]) all have some property, then we call that property generic. Example 34. Let £ : [0,9] -> [0,9] be the 'connect-the-dots' map with £(0) = 0, £(1) = 7, f (2) = £(5) = 4, £(6) = C(8) = 7, £(9) = 9, and which is linear (affine) on each of the intervals [0,1], [1, 2], [2, 5], [5, 6], [6, 8], and [8, 9]. Every map in C°([0,1]) has a fixed point. B y perturbation, we can replace that fixed point with a small invariant interval on which we insert a miniature homeomorphic copy of £. Call the resulting map / and let A , C, W denote the images under the homeomorphism of [5, 6], [3, 8], and [1,2], respectively, so that A c / ( A ) = / ( C ) = f(W) c C. These containments are stable under further perturbation, that is, for all g sufficiently close to / the three intervals g(A), g(C),g(W) all contain A and are contained in C. It follows that A C g"(W) C C for all n = 1,2,.... Then each x [0,1] be a continuous interval map and y € [0,1]. The set sa(y) is not closed if and only if y belongs to a maximal solenoidal set Q which contains a nonrecurrent point from the Birkhoff centre off. In this case srtyj\sa(y) = Qn ( R < 0 \ R e c ( / ) ) . The rest of section 5.1 is devoted to the proofs of these two theorems. For theorem 35 the main idea is that sa(y) c a(y), so we can apply theorems 6, 7, and 8. However, some extra care is needed for preperiodic points. Recall that x is preperiodic for / i f x ^ Per(/) but there exists n > 1 such thatf(x) 0. B y hypothesis either x x and times k„ > 0 such that /**> (y0 ) = y and fk "(yn) = yn-\ for n ^ 1. We also construct points a„ 1. For the base case, choose any point an e sa(y) with x < an < /(*)• Choose a small open interval UQ 3 an which is compactly contained in (x, f(x)). There exists yQ 0 such that f^iyo) = y. N o w we make the induction step. Suppose that x < yn_l < sup Un-\ and choose a„ G sa(y) with x < a„ < min{y„_i, inf U„-i,x + - } . Choose a small open interval U„ 3 a„ compactly contained in (x, f(x)) with sup U„ < min{yn_l, inf Un-{\. B y theorem 5 we know that a„ is a non-wandering point, so there exist b, c 0 such that fkn (b) = c. Thus fk "([x, b]) contains both f(x), c, but c < yn_l < f(x), so by the intermediate value theorem there is yn 0 t=l For e > 0 the set Wf is invariant for / . It is connected because the intervals f'((x — e, x]) all contain the common point f(x). Now choose a point a 0. Thus c € We, so by connectedness we get [x, f(x)] f(x) such that / ( [ / ( x ) , z]) C (x, 1]. Again by continuity there is a point w < x such that f([w,x]) C (x, z). Fix e > 0 arbitrary. Put U = (max{w, x — e}, x). Choose a e sa(y) n U. Since a is non-wandering there are points b,c 0. But f(b) > x, so there is some t > 1 such that fib) > x and f'+l (b) < x. Since W is invariant, / ' ( £ ) ^ W. Therefore > z, so it follows that z&We. Since e > 0 was arbitrary, this shows that Again fix e > 0 arbitrary and let £/, a, b, c, s be as they were before. We have f(b) e (x, z) c W and W is invariant, so f(b) = c e W, contradicting the assumption that x = min W. Finally, we construct inductively a monotone increasing sequence of points yn —> x and a sequence of times k„ > 0 such that /*°(yo) = yand fk "(y„) = y«-i forn ^ l.Let<5 > 0 be such that W contains the left-hand neighbourhood (x — S, x). For the base case we find a e sa(y) n (x — S, x). Then there is y 0 € (x — r5, x) and £n > 0 such that fk °(yo) = y. For the induction step, suppose we are givenyn _l e (x — S, x). Choose a positive number e < min{|x — y„|, ^}. Since (0, S) c W C W e , we see from the definition of W that there exist yn e (x — e, x] and fc„ > 0 such that fk "(y„) = y«-i- Clearly y„ 7^ x, since yn_l ^ f(x). Therefore yn /")- Since the closure of a finite union is the union of the closures, we find j such that x 0 such that g'((x — S, x]) C /„. Then by invariance we get g\{x - 8, x]) C I„ for all j ^ i. (2) Choose an arbitrary positive real number e < 5. Choose a periodic point p in (x — e, x). Let k be the period of p under the map g. Then glk (p) = p and by (2) g'k (p) n such that one of the components L of Orb(/,/,/) lies between b and h(x). B y the intermediate value theorem, h([p',x]) contains L. Let U be any right-hand neighbourhood of p1 , that is U = (p1 , z) with z — p' > 0 as small as we like. We consider the orbit of U. Since the graph of h lies above the diagonal on (/?', x] we get a monotone increasing sequence (/z; (z))^= 0 w i m ' ^ 1 minimal such that h'(z) > x (if there is no such I, then h] (z) converges to a fixed point of h in the interval [z, x], which contradicts the choice of p'). Then h'+l (U) D L. In particular, Orb(<7, / ) D Ovb(L, f) D Orb(/„/, / ) D Q. 7437 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth , ' L ' y h / x / 0 Figure 6. The graph of h\in. Since Orb(L, f)DQ and each right-hand neighbourhood of p1 eventually covers L, we see that p1 G a(y). Note: we do not claim that h is monotone on (p1 , x], only that the graph stays above the diagonal. We see that for every right-hand neighbourhood U of p' there is a point y' 0 can be arbitrarily small, we conclude that x G sa(y). This completes the proof. • 5.2. Properties of a non-closed special a-limit set As an application of theorems 36 and 35, we get the following results. T h e o r e m 39. If sa(y) is not closed, then it is uncountable and nowhere dense. Proof. Suppose sa(y) is not closed. B y theorem 36 we have y G Q for some solenoidal set Q = p| Orb(/„). B y theorem 6 we know that sa(y) contains Q n Rec(/). This set is perfect [4, theorem 3.1], and therefore uncountable. Let M be a transitive cycle of intervals for / . If Q n M =t 0, then by lemma 32, each cycle of intervals Orb(/„) has period at most twice the period of M. This contradicts the fact that the periods of the cycles of intervals tend to infinity. Therefore Q does not intersect any transitive cycle of intervals M for / . In particular, y does not belong to any transitive cycle of intervals M. B y corollary 20 we conclude that sa(y) does not contain any transitive cycle. B y theorem 24 it follows that sa(y) is nowhere dense. • A subset of a compact metric space X is called Fa i f it is a countable union of closed sets, and Gs if it is a countable intersection of open sets. These classes of sets make up the second level of the Borel hierarchy. Closed sets and open sets make up the first level of the Borel hierarchy and they are always both Fa and G$. The next result shows that the sa-limit sets of an interval map can never go past the second level of the Borel hierarchy in complexity. T h e o r e m 40. Each sa-limit set for a continuous interval map fis both FG and G$. Proof. We write Bas(/) for the union of all basic w-limit sets of / and Sol(/) for the union of all solenoidal w-limit sets of / . We continue to write Per(/) for the union of all periodic orbits of / . 7438 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth To prove that sa(y) is of type Fa we express it as the following union sa(y) = (sa(y) n Per(/)) U (sa(y) n Bas(/)) U (sa(y) n Sol(/)), and show that each of the three sets in the union is of type Fa. The set Per(/) = lj„{x : f"(x) = x} is clearly of type Fa. B y theorem 35, sa(y) n Per(/) is a relatively closed subset of Per(/), and is therefore of type Fa. Since an interval map has at most countably many basic sets [4, lemma 5.2], their union Bas(/) is of type Fa. B y theorem 35, we know that sa(y) n Bas(/) is a relatively closed subset of Bas(/), and is therefore of type Fa. B y theorem 6 and corollary 12, we know that sa(y) n Sol(/) is either the empty set, or a single minimal solenoidal set S, and minimal solenoidal sets are closed. Closed sets are trivially oftypeFC T . To prove that sa(y) is of type Gs it is enough to show that sa(y)\sa(y) is at most countable. B y theorem 36 we know that this set is of the form Myj\sa(y) = Q(1 (Re [0,1] be a continuous interval map. The following are equiv- alent: (a) For some y (b): suppose there is a point y G [0,1] with sa(y) not closed. Choose x e sa(y)\sa(y). B y theorem 36 we know that x is a non-recurrent point in a solenoidal set. B y theorem 6 it follows that no sa-limit set of / contains x. This shows that x G S A ( 7 ) \ S A ( / ) . Thus S A ( / ) is not closed. But A 2 ( / ) = S A ( / ) by theorem 5. (b) =>• (c): this follows immediately from the containments Rec(/) c A 2 ( / ) c R e c ( / ) in theorem 5. (c) (d): suppose S A ( / ) = A 2 ( / ) ^ Reef/). Then we can find x G R e c ( / ) \ S A ( / ) . B y theorem 5 we know that x G A (/). Because x is in an w-limit set, it must belong to a periodic orbit, a basic set, or a solenoidal w-limit set. 7439 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth Each periodic orbit is contained in the sa-limit set of any one of its points. Each basic set is also contained in a sa-limit set by corollary 16. Since we supposed that x is not in any sa-limit set, we must conclude that x is not in a periodic orbit or a basic set. Now we know that x belongs to a solenoidal w-limit set. Since x is not in any sa-limit set we may use theorem 5 to conclude that x is not recurrent. (d) => (a): suppose a solenoidal w-limit set to(z) contains a non-recurrent point x in the Birkhoff centre x [0,1] be a continuous interval map with zero topological entropy. Then all sa-limit sets off are closed if and only ifRec(f) is closed. Proof. For a continuous interval map / with zero topological entropy, the set of recurrent points is equal to A2 (f) by [29]. Now apply theorem 41. • Corollary 44. Suppose f: [0,1] —> [0,1] is not Li-Yorke chaotic. Then all sa-limit sets off are closed. Proof. When / is not L i - Y o r k e chaotic, Steele showed that Rec(/) is closed [25, corollary 3.4]. Moreover, it is well known that such a map has zero topological entropy, see e.g. [21]. Now apply corollary 43. • 5.4. Example of a non-closed special a-limit set In 1986 Chu and Xiong constructed a map / : [0,1] —> [0,1] with zero topological entropy such that Rec(/) is not closed [10]. This example appeared six years before the definition of sa-limit sets [15], but by corollary 43 it provides an example of a continuous interval map whose sa-limit sets are not all closed. In this section we give a short direct proof that one of the sa-limit sets of Chu and Xiong's map / is not closed. Here are the key properties of the map / from [10]. (a) There is a nested sequence of cycles of intervals [0,1] = M0 z> M\ D M2 D ... for / , where M„ = Orb(7„) has period 2". (b) For each n An U Jn+i U Bn is an increasing linear bijection, (2) f2 "\jn+1 : Jn+i -> Kn+\ is surjective, (3) f2 " \B„ : Bn —> Kn+l U B „ U / „ + i is a decreasing linear bijection, (4) f 2 " \ K „ + I '• Kn+i —> Jn+i is an increasing linear bijection, and (5) f2 "\c„ : C„ —> Bn U Kn+i U C„ is an increasing linear bijection. (e) The nested intersection = H^Lo-Ai is a non-degenerate interval = [x, y]. The graph of the map / : [0,1] —> [0,1] is shown in figure 7. Chu and Xiong showed that the left endpoint x of 7 ^ is not recurrent, but it is a limit of recurrent points. We give a short direct proof that sa(x) is not closed. T h e o r e m 45. Let f be the continuous interval map defined in [10] and let x be the left endpoint ofJoo as defined above. Then x € sa(x)\sa(x), and therefore sa(x) is not closed. Proof. Fix n An U Jn+\ U Bn is linear, say, with slope A„. B y property (e) we have x e Jn+\. Therefore there is a backward orbit branch { x , } ^ 0 of x such that xw = an + ^T*- for all k oo, then a„ —> x. This shows that x € sa(x). Now we will show that x £ sa(x). Let { x , } ^ 0 be any backward orbit branch of x. Let Q = n^Lo^n- ^ e distinguish two cases. First suppose that e Q for all i. For any given i > 1 we can choose n with 2" > i. Since e M„, f'(xi) e / „ , and Af„ is a cycle of intervals of period 2", we know that x; ^ 7„. Since Jn is the left-most component of M„ in [0,1], it follows that x; > y > x (recall that = f]J„ = [x,y] is non-degenerate). Since this holds for all ( 1 we see that the backward orbit branch does not accumulate on x. Now suppose there is z'o with x;0 e' Q. For each i ^ /0 there is n(i) e N such that x; € M„(;)\M„(;)+ i. Since / ( x ; + i ) = x; and each M„ is invariant, we get n(i+ 1) < n(i). 7441 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth A non-increasing sequence of natural numbers must eventually reach a minimum, say, n(i'i) = n{i\ + 1) = • • • = n. For i ^ z'i, x; ^ Mn+l, so in particular x; ^ J,,+i- But by properties (c) and (e) we know that Jn+\ is a neighbourhood of x. This shows that this backward orbit branch does not accumulate on x either. • 6. O p e n p r o b l e m s Only one problem concerning sa-limit sets of interval maps in [18] remains open: Problem 46 [18]. Characterize all subsets A of [0, 1] for which there exists a continuous map / : [0,1] -> [0,1] and a point x e [0,1] such that sa(x, f) = A. We have seen that even for interval maps, sa-limit sets need not be closed. If we want to work with closed limit sets, then there are several possible solutions. The first one, suggested to us by Snoha, is to answer the following question: what are some sufficient conditions on a topological dynamical system (X,f) so that all of its sa-limit sets are closed? In this regard we state one conjecture which we were not able to resolve. C o n j e c t u r e 47. If / : [0,1] -> [0,1] is continuously differentiable, then all sa-limit sets of / are closed. Another possibility is to ask whether the 'typical' continuous interval map has all sa-limit sets closed. Let C°([0,1]) be the complete metric space of all maps / : [0,1] —> [0,1] with the usual uniform metric d(f, g) = supx 6 [o,i] |/(x) — g(x)\. If some comeager subset of maps in C°([0,1]) all have some property, then we call that property generic. Problem 48. Is the property of having all sa-limit sets closed a generic property in C°([0,1])? Another possible solution is to work with the closures of sa-limit sets. Therefore we propose the following definition. Definition 49. Let (X,f) be a discrete dynamical system (i.e. a continuous self-map on a compact metric space) and let x G l The (3-limit set of x, denoted /3(x) or /3(x, / ) , is the smallest closed set such that d(x„, (3(x)) —> 0 as n —> oo for every backward orbit branch {xn}%L0 of the point x. The letter (3 here means 'backward', since /3-limit sets serve as attractors for backward orbit branches. It is clear from the definition that /3(x) = sa(x). P r o p o s i t i o n 50. If (X,f) is a discrete dynamical system and x [0,1] and a point x e [0,1] such that /3(x, / ) = A. If X is a compact metric space, then the space K(X) consisting of all nonempty compact subsets of X can be topologized with the Hausdorff metric and it is again compact. The map 7442 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth 1 3 x 4 cv(x, f) G K(X) associated to a dynamical system (X, f) is usually far from continuous, but its Baire class can be useful, see e.g. [25]. Therefore we propose the following question. Problem 52. O f what Baire class (if any) is the function [0, l ] 3 x i - > P ( x , f ) G A"([0,1]) when / is a surjective continuous interval map? Hero used sa-limit sets to characterize the attracting centre A2 (f) of a continuous interval map / , and some work has been done to extend his results to trees and graphs [15,26, 27]. We conjecture that for graph maps, the /3-limit sets can be used to characterize the Birkhoff centre Rec(/) as follows: C o n j e c t u r e 53. Let / : X —> X be a graph map, and let i e l The following are equivalent: (a) x G R e < f ) (b) x G p(x) (c) There exists y G X such that x G (3(y). The coexistence of periodic orbits for interval maps was studied by Sharkovsky [22]. A special case of his theorem, proved independently by L i and Yorke, says that if / : [0,1] —> [0,1] has a periodic orbit of period three, then it has periodic orbits of all periods [19]. This suggests the problem of studying the coexistence of periodic orbits within special alimit sets.5 In this spirit, we offer one conjecture and one open problem. C o n j e c t u r e 54. Let / : [0,1] -> [0,1] and x G [0,1]. If sa(x) contains a periodic orbit of period 3, then for every positive integer n, sa(x) contains a periodic orbit of period n or 2n. Problem 55. For which subsets A C N is there a map / : [0,1] —> [0,1] and a point x G [0,1] such that A is the set of periods of all periodic orbits of / contained in sa(x)? In support of conjecture 54, we show that the conclusion holds for n = 1. L e m m a 56. Letf: [0,1] —> [0, \]andx G [0, \]. If sa{x) contains a periodic orbit of period 3, then it contains a periodic orbit of period 1 or 2 as well. Proof. Let sa(x) contain the periodic orbit {a, b, c} for the interval map / with a < b < c. We may assume without loss of generality that f(a) = b, f(b) = c, and /(c) = a. B y continuity we may choose a closed interval U = [b — e, b + e] with e > 0 small enough that f2 (U) < U < f(U), that is to say, max f2 (U) < min U < max U < min f(U). Now find x\ G U and n\ ^ 1 such that fl (xi) = x. B y the intermediate value theorem we can find X2 G (max U, m i n / ( £ / ) ) such that f{x-i) = x\. Again, by the intermediate value theorem we can find xj, G (xi,X2) such that / ( X 3 ) = X2. In the next step we find X4 G te,X2) such that f(x4) = X3. Continuing inductively we find a whole sequence (x;) such that / ( x ! + i ) = x;, i ^ 1, arranged in the following order, X i < X 3 < X5 < • • • < • • • < X 6 < X 4 < X2Since a bounded monotone sequence of real numbers has a limit, we may put x~ = 1^-^00X2;+1 and x+ = l i m ^ x 2 i + 2 , and we have x~ < x + . Then / ( x ~ ) = lim / ( x 2 i + i ) = lim x 2 i = x+ and similarly / ( x + ) = x~ . This shows that {x+, x ^ } 5 Or equivalently, within /3-limit sets, since by theorem 8 a periodic orbit of an interval map / is contained in sa(x) if and only if it is contained in fi(x). 7443 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth is a periodic orbit contained in sa(x). The period is 2 if these points are distinct and 1 if they coincide. • Example 57. Let / : [0,5] -> [0,5] be the 'connect-the-dots' map with /(0) = 1, / ( l ) = 5, /(4) = 2, /(5) = 0, and which is linear (affine) on each of the intervals [0,1], [1,4], and [4, 5]. Then sa(0) contains the period-three orbit {0,1,5} and the period-two orbit {2,4}, but not the unique period-one orbit {3}. Example 58. Let / : [0,8] -> [0,8] be the 'connect-the-dots' map with /(0) = 4, /(4) = 8, /(5) = 3, /(8) = 0, and which is linear (affine) on each of the intervals [0,4], [4, 5], and [5, 8]. Then sa(0) contains the period-three orbit {0,4, 8}, the period-four orbit {1,5, 3,7} and the period-one orbit { y }, but not the unique period-two orbit {2,6}. A c k n o w l e d g m e n t s We wish to thank Piotr Oprocha for pointing us in the right direction at the beginning of this project, and telling us about several useful ideas from [1]. We also thank Lubo Snoha for his helpful feedback and warm encouragement. His insightful suggestions resulted in a much clearer paper, and in particular Theorem 36 would not exist without him. This research is part of a project that has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 883748 References [1] Balibrea F, Dvornikova G, Lampart M and Oprocha P 2012 On negative limit sets for onedimensional dynamics Nonlinear Anal. Theory Methods Appl. 75 3262-7 [2] Birkhoff G 1927 Dynamical Systems (Colloquium Publications vol 9) (Providence, RI: American Mathematical Society) [3] Block L and Coven E M 1986 w-limit sets for maps of the interval Ergod. Theor. Dynam. Syst. 6 335-44 [4] Blokh A M 1995 The 'spectral' decomposition for one-dimensional maps Dynamics Reported (Expositions in Dynamical Systems vol 4) (Berlin: Springer) pp 1-59 [5] Blokh A, Bruckner A M, Humke P D and Smital J 1996 The space of w-limit sets of a continuous map of the interval Trans. Am. Math. Soc. 348 1357-72 [6] Bowen R 1975 w-Limit sets for Axiom A diffeomorphisms J. Differ. Equ. 18 333-9 [7] Conley C 1978 Isolated Invariant Sets and the Morse Index (Regional Conference Series in Mathematics vol 38) (Providence, RI: American Mathematical Society) [8] Coven E M and Hedlund G A 1980 P = R for maps of the interval Proc. Am. Math. Soc. 79 316 [9] Coven E M and Nitecki Z 1981 Non-wandering sets of the powers of maps of the interval Ergod. Theor. Dynam. Syst. 1 9-31 [10] Chu H and Xiong J C 1986 A counterexample in dynamical systems of the interval Proc. Am. Math. Soc. 97 361 [11] Cui H and Ding Y 2010 The a-limit sets of a unimodal map without homtervals Topol. Appl. 157 22-8 [12] Dowker Y N and Friedlander F G 1954 On limits sets in dynamical systems Proc. London Math. Soc. 3-4 168-76 [13] Forys-Krawiec M, Hantakova J and Oprocha P 2021 On the structure of a-limit sets of backward trajectories for graph maps (arXiv:2106.05539 [math.DS]) [ 14] Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences vol 42) (Berlin: Springer) [15] Hero M W 1992 Special a-limit points for maps of the interval Proc. Am. Math. Soc. 116 1015 7444 Nonlinearity 34 (2021) 7415 J Hantakova and S Roth [16] Hirsch M W, Smith H L and Zhao X-Q 2001 Chain transitivity, attractivity, and strong repellors for semidynamical systems J. Dyn. Differ. Equ. 13 107-31 [17] Jackson S, Mance B and Roth S 2021 A non-Borel special alpha-limit set in the square Ergod. Theor. Dynam. Syst. 1-11 [18] Kolyada S, Misiurewicz M and Snoha L 2020 Special a-limit sets Dynamics: Topology and Numbers (Contemporary Mathematics vol 744) (Providence, RI: American Mathematical Society) pp 157-73 [19] Li T-Y and Yorke J A 1975 Period three implies chaos Am. Math. Mon. 82 985-92 [20] Przytycki F 1999 Conical limit set and Poincare exponent for iterations of rational functions Trans. Am. Math. Soc. 351 2081-99 [21] Ruette S 2017 Chaos on the Interval (University Lecture Series vol 67) (Providence, RI: American Mathematical Society) [22] Sharkovskii A N 1995 Co-existence of cycles of a continuous mapping of the line into itself Internationl Journal of Bifurcation and Chaos 5 1263-73 [23] Sharkovskii A N 1965 On attracting and attracted sets Dokl. Akad. Nauk SSSR 160 1036-8 [24] Sharkovskii A N 1966 Continuous maps on the set of limit points of an iterated sequence Ukr. Math. J. 18 127-30 [25] Steele T H 2003 Chaos and the recurrent set Real Anal. Exch. 29 79-87 [26] Sun T, Xi H, Chen Z and Zhang Y 2004 The attracting centre and the topological entropy of a graph map Advances in Mathematics (China) 33 540-6 [27] Sun T, Xi H and Liang H 2011 Special a-limit points and unilateral 7-limit points for graph maps Sci. China Math. 54 2013-8 [28] Sun T, Tang Y, Su G, Xi H and Qin B 2018 Special a-limit points and 7-limit points of a dendrite map Qual. Theory Dyn. Syst. 17 245-57 [29] Xiong J C 1988 The attracting centre of a continuous self-map of the interval Ergod. Theor. Dynam. Syst. 8 205-13 7445