@ Appl. Gen. Topol. 20, no. 2 (2019), 307-324 doi:10.4995/agt.2019.7910 c© AGT, UPV, 2019 Simple dynamical systems K. Ali Akbar a, V. Kannan b and I. Subramania Pillai c a K. Ali Akbar, Department of Mathematics, Central University of Kerala, Kasaragod - 671320, Kerala, India. (aliakbar.pkd@gmail.com, aliakbar@cukerala.ac.in) b V. Kannan, School of Mathematics and Statistics, University of Hyderabad, Hyderabad - 500 046, Telangana, India. (vksm@uohyd.ernet.in) c I. Subramania Pillai, Department of Mathematics, Pondicherry University, Puducherry-605014, India. (ispillai@gmail.com) Communicated by M. Sanchis Abstract In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We char- acterize the family of homeomorphisms on R having finitely many non- ordinary points upto (order) conjugacy. For x, y ∈ R, we say x ∼ y on a dynamical system (R, f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x ∼ y if there exists an increasing homeomorphism h : R → R such that h ◦ f = f ◦ h and h(x) = y. An element x ∈ R is ordinary in (R, f) if its equivalence class [x] is a neighbourhood of it. 2010 MSC: 54H20; 26A21; 26A48. Keywords: special points; non-ordinary points; critical points; order conju- gacy. 1. Introduction A dynamical system is a pair (X,f) where X is a metric space and f is a continuous self map on X. Two dynamical systems (X,f), (Y,g) are said to be topological conjugate if there exists a homeomorphism h : X → Y (called topo- logical conjugacy) such that h◦f = g◦h. The properties of dynamical systems which are preserved by topological conjugacies are called dynamical properties. Received 26 July 2017 – Accepted 30 July 2019 http://dx.doi.org/10.4995/agt.2019.7910 K. Ali Akbar, V. Kannan and I. Subramania Pillai The points which are unique upto some dynamical property are called dynam- ically special points. Said differently, a special point has a dynamical property which no other point has. The idea of special points is relatively new to the literature (see [7]). In this paper, we introduce the notion of non-ordinary point. Throughout this paper we will be working with continuous self homeomor- phisms of the real line. Since R has order structure, we would like to consider the topological conjugacies (simply we call conjugacies) preserving the order. Hence the conjugacies which we mainly consider in this paper are order pre- serving conjugacies (increasing conjugacies). The increasing conjugacies are usually called order conjugacies. When we are working with a single system, any self conjugacy can utmost shuffle points with same dynamical behavior. Therefore a point which is unique upto its behavior must be fixed by every self conjugacy. On the other hand, if a point is fixed by all self conjugacies then it must have a special property (some times it may not be known explicitly). These ideas motivated us to call the set of all points fixed by all self conjugacies as set of special points. For x,y ∈ R, we write x ∼ y if x and y have the same dynamical properties in the dynamical system (R,f). Said precisely, x ∼ y if there exists an increasing homeomorphism h : R → R such that h◦f = f ◦h and h(x) = y. It is easy to see that ∼ is an equivalence relation. Since the equivalence relation is coming from self conjugacy it is important in the field of topological dynamics. Let [x] denote the equivalence class of x ∈ R. In a dynamical system (X,f), we say that a point x is ordinary if its “like” points are near to it. That is, an element x ∈ R is ordinary in (R,f) if its equivalence class [x] is a neighbourhood of it, i.e., the equivalence class of x contains an open interval around x. A point which is not ordinary is called non-ordinary . Let N(f) be the set of all non-ordinary points of f. We call a point to be special if [x] = {x}. Let S(f) be the set of all special points of f. A point x in a topological space X is said to be rigid if it is fixed by every self homeomorphism of X. For example, the point 1 is rigid in (0, 1]. According to the above definition all rigid points are special even though there is no role for the map f. We make this as a convention. By definition, the points of [x] are dynamically same. We consider systems for which there are only finitely many equivalence classes. This means there are only finitely many kinds of orbits upto conjugacy. In particular, their sets of periods Per(f) are contained in {1, 2, 22, ...}. If f : R → R is continuous and Per(f) properly contained in {1, 2, 22, ...} then f is not Li-Yorke chaotic (see [1]). Also note that if f : R → R is Devaney chaotic then 6 ∈ Per(f) (see [2]). Therefore, if f : R → R is a continuous map having finitely many non-ordinary points then it is neither Li-Yorke chaotic nor Devaney chaotic due to Sharkovskii’s theorem. For these reasons, we call such systems as simple systems. These are the system in which the phase portrait can be drawn. Phase portraits (see [5]) are frequently used to graphically represent the dynamics of a system. A phase portrait consists of a diagram representing possible beginning positions in the system and arrows that indicate the change in these positions under iteration c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 308 Simple dynamical systems of the function. The drawable systems are interesting to physicists and for this reason the study of the class of simple dynamical systems can be useful. Our main results characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. In particular, we prove that: (i) The number of all increasing continuous bijections (upto order conjugacy) on R with exactly n non-ordinary points is equal to an = C1(1 + √ 3)n +C2(1−√ 3)n, where C1 = (5+3 √ 3) 2 √ 3 and C2 = (3 √ 3−5) 2 √ 3 . (ii) If tn denotes the number of increasing homeomorphisms (upto conju- gacy) on R with exactly n non-ordinary points, then t0 = 2, t1 = 5 and t2 = 12, and we have tn =   an+2a n−4 2 2 if n is even an+2a n−3 2 2 if n is odd for n ≥ 3. (iii) If sn denotes the number of decreasing homeomorphisms (upto order conjugacy) on R with exactly n non-ordinary points, then sn = { 0 if n is even an−1 2 if n is odd for all n. (iv) If kn denotes the number of decreasing homeomorphisms (upto conju- gacy) on R with exactly n non-ordinary points, then kn = { 0 if n is even tn−1 2 if n is odd for all n. 2. Basic Results Let (X,f) be a dynamical system. We denote the full orbit of a point x ∈ X by the set Õ(x) = {y ∈ X : fn(x) = fm(y) for some m,n ∈ N}. For any subset A ⊂ R, let Õ(A) = ⋃ x∈A Õ(x) = ⋃ x∈A {y ∈ R : fn(y) = fm(x) for some m, n ∈ N}. A point x in a dynamical system (X,f) is said to be a critical point if f fails to be one-one in every neighbourhood of x. The set of all critical points of f is denoted by C(f), and by P(f) we denote the set of all periodic points of f. Recall that a point x in a dynamical system (X,f) is said to be periodic if fn(x) = x for some n ∈ N. Let f : R → R and D(f) be the set Õ(C(f) ∪ P(f)∪{f(∞),f(−∞)}), where f(∞) and f(−∞) are the limits of f at ∞ and −∞ respectively, provide they are finite. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 309 K. Ali Akbar, V. Kannan and I. Subramania Pillai We will prove below (see Proposition 2.13) that if the map has only finitely many non-ordinary points then N(f) = S(f). Hence the following characteri- zation theorem holds for the set N(f) since a similar type of characterization holds for S(f). For a continuous map f : R → R, we define Sp(R,f) = ⋂ h{x ∈ R : h(x) = x, h : R → R is a homeomorphism such that h◦f = f ◦h}. Theorem 2.1. For continuous self maps of the real line R, the set of all special points is contained in the closure of the union of full orbits of critical points, periodic points and the limits at infinity (if they exist and finite). That is, N(f) is a subset of the closure of D(f). Proof. For a continuous map f : R → R, first observe that S(f) ⊂ Sp(R,f). By Theorem 1 of [7], we have Sp(R,f) ⊂ D(f). Hence the proof follows. � Remark 2.2. Consider the map f(x) = x + sin(x) for all x ∈ R. Observe that all integral multiples of π are fixed points for f but the increasing bijection x 7→ x + 2π commutes with f and fixes none of them. Hence in this case N(f) is properly contained in the closure of D(f). Now consider the following theorems. Theorem 2.3. For polynomials of even degree the equality D(f) = S(f) holds. Proof. Let f : R → R be a polynomial map of even degree. By Theorem 2 of [7], we have Sp(R,f) = S(f) = D(f). � Theorem 2.4. For polynomial maps f of R, S(f) has to be either empty or a singleton or the closure of D(f). Proof. The ideas involved in the proof of Theorem 3 as in [7] can be adapted to order conjugacies. Hence the proof follows. � From the definition of special points, it is clear that the set of special points S(f) is always closed. The following theorem is about the converse. Theorem 2.5. Given any closed subset F of R, there exists a continuous map f : R → R such that S(f) = F . Proof. The ideas involved in the proof of Lemma 2, Lemma 3, and Theorem 4 can be adapted to order conjugacies. Hence the proof follows. � The following total order on N is called the Sharkovskii’s ordering: 3 � 5 � 7 � 9 � ... � 2 × 3 � 2 × 5 � 2 × 7 � ... � 2n × 3 � 2n × 5 � 2n × 7 � ... ... � 2n � .... � 22 � 2 � 1. We write m � n if m precedes n (not necessarily immediately) in this order. An n-cycle means a cycle of length n. Theorem 2.6 (Sharkovskii’s Theorem, see[8]). Let m � n in the Sharkovskii’s ordering. For every continuous self map of R, if there is an m-cycle, then there is an n-cycle. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 310 Simple dynamical systems For any continuous map f : R → R, topological conjugacy (respectively order conjugacy) of f is a homeomorphism (respectively increasing homeomor- phism) h : R → R such that h◦f = f ◦h. For any continuous map f : R → R, let Gf be the set of all topological conjugacies of f and let Gf↑ be the set of all order conjugacies of f. Proposition 2.7. If x is an ordinary point of f and if h is a self conjugacy of f, then h(x) is ordinary. Proof. Since x is ordinary there exists an open interval V contained in [x]. We prove that the open interval (since h is a homeomorphism) h(V ) is contained in [h(x)]. Take s ∈ h(V ). Then s = h(t) for some t ∈ V. Since V ⊂ [x], there exists ϕ ∈ Gf↑ such that ϕ(t) = x. Then the increasing homeomorphism ψ = hϕh−1 carries s to h(x) and commutes with f. � Proposition 2.8. If x is a non-ordinary point of f and if h is a self conjugacy of f, then h(x) is non-ordinary. Proof. Note that if h is a self conjugacy of f then h−1 is also a self conjugacy of f. Now, the proof follows from Proposition 2.7. � For any subset A of R, we write ∂A = A ∩ (X −A) for the boundary of A, where A denotes the closure of A in R. Recall that the properties which are preserved under topological conjugacies are called dynamical properties. Hence, if two points x,y in the dynamical system (X,f) differ by a dynamical property, then no conjugacy can map one to the other. Hence the following proposition follows. Proposition 2.9. The points of ∂SP are non-ordinary for any dynamical prop- erty P , where SP denotes the set of all points in (X,f) having the dynamical property P . Corollary 2.10. Let f : R → R be constant in a neighbourhood of a point x0. Then the end points of the maximal interval around x0 on which f is constant are non-ordinary. Remark 2.11. Note that, being a point in a particular equivalence class [x] is a dynamical property. Therefore when there are n non-ordinary points then there are n + 1 equivalence classes. But the converse is not true. Consider the map x 7→ x + sin(x) on R. There are two equivalence classes but infinitely many non-ordinary points. Now we ask: For a continuous map f : R → R, how the equivalence classes look like? The following lemma answers this question. Lemma 2.12. Let f : R → R be continuous. Suppose a < b and (a,b)∩N(f) = ∅. Then x ∼ y for all x,y ∈ (a,b). Proof. Assume without loss of generality that x < y. Suppose x � y, so z = sup([x] ∩ (−∞,y]) exists. Clearly z ∈ [x]. If z = y then z ∈ [y] ⊂ R\ [x]. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 311 K. Ali Akbar, V. Kannan and I. Subramania Pillai Otherwise z < y and [z,y) ∩ (R\ [x]) 6= ∅ for every y −x > � > 0 which again shows z ∈ R\ [x]. Then z ∈ ∂([x]) and hence z ∈ N(f) by Proposition 2.9. But a < x ≤ z ≤ y < b. Hence z ∈ (a,b) ∩N(f) contradicting our hypothesis. � Proposition 2.13. If f : R → R has only finitely many non-ordinary points then every non-ordinary point is special. Proof. Since the set of all non-ordinary points N(f) is finite, it follows from Proposition 2.7 and Proposition 2.8 that h(N(f)) = N(f) for all h ∈ Gf↑. Then we must have h(x) = x for all x ∈ N(f) because of the order preserving nature of h. Hence all points of N(f) are special. � Thus by above proposition the idea of special points and the idea of non-ordinary points coincide in the class of maps with finitely many non-ordinary points. For a set A, we denote |A| for the cardinality of A. Now we consider the following theorem. Theorem 2.14. Let f : R → R be continuous. If |N(f)| = n then |{[x] : x ∈ R}| = 2n + 1. Proof. Let N(f) = {x1,x2, · · · ,xn} where x1 < x2 < · · · < xn. By Propo- sition 2.13, each {xi} is an equivalence class. Then each of these intervals (−∞,x1), (x1,x2), · · · , (xn−1,xn), (xn,∞) is invariant under every element of Gf↑. Hence all the remaining equivalence classes are contained in one of these intervals. Lemma 2.12 above now shows that each of these interval is an equiv- alence class, giving |{[x] : x ∈ R}| = 2n + 1. � Remark 2.15. Note that, being a point in a particular equivalence class [x] is a dynamical property. Remark 2.16. If f : R → R has a unique fixed point then it is non-ordinary and vice-versa. Proof. Since the topological conjugacies carry fixed points to fixed points, the unique fixed point must be fixed by every self conjugacy and hence special. Next suppose x0 ∈ R is the unique non-ordinary point of f. Then h(x0) = x0 for all h ∈ Gf↑. Now, for any h ∈ Gf↑ we have h(f(x0)) = f(h(x0)) = f(x0). That is, the point f(x0) is special. Then we have f(x0) = x0 since x0 is the only special point. � Remark 2.17. If f : R → R has finitely many fixed points (critical points) then all fixed (critical) points are special and hence non-ordinary. Proof. This remark follows from the fact that under a topological conjugacy fixed points will be mapped to fixed points and critical points will be mapped to critical points and the fact that it takes the finite set F (of fixed points) to F bijectively preserving the order. � Remark 2.18. If there are only finitely many periodic cycles then all periodic points are special. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 312 Simple dynamical systems The following remark says that in general non-ordinary points need not be special. Remark 2.19. It is immediate from the definition that every special point is non-ordinary. But every non-ordinary point may not be special. For example, consider the map x 7→ x + sin x on R which has countably many fixed points. Note that all the fixed points are non-ordinary and they all together form a single equivalence class. Hence they are not special. Proposition 2.20. For maps with finitely many non-ordinary points, f(x) is non-ordinary whenever x is non-ordinary. Proof. Since x is non-ordinary and since there are only finitely many non- ordinary points, we have h(x) = x for all h ∈ Gf↑. Now for any h ∈ Gf↑, we have h(f(x)) = f(h(x)) = f(x). Hence f(x) is non-ordinary. � For a map f : A → R, A ⊂ R, we denote sup f and inf f for the supremum of f(A) and infimum of f(A) respectively. Recall that, if f : R → R has a unique non-ordinary point then it must be a fixed point. Proposition 2.21. Let f : R → R be a continuous function and let x ∈ R. Then (i) If x ∈ R is both critical and ordinary then f is locally constant at x. (ii) If x is ordinary then so is f(x) unless f is constant in a neighbourhood of x. Proof. (i) Let x0 ∈ R be both critical and ordinary. Claim: f is constant in some neighbourhood of x0. Since x0 is ordinary, there exists η > 0 such that all points in (x0−η,x0 +η) will look alike. So it is enough to prove that f is somewhere constant in (x0 −η,x0 + η). Case 1: Suppose some point in (x0 − η, x0 + η) is point of local maximum of f. Then we can prove easily that every point of (x0 − η, x0 + η) is a point of local maximum. That is there exists � > 0 such that f(x0) ≥ f(t) ∀ t ∈ (x0 − �, x0 + �). Next choose δ < �, η. Then there exists y ∈ [x0 − δ, x0 + δ] such that (2.1) f(y) ≤ f(t)∀ t ∈ [x0 −δ,x0 + δ]. But y is a point of local maximum (since δ < η). That is there exists α > 0 such that (2.2) f(y) ≥ f(s) ∀ s ∈ (y −α, y + α) From (1) and (2), it follows that f is constant in some neighbourhood y and hence constant in some neighbourhood of x0. Case 2: No point is a point of local maximum of f. That is, f attains its maximum at one of the endpoints in every subinterval. If f assumes supremum always on c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 313 K. Ali Akbar, V. Kannan and I. Subramania Pillai the right end (or always on the left end) then f is strictly monotone. Note that, it is enough if we prove monotone somewhere. Take a neighbourhood (α,β) of x0 such that (α,β) ⊂ (x0 − η, x0 + η) and let sup f on (α,β) be attained at the right end point β. Suppose sup f is attained at the right end point in every subinterval of (α,β) containing x0. Then f is increasing in (x0, β). We are done. Suppose there is a subinterval say (x0 − �1, x0 + �2) of (α,β) on which f attains its supremum at the left end point. Then f attains its infimum on (x0 − �1,β) at some interior point. We now argue as in Case 1. This ends the proof of Part (i). (ii) We make use of (i). Assume that f is not constant on any neighbourhood of x. Because x is ordinary, there exists an open interval J around x, in which all points are equivalent to x, such that f is not constant on J. It follows that f is not constant on any non-trivial subinterval of J, because the endpoints of intervals of constancy are non-ordinary. From (i), it follows that J has no critical point. Therefore f(J) is an open interval. We claim that any two elements of f(J) are equivalent. Let f(y) be a general element of f(J), where y ∈ J, y 6= x. By choice of J, there exists a self conjugacy h of f such that h(y) = x, which implies h◦f(y) = f ◦h(y) = f(x). Therefore f(y) is equivalent to f(x). This proves f(x) is ordinary. � Remark 2.22. Let f : R → R be continuous. Then sup f(R), inf f(R), limx→∞f(x) and limx→−∞f(x) are non-ordinary provided they are finite. Note that, for maps with finitely many non-ordinary points both limx→∞f(x) and limx→−∞f(x) always exist in R∪{∞,−∞}. Proof. For any h ∈ Gf↑, h(f(R)) = f(h(R)) = f(R). That is, h takes the range of f to itself. Since h is increasing, h(sup f) = sup f and h(inf f) = inf f. Now we prove that for maps with finitely many non-ordinary points, limx→∞f(x) always exists in R ∪ {∞,−∞}. For this, let t0 be the largest non-ordinary point and let A be the set of all critical points > t0. Suppose A is empty. Then f is monotone on [t0,∞) and hence limx→∞f(x) exists. Suppose A is nonempty. Then ∂A is nonempty. But every element of ∂A is non-ordinary. Hence ∂A = {t0}. Therefore A = (t0,∞). Therefore f is constant on A (we argue as in the proof of Case 2 of (i) in Proposition 2.21). Hence limx→∞f(x) exists. Next we will prove limx→∞f(x) is special. We denote limx→∞f(x) by l. Let h ∈ Gf↑. Note that for any sequence (xn) → ∞, we have f(xn) → l and h(xn) → ∞. Hence h(f(xn)) = f(h(xn)) → h(l). Being h(xn) → ∞, by the definition of l we find f(h(xn)) = h(f(xn)) → l. Hence h(l) = l. This completes the proof. � Proposition 2.23. The maps x+ 1 and x−1 on R are topologically conjugate; but not order conjugate. Proof. The maps x+ 1 and x−1 are conjugate to each other through the order conjugacy −x + 1 2 . If possible, let h be an order conjugacy from f(x) = x + 1 to g(x) = x − 1. Then h(x + 1) = h(f(x)) = g(h(x)) = h(x) − 1. i.e., c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 314 Simple dynamical systems h(x + 1) −h(x) = −1 < 0, which is a contradiction to the assumption that h is increasing. � Remark 2.24. Note that for the map x + 1 on R, all points are ordinary. This is because, if a,b ∈ R then the map x + b−a is the order conjugacy of x + 1 which maps a to b. The following proposition is proved in [6]. For the sake of completeness, we included its proof. Proposition 2.25. Let f : R → R be a homeomorphism without fixed points. Then (i) If f(0) > 0 then f is order conjugate to x + 1. (ii) If f(0) < 0 then f is order conjugate to x− 1. Proof. Define h : R → R as follows. Assume f(0) > 0. Define h(t) = t f(0) , 0 ≤ t < f(0). We know that (fn(0)) increases and diverges to ∞ and (f−n(0)) decreases and diverges to −∞ for all n ∈ N. Moreover for t ∈ R there exists unique n ∈ Z such that, fn(0) ≤ t < fn+1(0). Define h(t) = h(f−n(t)) + n. Then h◦f(t) = h(t) + 1 for all t ∈ R. This h gives a conjugacy from f to x + 1. If f(0) < 0 then we can give a similar proof. � For a map f : A → R, A ⊂ R, we define graph(f) := {(x,f(x)) : x ∈ A}. For continuous maps f,g : A → R, we say that graph(f) and graph(g) are above the diagonal if f(x) > x and g(x) > x for all x ∈ A. Similarly, graph(f) and graph(g) are said to be below the diagonal if f(x) < x and g(x) < x for all x ∈ A, and graph(f) and graph(g) are said to be on the diagonal if f(x) = g(x) = x for all x ∈ A. We say that graph(f) and graph(g) are on the same side of the diagonal if it is either above the diagonal or below the diagonal or on the diagonal. Corollary 2.26. Let f,g : (a,b) → (a,b) be homeomorphisms without fixed points. Then f is order conjugate to g if and only if both graph(f) and graph(g) are on the same side of the diagonal. In particular: (i) If f(x) > x for all x ∈ (a,b) then f is order conjugate to x + 1. (ii) If f(x) < x for all x ∈ (a,b) then f is order conjugate to x− 1. Remark 2.27. In fact, the interval (a,b) involved in Corollary 2.26 can be replaced by any open ray in R. Remark 2.28. If f : R → R is an increasing bijection with finitely many non- ordinary points then all non-ordinary points are fixed points. Proof. We know that for maps with finitely many non-ordinary points all non- ordinary points are fixed by every order conjugacy. Here f itself is a self conjugacy. � c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 315 K. Ali Akbar, V. Kannan and I. Subramania Pillai For a continuous map f : R → R, let Fix(f) denote the set of all fixed points of f. It follows from the continuity of f that Fix(f) is closed. For any subset A of a metric space X, we denote Ac for the complement of A and int(A) for the interior of A. Recall that (2.3) (∂A)c = int(A) ∪ int(Ac). The following proposition provides a characterization for the non-ordinary points of increasing homeomorphisms. Proposition 2.29. Let f : R → R be an increasing bijection and let x ∈ R. Then x is non-ordinary if and only if x ∈ ∂(Fix(f)). Proof. Let x ∈ ∂Fix(f). Then x is non-ordinary since every open interval around x contains fixed and non-fixed points. Now suppose x /∈ ∂Fix(f). We shall prove that x is ordinary. Now, x /∈ ∂Fix(f) implies x ∈ (∂Fix(f))c = int(Fix(f)) ∪ int((Fix(f))c) by equation (1). Hence x ∈ int(Fix(f)) or x ∈ int(Fix(f)c). Case 1: x ∈ int(Fix(f)) Suppose x ∈ int(Fix(f)). Then choose a,b ∈ R such that x ∈ (a, b) ⊂ Fix(f). Let y ∈ (a,b) be such that y 6= x. Then define an increasing continuous bijection φy : R → R such that φy(t) =   t if t /∈ (a,b) y if t = x piecewise linear otherwise. This φy maps x to y. Both [a,b] and its complement are invariant under both φy and f. Note that f is identity on [a,b] and φy is identity on the complement of [a,b]. Hence φy commutes with f on [a,b]. This proves x is an ordinary point. Case 2: x ∈ int(Fix(f)c) Suppose x ∈ int(Fix(f)c). Let (a,b) be the component interval (open) of (Fix(f))c containing x. Then f(a) = a and f(b) = b, and the map f|(a,b) is a fixed point free self map of (a,b) since f is increasing. Hence by Corollary 2.26, the map f|(a,b) is order conjugate to either x + 1 or x− 1, for which all points are ordinary. This completes the proof. � Remark 2.30. There are continuous maps f : R → R having finitely many equivalence classes (hence only finitely many special points) but infinitely many non-ordinary points. For example, consider the map f(x) = x + sin x on R. There are two classes of fixed points. Since increasing orbits must map to increasing orbits under increasing conjugacies, points like π 2 (increasing orbit) and 3π 2 (decreasing orbit) cannot be equivalent. Hence there must be at least four equivalence classes. To see that there are exactly four equivalence classes, let Ik = (2kπ, (2k + 1)π),Dk = ((2k + 1)π, 2(k + 1)π) and observe that Ik ∩ N(f) = ∅ = Dk ∩N(f) for each k ∈ Z by Proposition 2.29. Hence by Lemma 2.12, each Ik and Dk is contained in a single equivalence class. Conjugacies of the form x 7→ x + 2kπ complete the argument. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 316 Simple dynamical systems 3. Main Results: Class of homeomorphisms Note that, under a topological conjugacy a point can be mapped to a point with similar dynamics. By definition, the points of [x] are dynamically the same, i.e., all have the dynamics similar to that of x. We now consider the systems for which there are only finitely many equivalence classes. This means there are only finitely many kinds of orbits upto conjugacy. For this reason, we call such systems as simple systems. In this paper, we try to understand some simple systems on R. Recall that, if SP denote the set of all points having the dynamical property P then the points of ∂SP (the boundary of SP ) are non-ordinary. In particular, being a point in a particular equivalence class is a dynamical property of the point. Hence, by the very nature of the order conjugacies, it follows that when there are finitely many non-ordinary points (therefore special points) there are only finitely many equivalence classes. These are the simple systems we study in this paper. We describe completely, the homeomorphisms on R, having finitely many non-ordinary points and give a general formula for counting. By Remark 2.11, for systems with finitely many non-ordinary points there are only finitely many equivalence classes. We now study, in the next subsections, the class of simple systems induced by homeomorphisms having finitely many non-ordinary points. 3.1. Class of increasing homeomorphisms. Note that the complement of Fix(f) is a countable union of open intervals (including rays) whose end points are fixed points. Since f is increasing and the end points are fixed, no point in a component interval can be mapped to a point in any other component interval by f. Hence, it is observed that, for an increasing bijection f on R, if Fix(f) c = tIn then f|In is a self map of In, where t denotes the disjoint union. Proposition 3.1. Let f,g be two increasing bijections such that Fix(f) = Fix(g) and let Fix(f) c = tIn. If f|In is order conjugate to g|In for every n, then f is order conjugate to g. Proof. For each n ∈ N, let hn : In → In be an order conjugacy from f|In to g|In . Define h : R → R by h(x) = { hn(x) if x ∈ In x otherwise. Then h is an increasing bijection such that h◦f = g ◦h. � An alphabet is a finite set of letters with at least two elements. A finite sequence of letters from an alphabet is often referred to as a word. For example, if Σ = {a,b} be an alphabet then abab, aaabbbab are words over Σ. Number of letters (may not be distinct) in a word is called its length. Any word of consecutive characters in a word w is said to be a subword of w. Throughout this section we will be working with the alphabet {A, B, O}. Let à = B, c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 317 K. Ali Akbar, V. Kannan and I. Subramania Pillai B̃ = A and Õ = O. If w = w1w2...wn then the dual of w is defined as w̃ = w̃nw̃n−1...w̃1. If w̃ = w then the word w is said to be self conjugate. Here A stands for “above the diagonal” and B stands for “below the diagonal” and O stands for “on the diagonal”. Let f : R → R be an increasing homeomorphism with finitely many non-ordinary (hence special) x1, ...,xn for some n ∈ N. Without loss of generality, assume that x1 < x2 < ... < xn for some n ∈ N. This finite set of points gives rise to an ordered partition {(−∞,x1), (x1,x2), ..., (xn,∞)} of R\{x1,x2, ...,xn}. Note that, on each component interval exactly one of the following holds by Proposition 2.29 (since the only subsets of R with empty boundary are the empty set and R): (i) f(t) > t ∀ t (ii) f(t) < t ∀ t (iii) f(t) = t ∀ t. This gives rise to a word w(f) over {A, B, O} of length n + 1 by associating A to (i), B to (ii) and O to (iii). Note that the subword OO is forbidden. For this, suppose O is occurring at ith and (i+ 1)th place then in both(xi,xi+1) and (xi+1,xi+2) all points are fixed. Then xi+1 becomes ordinary, a contradiction to the assumption that xi+1 is a non-ordinary point. Conversely, suppose a word w of length n + 1 in which OO is forbidden is given. Then we can construct an increasing bijection on R such that its associated word is w, as follows: Take the points 0, 1, 2, ...,n and consider {(−∞, 0), (0, 1), (1, 2), ....., (n,∞)}, a partition of R. If w = w1w2.....wn+1 then associate w1 to (−∞, 0), w2 to (0, 1), ......, and wn+1 to (n,∞). Now it is easy to construct an increasing bijection f : R → R such that w(f) = w. Firstly, define f(t) on (−∞, 0) according to the value of w1 as follows: f(t) =   1 2 t if w1 = A 3 2 t if w1 = B t if w1 = O. Secondly, define f(t) on the remaining subintervals (i−2, i−1) for i = 2, ...,n as follows: To be precise, if i− 2 < t < i− 1, i = 2, 3, ...,n, then consider f(t) =   i− 2 + (t− i + 2)2 if wi = B i− 2 + √ t− i + 2 if wi = A t if wi = O. And, finally, define f(t) on (n,∞) according to the value of wn+1 as follows: f(t) =   1 2 (t−n) + n if wn+1 = B 3 2 (t−n) + n if wn+1 = A t if wn+1 = O. Now by the following proposition the increasing bijection constructed above is unique upto order conjugacy. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 318 Simple dynamical systems Proposition 3.2. Let f, g be two increasing bijection on R with finitely many (same number of ) non-ordinary points. Then f and g are order conjugate if and only if w(f) = w(g). Proof. Suppose w(f) = w(g) = w1w2...wn. Let x1 < x2 < ... < xn and y1 < y2 < ... < yn be the non-ordinary points f and g respectively. The former gives the ordered partition {(−∞,x1), (x1,x2), ...(xn,∞)} of R\{x1,x2, ...,xn} and the later gives the ordered partition {(−∞,y1), (y1,y2), ..., (yn,∞)} of R\{y1,y2, ...,yn}. Now by Proposition 2.29, it follows that both f|(xi,xi+1) and g|(yi,yi+1) are fixed point free self maps (homeomorphisms) for each i and hence by Corollary 2.26, both are order conjugate to x + 1 if wi+1 = A, and order conjugate to x − 1 if wi+1 = B. Hence, by Proposition 3.1, f is order conjugate to g. Converse follows from Corollary 2.26. � Thus we have proved: Proposition 3.3. There is a one to one correspondence between the set of all increasing continuous bijections (upto order conjugacy) on R with exactly n non-ordinary points and the set of all words of length n + 1 on three symbols A, B, O such that OO is forbidden. Now we consider the following proposition. Proposition 3.4. Let an be the number of words of length n + 1 over {A, B, O}, where OO is forbidden. Then an = C1(1 + √ 3)n + C2(1 − √ 3)n, where C1 = (5+3 √ 3) 2 √ 3 and C2 = (3 √ 3−5) 2 √ 3 . Proof. Let An be the set of all words of length n + 1 over {A, B, O} in which OO is forbidden. A general element in An+2 is of the form (i) Aw or Bw for some w ∈ An+1 or (ii) OAv or OBv for some v ∈ An. Therefore an+2 = an+1 + an+1 + an + an since An+2 is the disjoint union of four types of the elements described above. Hence an+2 = 2(an + an+1). This is a linear homogeneous recurrence relation with constant coefficients. The corresponding characteristic equation is α2 − 2α − 2 = 0 which has the two distinct roots α1 = 1 + √ 3 and α2 = 1 − √ 3. It follows that an = C1(1 + √ 3)n + C2(1− √ 3)n, where the constants C1 and C2 can be determined by using the boundary conditions a0 = 3 and a1 = 8. Here C1 = (5+3 √ 3) 2 √ 3 and C2 = (3 √ 3−5) 2 √ 3 . � The following result is one of our principal theorems. It follows from Propo- sitions 3.3 and 3.4. Theorem 3.5. The number of all increasing continuous bijections (upto order conjugacy) on R with exactly n non-ordinary points is an = C1(1 + √ 3)n + C2(1 − √ 3)n, where C1 = (5+3 √ 3) 2 √ 3 and C2 = (3 √ 3−5) 2 √ 3 . For any two continuous map f, g : R → R, we say that f is decreasingly conjugate to g if there is a decreasing homeomorphism h : R → R such that c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 319 K. Ali Akbar, V. Kannan and I. Subramania Pillai h◦f = f ◦h. The following proposition is an analogue of Proposition 3.2, and its proof will be omitted. Proposition 3.6. Let f, g be two increasing bijections on R with finitely many (same number of ) non-ordinary points. Then f and g are decreasingly conjugate if and only if w(g) = w(f). Let tn denote the number of increasing homeomorphisms on R with exactly n non-ordinary points upto topological conjugacy, an denote the number of in- creasing homeomorphisms on R with exactly n non-ordinary points upto order conjugacy, and νn+1 denote the number of self conjugate words of length n + 1 over {A, B, O}. If f, g : R → R are two increasing homeomorphisms such that they are topologically conjugate then either w(g) = w(f) or w(g) = w(f) by Proposition 3.2 and Proposition 3.6. First observe that t0 = 2, t1 = 5, and t2 = 12. In general, we have tn = 1 2 (an −νn+1) + νn+1 for all n ∈ N. Case 1: When n is even. Say n = 2m. A self conjugate word w of length 2m + 1 (OO is forbidden) is of the form w1w2...wmwm+1wm+2...w2m+1 such that wm+1 = O and wm,wm+2 ∈ {A, B} such that wm 6= wm+2. Therefore the number of self conjugate words is 2am−2. Hence t2m = a2m+2am−2 2 for all m ≥ 2. Case 2: When n is odd. Say n = 2m+1. In this case any self conjugate word of length 2m+2 (OO is forbidden) is of the form w1w2...wmwm+1wm+2...w2m+2 such that wm+1,wm+2 ∈{A, B} and wm+1 6= wm+2. Hence the number of self conjugate words of length 2m + 2 is 2am−1. Therefore t2m+1 = a2m+1+2am−1 2 for all m ≥ 1. Thus we have proved: Theorem 3.7. If tn denotes the number of increasing homeomorphisms upto topological conjugacy. Then t0 = 2, t1 = 5 and t2 = 12 by direct computation and for all n ≥ 3 we have: tn =   an+2a n−4 2 2 if n is even an+2a n−3 2 2 if n is odd 3.2. Class of decreasing homeomorphisms. We now ask: Given a whole number n, how many decreasing bijections are there on R upto order conjugacy having exactly n non-ordinary points? For a map f : R → R, we denote the composition of f with itself by f2. Proposition 3.8. Two decreasing bijections f and g are order conjugate (re- spectively topologically conjugate) if and only if f2|[a,∞) and g2|[b,∞) are order conjugate (respectively topologically conjugate), where a and b are the fixed points of f and g respectively. Note that every decreasing homeomorphism has a unique fixed point. Proof. Suppose f and g are order conjugate (respectively topologically con- jugate). Then the same conjugacy between f and g when we restrict forms an order conjugacy (respectively topological conjugacy) between f2|[a,∞) and c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 320 Simple dynamical systems g2|[b,∞). Conversely, suppose f2|[a,∞) and g2|[b,∞) are order conjugate through the increasing homeomorphism h1. Then h1([a,∞)) = [b,∞) and h1(a) = b. Also note that f((−∞,a]) = [a,∞) and g((−∞,b]) = [b,∞). That is, f−1([a,∞)) = (−∞,a] and g−1([b,∞)) = (−∞,b]. Define h : R → R such that h(x) = { h1(x) if x ∈ [a,∞) g−1hf(x) if x < a. If t < a then by definition h◦f(t) = g◦h(t). If t > a then f(t) < a. Therefore h(f(t)) = g−1(h1(f(f(t)))) = g −1(h1(f 2(t))) = g−1(g2(h1(t))) = g(h1(t)) = g(h(t)). Hence h forms an order conjugacy from f to g. � The following proposition is analogous to Proposition 3.8, and its proof will be omitted. Proposition 3.9. Two decreasing bijections f and g are order conjugate (re- spectively topologically conjugate) if and only if f2|(−∞,a] and g2|(−∞,b] are order conjugate (respectively topologically conjugate), where a and b are the fixed points of f and g respectively. A map f : R → R is said to be odd if f(−x) = −f(x) for all x ∈ R. Proposition 3.10. Let f : R → R be an odd increasing bijection. Then, there exist a decreasing homeomorphism fr such that f 2 r = f. Such an fr is called a decreasing square root of f. Proof. Note that f(0) = 0. Define fr such that fr(x) = { −f(x) if x ≥ 0 −x if x < 0. Clearly, fr is a decreasing bijection. Then fr(x) ≤ 0 for all x ≥ 0. Therefore fr(fr(x)) = −fr(x) = f(x). Also we have fr(fr(x)) = fr(−x) = −f(−x) = f(x) for all x < 0. � Remark 3.11. The conclusion of the above proposition is not true in general. For this, let h(x) = { x 2 if x ≥ 0 x if x < 0. Clearly, h is an increasing bijection from R to R. There is no decreasing bijection f : R → R such that f ◦f = h. Let if possible f be one such function. Then we have f(f(x)) = h(x) = x for all x < 0. Choose y > 0 such that f(y) < 0. Therefore f2(f(y)) = f(y) = f(f2(y)). Then f2(y) = y since f is one-one. Therefore h(y) = y, a contradiction since h(y) = y 2 . Proposition 3.12. Let f : (0,∞) → (0,∞) be an increasing bijection. Then there exists a unique decreasing bijection fr : R → R upto order conjugacy such that f2r |(0,∞) = f. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 321 K. Ali Akbar, V. Kannan and I. Subramania Pillai Proof. Let f : (0,∞) → (0,∞) be an increasing bijection. This forces that f(0) := limx→0+f(x) = 0. Any map f : (0,∞) → (0,∞) can be extended uniquely to an odd function f̃ : R → R. Then there exists fr : R → R such that f2r |(0,∞) = f, by Proposition 3.10. This f is unique upto order conjugacy by Proposition 3.9. � Proposition 3.13. Let f : R → R be a decreasing bijection. Then the non- ordinary points of f2 are precisely the non-ordinary points of f. Proof. Suppose x is an ordinary point for f. Then the resut follows from the fact that if h commutes with f then it commutes with f2 also. Conversely, suppose x is an ordinary point of f2. Let the unique fixed point of f be zero, i.e., f(x) = 0 if and only if x = 0. Now let x > 0. Then there exists a neighbourhood (x−δ,x+δ) such that for y in (x−δ,x+δ) there is an h ∈ Gf◦f for which h(x) = y. Then h|(0,∞) is a topological conjugacy between f◦f|(0,∞) and f ◦f|(0,∞). Then h induces h̃ a conjugacy between f and f by Proposition 3.9. By the way h̃ is defined, we have h̃(x) = h(x) = y. Therefore x is an ordinary point of f. � Proposition 3.14. Let f be a decreasing bijection from R to R with fixed point a. Then, f has 2n+1 non-ordinary points if and only if f2|(a,∞) : (a,∞) → (a,∞) has n non-ordinary points. Proof. Suppose that f has 2n + 1 non-ordinary points. Let them be x1 < x2 < ... < xn < xn+1 < xn+2 < ... < x2n+1. Let N = {x1,x2, ...,x2n+1}. Then f(N) ⊂ N by Proposition 2.20. Since f is a decreasing bijection, we have f(N) = N and a = xn. Hence f 2|(a,∞) : (a,∞) → (a,∞) has n non- ordinary points. Conversely, suppose f2|(a,∞) : (a,∞) → (a,∞) has n non- ordinary points. Then by Proposition 3.14, we have N(f) = N(f2|(a,∞)) ∪ f(N(f2|(a,∞))) ∪{a}. Thus f has 2n + 1 non-ordinary points. � Remark 3.15. From the above proposition it follows that there does not exist a decreasing homeomorphism with even number of non-ordinary points. Theorem 3.16. If sn denotes the number of decreasing homeomorphisms upto order conjugacy, then sn = { 0 if n is even an−1 2 if n is odd for all n. Proof. By Proposition 3.14, we have s2n = 0 for each n ∈ N. Now we will prove that s2n+1 = an for all n ∈ N. Let f : R → R be a decreasing bijection with 2n + 1 non-ordinary points. Without loss of generality we can assume that 0 is the unique fixed point. Then g = f2|(0,∞) : (0,∞) → (0,∞) is an increasing bijection with n non-ordinary points. Since (0,∞) is homeomorphic to R, we get an increasing homeomorphism g′ : R → R (unique upto order conjugacy) with n non-ordinary points and which is order conjugate to g. On the other c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 322 Simple dynamical systems hand, suppose h : R → R is an increasing bijection with n non-ordinary points. Since (0,∞) is homeomorphic to R, corresponding to each h we have a unique (upto order conjugacy) increasing bijection h′ : (0,∞) → (0,∞) with n non- ordinary points. Then by Proposition 3.10, there exists unique (upto order conjugacy) decreasing square root f : R → R for h′ such that f ◦f|(0,∞) = h′. By proposition 3.14, f has 2n+ 1 non-ordinary points. Thus, there is a one-one correspondence between the set of all increasing bijections with n non-ordinary points (upto order conjugacy) and the set of all decreasing bijections with 2n+1 non-ordinary points (upto order conjugacy). Hence s2n+1 = an. � Theorem 3.17. If kn denotes the number of decreasing homeomorphisms having n non-ordinary points upto topological conjugacy, then kn = { 0 if n is even tn−1 2 if n is odd for all n. Proof. If n is even then we have kn = 0 since sn = 0. If n is odd then we will argue as in Theorem 3.16 to prove that there is a one-one correspondence between the set of all increasing bijections (upto topological conjugacy) on R having n non-ordinary points and the set of all decreasing bijections (upto topological conjugacy) on R with 2n + 1 non-ordinary points. Thus k2n+1 = tn. � 4. Summary We conclude this paper with the following table: n an sn tn kn 0 3 0 2 0 1 8 0 5 2 2 22 0 12 0 3 60 8 33 5 4 164 0 85 0 5 448 22 232 12 where an be the number of increasing bijections on R with exactly n non- ordinary points upto order conjugacy, tn be the number of increasing bijections on R with exactly n non-ordinary points upto topological conjugacy, sn be the number of decreasing bijections on R with exactly n non-ordinary points upto order conjugacy, and kn be the number of decreasing bijections on R with exactly n non-ordinary points upto topological conjugacy. Acknowledgements. The authors are thankful to the referee for his/her valuable suggestions. The first author acknowledges UGC, INDIA for financial support. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 323 K. Ali Akbar, V. Kannan and I. Subramania Pillai References [1] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Volume 1513 of Lecture Notes in Mathematics, Springer-Verlag, Berline, 1992. [2] L. Block and E. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987), 297–306. [3] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002. [4] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Pub- lishing Company Advanced Book Program, Redwood City, CA, second edition, 1989. [5] R. A. Holmgren, A First Course in Discrete Dynamical Systems, Springer-Verlag, New York, 1996. [6] S. Sai, Symbolic dynamics for complete classification, Ph.D Thesis, University of Hy- derabad, 2000. [7] B. Sankara Rao, I. Subramania Pillai and V. Kannan, The set of dynamically special points, Aequationes Mathematicae 82, no. 1-2 (2011), 81–90. [8] A. N. Sharkovskii, Coexistence of cycles of a continuous map of a line into itself, Ukr. Math. Z. 16 (1964), 61–71. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 2 324