@ Appl. Gen. Topol. 23, no. 1 (2022), 135-143 doi:10.4995/agt.2022.15613 © AGT, UPV, 2022 Topological transitivity of the normalized maps induced by linear operators Pabitra Narayan Mandal School of Mathematics and Statistics, University of Hyderabad, Hyderabad - 500046, India. (pabitranarayanm@gmail.com) Communicated by F. Balibrea Abstract In this article, we provide a simple geometric proof of the following fact: The existence of topologically transitive normalized maps induced by linear operators is possible only when the underlying space’s real dimension is either 1 or 2 or infinity. A similar result holds for projective transformation as well. 2020 MSC: 47A16; 37B05. Keywords: topological transitivity; supercyclicity; projective transforma- tion; linear transformation; cone transitivity. 1. Introduction In general, a topological dynamical system is a pair (X,f), where X is a topological space and f is a self map on X. The study of dynamics is mainly about the eventual behavior of orbits. In our setting, we take X as a metric space and f as a continuous self map on X. For x ∈ X, the f- orbit of x is denoted by O(f,x) and is defined by {fn(x) : n ∈ N0} where N0 = N ∪{0} and N = {1, 2, 3, ...}. Here fn(x) = f ◦ f ◦ ... ◦ f(x) (n-times) and f0(x) = I(x) = x (I denotes the identity map). Now f is said to have dense orbit if there exists x ∈ X such that O(f,x) is dense in X and f is said to be topologically transitive if for any two non-empty open sets U, V in X there exists n ∈ N such that fn(U) ∩ V 6= ∅. For several equivalent formulations of topological transitivity, see [3], [9]. Having dense orbit and Received 13 May 2021 – Accepted 28 December 2021 http://dx.doi.org/10.4995/agt.2022.15613 https://orcid.org/0000-0002-8020-3371 P. Narayan Mandal topological transitivity are not equivalent notions in general. However, in some ‘nice spaces’ both the notions are equivalent. For example, if X is a separable complete metric space without isolated points. Then the following assertions are equivalent: (i) f is topologically transitive; (ii) there exists x ∈ X such that O(f,x) is dense in X. This is known as Birkhoff transitivity theorem (see [6]). Let T : X → X be an invertible continuous linear operator, where X is a real separable Hilbert space. Hereafter, in this article, X denotes a real separable Hilbert space, unless otherwise mentioned. Let L(X) be the set of all continuous linear operators on X and GL(X) be the set of all invertible continuous linear operators on X. The map T : X → X induces a map T : SX → SX, where SX := {x ∈ X : ||x|| = 1} and is defined by Tx = Tx ||Tx||, whenever T ∈ GL(X). We call T as the normalized map induced by T . There are some excellent monographs and expository articles which dealt with dynamics of linear operators in great detail (see [2], [4], [6]). Here we are interested to study a particular notion namely, the topological transitivity of the map T on SX. More precisely, we ask the following questions: Question 1.1. For a given X, does there exist T ∈ GL(X) such that T is topologically transitive on SX? If the answer to the above is positive, we would like to investigate the fol- lowing also. Question 1.2. What are all invertible continuous linear operators whose nor- malized maps are topologically transitive? Let dim(X) be the Hilbert dimension of X i.e., the cardinality of an or- thonormal basis of X. In this article, as an answer to the Question 1.1, we show the following: Theorem 1.3. There exists T ∈ GL(X) such that T is topologically transitive on SX if and only if dim(X) ∈{1, 2,∞}. In the context of linear dynamics, topological transitivity and having dense orbit are equivalent (due to Birkhoff transitivity theorem) and it is known as hy- percyclicity. Supercyclicity and Positive supercyclicity are two weaker notions than the notion of hypercyclicity. A linear operator T is said to be supercyclic (resp. positive supercyclic) if there exists a vector x ∈ X whose projective orbit (resp. positive projective orbit) R.O(T,x) (resp. R+.O(T,x)) defined by {λTn(x) : n ∈ N0 and λ ∈ R} (resp. {λTn(x) : n ∈ N0 and λ ∈ R+}) is dense in X, where R and R+ denote the set of all real numbers and the set of all positive real numbers respectively. For equivalent formulations on these notions, one can refer [2], [6]. In [7], G. Herzog proved that supercyclic vector exists if and only if the dimension of the space is 1, 2, ∞ (over real) and 1, ∞ (over complex). Proof of Herzog’s theorem is based on the techniques from functional analysis and operator theory. Here we essentially reprove the Her- zog’s theorem in the light of dynamics on unit sphere. Although our proof is © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 136 Topological transitivity of the normalized maps induced by linear operators relatively longer, it is still simple. It is intuitive according to our expositions and based on basic and well-known techniques from linear algebra and theory of dynamical systems. In the end, we also list all such invertible continuous lin- ear operators whose normalized maps are topologically transitive as a possible answer to the Question 1.2. Similarly, the map T ∈ GL(X) induces a map T̃ : SX/∼→ SX/∼, where SX/∼ is the quotient space of SX under the relation ∼. Here ‘∼’ identifies the antipodal points. The map T̃ is known as real projective transformation. For a detailed study on dynamics of projective transformation, see [4], [8]. One can ask similar questions as above and expect that similar results also hold for real projective transformation. 2. Cone Transitivity and Basic Properties In this section, we introduce a weak notion of topological transitivity, which we call cone transitivity. This notion helps us to study the dynamics of the normalized maps induced by linear operators. First, we start with a definition. Definition 2.1 (Open cone). An open set V ⊂ X is said to be an open cone if λV ⊂ V for any λ > 0. Remark 2.2. For any non-empty open cone V , there exists a non-empty open subset S (namely SX ∩ V ) of SX (in the subspace topology) such that V = ∪x∈SLx, where Lx := {λx : λ > 0}. Definition 2.3 (Cone transitivity). A T ∈ L(X) is said to be cone transitive, if for any two non-empty open cones U, V in X, there exists n ∈ N such that Tn(U) ∩V 6= ∅. We now provide an alternative definition for the cone transitivity in the following theorem, which is analogous to Birkhoff transitivity theorem. Theorem 2.4. Let T ∈ L(X). Then the following are equivalent: (i) There exists x ∈ X such that for any non-empty open cone V , Tnx ∈ V for some n ∈ N. (ii) T is cone transitive. Proof. Let U and V be two non-empty open cones. By (i), there exists x ∈ X such that Tn(x) ∈ V and Tm(x) ∈ U for some m,n ∈ N. Since X does not contain any isolated point, without loss of generality we can choose n > m. Therefore Tn−m(U) ∩V 6= ∅. Hence, (i) implies (ii). Since X is separable, it has a countable dense set say {v1,v2,v3, ...}. Con- sider the open balls V ′k of radius � > 0 around vk for each k ≥ 1, form a countable base of the topology of X. Let Vk be the smallest open cone con- taining V ′k. If T −n(Vk) is the n-th pre-images of Vk, then ∪∞n=0T−n(Vk) is the set of points which visit Vk at least once. We claim that ∪∞n=0T−n(Vk) is dense in X. If not, then there exists an non-empty open set U′ in X such that U′ ∩∪∞n=0T−n(Vk) = ∅. Therefore for each n ∈ N, U′ ∩T−n(Vk) = ∅. Since © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 137 P. Narayan Mandal Vk is a cone and T is linear, we have for any λ > 0, λU ′∩T−n(Vk) = ∅. There- fore if U is the smallest open cone containing U′, then U ∩T−n(Vk) = ∅ i.e., Tn(U)∩Vk = ∅. This is a contradiction to the fact that T is a cone transitive operator. Hence ∪∞n=0T−n(Vk) is dense in X. Therefore ∩∞k=1 ∪ ∞ n=0 T −n(Vk) is non-empty, by Baire’s category theorem. Hence, (ii) implies (i). � Remark 2.5. If T is a cone transitive operator, then the x defined in Theorem 2.4 is called a cone transitive vector of T . We denote by CV (T) the set of all cone transitive vectors of T . Here X does not contain any isolated point. Therefore there exists a sequence of natural numbers (nk) such that T nkx ∈ V where nk →∞ and O(x,T) ⊂ CV (T). One can show that the notions of cone transitivity and positive supercyclic- ity are equivalent. To keep our exposition self-contained and geometrically intuitive, we prefer to use cone transitivity instead of positive supercyclicity. Let us see some examples of cone transitivity: Example 2.6. Any topologically transitive continuous linear operator in infi- nite dimensional space (known as Hypercyclic operator) is cone transitive. To see this: take any two non-empty open sets as any two non-empty open cones. In the next example, we see that there exists a cone transitive linear op- erator which is not topologically transitive. Furthermore, cone transitivity is not an infinite dimensional property like the topological transitivity for linear operators. Example 2.7. Let T := ( cos θ sin θ −sin θ cos θ ) , where θ π ∈ R\Q. Here T = T |SR2 . Let U and V be two open cones in R 2. Then U′ := U∩SR2 and V ′ = V ∩SR2 are two non-empty open sets in SR2 . It is well-known that any irrational circular rotation is topologically transitive on SR2 . Therefore there exists n ∈ N such that T n (U′)∩V ′ 6= ∅. Hence Tn(U)∩V 6= ∅ for some n ∈ N. Therefore T is a cone transitive operator on R2. We ask the following question: Question 2.8. Does there exist any relation among the cone transitivity of T on X and the topological transitivity of T on SX? First, observe that a non-invertible linear operator can not be cone transitive as T(X) is a proper subspace of X, when dim(X) < ∞. However, we find an affirmative answer to Question 2.8. In fact, both the notions are equivalent. In this section, we prove the equivalence together with some basic properties of cone transitivity, which are useful throughout our discussions. Theorem 2.9. Let T ∈ GL(X). Then T : SX → SX is topologically transitive if and only if T : X → X is cone transitive. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 138 Topological transitivity of the normalized maps induced by linear operators Proof. (⇒) Let U and V be two non-empty open cones in X. In view of Remark 2.2, U′ := U ∩SX and V ′ := V ∩SX are two non-empty open sets in SX. Since T : SX → SX is topologically transitive, there exists n ∈ N such that T n (U′) ∩V ′ 6= ∅, i.e., there exists x ∈ U′ such that T n x ∈ V ′. Since V is an open cone containing V ′, we have Tnx ∈ V . This shows that Tn(U) ∩V 6= ∅. Hence T is cone transitive. (⇐) Let U′ and V ′ be two non-empty open sets in SX. Take U := ∪λ>0λU′, the smallest open cone containing U′ and V := ∪λ>0λV ′, the smallest open cone containing V ′. Since T is cone transitive, there exists n ∈ N such that Tn(U)∩V 6= ∅ i.e., there exists x ∈ U such that Tnx ∈ V . Consider x′ := x||x||. In view of Remark 2.2, x′ ∈ U′. Since V is an open cone, we have Tnx′ ∈ V . Therefore T n (x′) = T nx′ ||Tnx′|| ∈ V ′. This shows that T n (U′) ∩V ′ 6= ∅. Hence T is topologically transitive. � Proposition 2.10. Let T ∈ L(X). If x is a cone transitive vector of T, then λx is also a cone transitive vector of T for any λ > 0. Proof. Let V be any non-empty open cone. Since x is cone transitive vector of T , there exists n ∈ N such that Tnx ∈ V . Therefore Tn(λx) = λTnx ∈ λV ⊂ V for λ > 0. Hence λx is also a cone transitive vector of T for any λ > 0. � Remark 2.11. For any T ∈ L(X), TV (T) = { x||x|| : x ∈ CV (T)} and CV (T) := {λx : x ∈ TV (T) and λ > 0}, where TV (T) is the set of all transitive vectors for T . Moreover CV (T) is either empty or dense in X. Definition 2.12 (Linear conjugacy). Let T,S ∈ L(X). Now T and S are said to be linearly conjugate if there exists P ∈ GL(X) such that S = P−1TP. Proposition 2.13. Cone transitivity is preserved by linear conjugacy. Proof. Let T ∈ L(X) be cone transitive. We claim that for any P ∈ GL(X), S = P−1TP is also cone transitive. Let U be a non-empty open set and V be a non-empty open cone. Then observe that P(U) is a non-empty open set and P(V ) is a non-empty open cone. Since T is cone transitive, we have Tn(P(U)) ∩P(V ) 6= ∅ for some n ∈ N. On the other hand, PSn = TnP . We conclude that PSn(U)∩P(V ) 6= ∅, which readily gives Sn(U)∩V 6= ∅. Hence S is cone transitive. � Proposition 2.14. If T,S ∈ GL(X) are linearly conjugate, then T, S are topologically conjugate. Proof. Let S and T be two invertible linear operators which are linearly con- jugate. We claim that S and T are topologically conjugate. Since S and T are linearly conjugate, there exists a invertible linear operator P on X such that S = P−1TP. For x ∈ X, P−1 T Px = P−1 T( Px||Px||) = P −1( T(Px)/||Px|| ||TPx||/||Px||) = P−1TPx/||TPx|| ||P−1TPx||/||TPx|| = P−1TPx ||P−1TPx|| = Sx. Take h = P , then S = h −1Th. � Theorem 2.15. If T ∈ GL(X) is cone transitive, then T−1 is also cone tran- sitive. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 139 P. Narayan Mandal Proof. Observe that T −1 = T−1. Again the proof goes via topological transi- tivity of T on SX, by Theorem 2.9 and Proposition 1.14 in [6]. � Remark 2.16. If dim(X) ≥ 2, we may use Birkhoff transitivity theorem to show that topological transitivity of T and having dense orbit of T are equivalent. We claim that both the notions are also equivalent for dim(X) = 1. In this case, X ∼= R and SX ∼= {−1, +1}. If T is not topologically transitive but it contains a dense orbit, then we get ||T(1)|| = −||T(−1)||, which is absurd. It is obvious that SX is a Baire space with a countable basis of open sets. Therefore topological transitivity of T ensures a dense orbit of T. Hence in our setting both the notions are equivalent. Theorem 2.17 (A necessary condition). Let Ti ∈ L(Xi) for i = 1, 2 and T := T1 ⊕ T2 : X1 ⊕ X2 → X1 ⊕ X2, where each Xi is real separable Hilbert space. If T is topologically transitive on SX1 ⊕ X2, then Ti is also topologically transitive on SXi for i = 1, 2. Proof. Since T is topologically transitive on SX1 ⊕ X2 , there exists (x1,x2) ∈ SX1 ⊕ X2 such that {T n (x1,x2) : n ∈ N} is dense in SX1 ⊕ X2 . It is enough to prove that Ti is cone transitive on Xi for i = 1, 2. First, we claim that x1 is cone transitive vector of T1. If not, then there exists an open cone V1 in X1 such that V1 ∩{Tn1 x1 : n ∈ N} = ∅. This implies λTn1 x1 /∈ V1 for any λ > 0. Let V2 be any open cone in X2. Then V1 × V2 is an open cone in X1 ⊕ X2. Since λTn1 x1 /∈ V1 for any λ > 0, we have (Tn1 x1,T n 2 x2) ||(Tn1 x1,T n 2 x2)|| /∈ V1 ×V2. Therefore T is not topologically transitive on SX1 ⊕ X2 , which is a contradiction. � The converse of the above theorem is not true in general. We encounter this on several occasions in the proof of the main result. 3. Main result Theorem 3.1 (Main theorem). If dim(X) ∈{1, 2,∞}, then there exists a T ∈ GL(X) such that T is topologically transitive on SX. If dim(X) /∈ {1, 2,∞}, then there exists no T ∈ GL(X) such that T is topologically transitive on SX. Proof. The proof depends on the dimension of the space X. Thus we prove the result by considering various cases. Case A: Let dim(X) = 1. Then X ∼= R and SX ∼= {+1,−1}. Take Tx = −x, where x ∈ X. Then T is topologically transitive. Case B: Let dim(X) = 2. Then X ∼= R2 and SX ∼= SR2 := {(x1,x2) ∈ R2 : |x1|2 + |x2|2 = 1} = S1. Take T := ( cos θ sin θ −sin θ cos θ ) , where θ π is an irrational number. Then T : S1 → S1 is an irrational rotation and it is well-known that irrational rotation is topologically transitive on SR2 (for a proof, see Example 1.12 in [6]). Case C: Let dim(X) = ∞. Since X is an infinite dimensional separable Hilbert space, we have X ∼= l2(Z) (see [5]). Let T be an invertible hypercyclic operator (see [6]). Then T : SX → SX is topologically transitive. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 140 Topological transitivity of the normalized maps induced by linear operators Case D: Let 2 < dim(X) < ∞. If S is a linear operator on X, then using real Jordan canonical form, S has at least one block-diagonal which is linearly conjugate to one of the following: (1)   r cos θ r sin θ 0 0 −r sin θ r cos θ 0 0 0 0 s cos φ s sin φ 0 0 −s sin φ s cos φ   on R4, where r,s ∈ R\{0}; (2)   λ 1 0 ... 0 0 λ 1 ... 0 .. .. .. .. .. .. .. .. .. .. 0 0 .. λ 1 0 0 0 ... λ   on R n, where λ ∈ R\{0}; (3)   J1 I2 0 ... 0 0 J2 I2 ... 0 .. .. .. .. .. .. .. .. .. .. 0 0 .. Jn−1 I2 0 0 0 ... Jn   on R 2n, where Ji = ( cos θ sin θ −sin θ cos θ ) and I2 = ( 1 0 0 1 ) ; (4)  α 0 00 r cos φ r sin φ 0 −r sin φ r cos φ   on R3, where α,r ∈ R\{0}; (5) ( α 0 0 β ) on R2, where α,β ∈ R\{0}. We now show that each of the above representation can not be cone transitive on their respective invariant subspaces. For (1): Let S |R4 = T . By Theorem 2.15, we may assume either |s| ≥ |r|, |s| > 1 or |s| = |r| = 1. If possible, let us assume (x,y,u,v) be a cone transitive vector of T such that ||(x,y, 0, 0)|| ≤ ||(0, 0,u,v)||. Identify R4 as xyuv-space and Tn(x,y,u,v) = (xrn cos nθ + yrn sin nθ,−xrn sin nθ + yrn cos nθ,usn cos nφ + vsn sin nφ,−usn sin nφ + vsn cos nφ). It is clear from the definition that the orbit of (x0,y0, 0, 0) lies on xy-plane and the orbit of (0, 0,u0,v0) lies on uv- plane. Since (x,y,u,v) is a cone transitive vector of T , we have (x,y) 6= (0, 0) and (u,v) 6= (0, 0). Let (x′,y′, 0, 0) be any point in the xy-plane. Then ||Tn(x,y,u,v) − (x′,y′, 0, 0)|| ≥ ||(0, 0,usn cos nφ + vsn sin nφ,−usn sin nφ + vsn cos nφ)|| = |s|n||(0, 0,u,v)|| and ||Tn(x,y,u,v)|| ≤ 2 × max{|r|n, |s|n} ≤ Mn, for some M. If |s| > 1, then we take an open 4-ball centered at (x,y, 0, 0) with radius r′, where r′M < ||(x,y,0,0)|| 2 . Let V be the smallest cone containing the 4-ball. Then Tn(x,y,u,v)) /∈ V , for any n ∈ N. If |s| = |r| = 1, then observe that T = T on SR4 . If (x′,y′, 0, 0) is any point in SR4 , similarly, we have ||T n (x,y,u,v) − (x′,y′, 0, 0)|| ≥ ||(0, 0,u,v)||. This means T is not topologically transitive on SR4 . © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 141 P. Narayan Mandal For (2): Let S |Rn = T. By Theorem 2.15, we may assume that |λ| ≥ 1. Let (x1,x2, ...,xn) be any point in Rn. Let B be an open ball centered at (0, 0, ...,xn) with radius r < ||(0, 0, ...,xn)|| and V be the smallest open cone B. It is straightforward to verify that there exists C > 0 such that for every (α1,α2, ...,αn) ∈ V , we have |α1| |αn| < C. In particular, if r < ||(0,0,...,xn)|| 2 , then C < 1. If for some N ≥ n0, we have TN (x) ∈ V , then |λNx1 + NλN−1x2 + ... + ( N n− 1 ) λN−n+1xn| < C|λNxn| or, |x1 + N λ x2 + ... + ( N n− 1 ) 1 λn−1 xn| < C|xn|. Observe that it is not true, if we choose n0 large enough. Therefore there exists a n0 ∈ N, such that Tm(x) /∈ V for m ≥ n0. For (3): The proof is in similar lines as that of the previous case (i.e., (2)). Thus we omit the details. For (4): Let S |R3 = T . Here Tn(x1,x2,x3) = (αnx1,rnx2 cos nφ+rnx3 sin nφ, −rnx2 sin nφ + rnx3 cos nφ) for any (x1,x2,x3) ∈ R3. If |α| > |r|, then take a sphere centered at (0,x2,x3) with radius r ′, where r′ < ||(0,x2,x3)|| 2 . If V be the smallest cone containing the sphere, then we claim that there exists a n0 ∈ N such that Tn(x1,x2,x3) /∈ V for n ≥ n0. If possible, for some N ≥ n0, we have TN (x1,x2,x3) ∈ V . Then |α|N||(x1,0,0)|| |r|N||(0,x2,x3)|| < 1, which is not true if we take n0 large enough. If |r| > |α|, then we take the sphere centered at (x1, 0, 0) with radius r′′ < ||(x1,0,0)|| 2 . If |α| = |r|, then ||T n(x1,0,0)|| ||Tn(0,x2,x2)|| = ||(x1,0,0)|| ||(0,x2,x3)|| . Hence T is not cone transitive. For (5): Let S |R2 = T . Observe that for any point (x,y), there exists an arbi- trarily small � > 0 such that the smallest open cone containing B((x,y),�) con- tains only finitely many Tn(x,y) when |α| 6= |β|. If |α| = |β|, then all Tn(x,y) is in the smallest cone containing B((x,y),�), B((−x,y),�), B((x,−y),�) and B((−x,−y),�). Hence in both the cases, T is not cone transitive. By Theorem 2.15, S is not cone transitive, whenever 2 < dim(X) < ∞. Hence the result. � Remark 3.2. The proof of Theorem 3.1 suggests that a similar argument can also ensure the non-existence of topologically transitive projective transforma- tion on SX/∼, when 2 < dim(X) < ∞. In this case instead of taking a positive open cone, we may need to take a full open cone i.e., instead of λ > 0, we take λ 6= 0. Remark 3.3. In contrast with the above result, for n ≥ 2, every n-dimensional compact manifold admits a chaotic homeomorphism. For a proof, see [1]. In particular, for each n ∈ N, there exists a homeomorphism hn on SRn such that hn is topologically transitive on SRn . © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 142 Topological transitivity of the normalized maps induced by linear operators We conclude by providing a complete list of linear operators whose normal- ized maps are topologically transitive (similarly, for real projective transfor- mation), which is apparent from the above discussion. Since linear conjugacy preserves cone transitivity, we make a list as follows: dim(X) T ∈ GL(X) such that (T,SX) is topologically transitive T ∈ GL(X) such that (T̃,SX/ ∼) is topologically transitive 1 rI, for r < 0. rI, for r 6= 0. 2 ( r cos θ r sin θ −r sin θ r cos θ ) , where θ π ∈ R\Q and r 6= 0. ( r cos θ r sin θ −r sin θ r cos θ ) , where θ π ∈ R\Q and r 6= 0. ∞ Invertible positive supercyclic operators. Invertible supercyclic operator. Acknowledgements. I profusely thank the anonymous referee for a careful reading of the manuscript and for providing helpful suggestions that signifi- cantly improved the original manuscript. I sincerely thank Prof. V. Kannan and Dr. T. 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