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An example of a chaotic group action on Euclidean space by compactly supported homeomorphisms
Topology and its Applications 155 (2007) 161–164 www.elsevier.com/locate/topol
An example of a chaotic group action on Euclidean space by compactly supported homeomorphisms Grant Cairns ∗ , Thanh Duong Pham Department of Mathematics, La Trobe University, Melbourne, Australia 3086 Received 7 May 2007; received in revised form 13 October 2007; accepted 13 October 2007
Abstract We give an example of a group action on Euclidean space for which each map in the action is compactly supported, and such that the action is chaotic in the sense that it is topologically transitive and the set of points with finite orbit is dense. This resolves a conjecture of Naolekar and Sankaran. © 2007 Elsevier B.V. All rights reserved. Keywords: Chaos; Group action
1. Introduction We say that an action by homeomorphisms of a group G on a topological space X is chaotic if the following two conditions are satisfied: (i) the set of points with finite G-orbit is dense in X, (ii) the action is topologically transitive in the sense that for all nonempty open sets U, V ⊆ X, there exists g ∈ G for which gU ∩ V is nonempty. This is the natural generalization to group actions of Devaney’s definition of chaos for functions [1] and has been studied in [2–4]. In [6], Naolekar and Sankaran conjectured that on Euclidean space, there are no chaotic group actions via compactly supported homeomorphisms. They established this in several cases, including the cases of finitely generated groups and solvable groups. We give counterexamples to this conjecture on Rn , for all n 2. The n = 2 counterexample is quite easy to describe so we will treat this case before turning to the general case. 2. Counterexample in the plane Consider the torus T2 = R2 /Z2 , which for convenience we identify with the square with vertices (± 12 , ± 12 ) and opposing edges identified in the usual manner. For each i ∈ N, consider the following sets (see Fig. 1): Hi = (x, y) ∈ T2 : 2−1 − 2−i |y| 2−1 − 2−i−1 , Vi = (x, y) ∈ T2 : 2−1 − 2−i |x| 2−1 − 2−i−1 . * Corresponding author.
E-mail addresses: [email protected] (G. Cairns), [email protected] (T.D. Pham). 0166-8641/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2007.10.002
162
G. Cairns, T.D. Pham / Topology and its Applications 155 (2007) 161–164
Fig. 1.
Let φ and ψ be the linear automorphisms of T2 defined by the matrices
1 2 01
and
1 0 21
respectively. It is well known i
[3]. For i ∈ N, set φi = φ 2 on Hi and set φi that the group Γ = φ, ψ generated by φ and ψ acts chaotically on i to be the identity map elsewhere; similarly, set ψi = ψ 2 on Vi and set ψi to be the identity map elsewhere. Notice that φi (resp. ψi ) is the identity on the boundary of Hi (resp., Vi ); for example, x 1 2i+1 x x = = (mod 1). φi 2−1 − 2−i−1 2−1 − 2−i−1 0 1 2−1 − 2−i−1 T2
It follows that the maps φi and ψi are homeomorphisms. Consider the group G = φi , ψi : i ∈ N generated by all the φi and ψi . Since each generator of G is the restriction of an element of Γ , every point with finite Γ -orbit also has finite G-orbit. Thus the points with finite G-orbit form a dense subset of T2 . The action of G on T2 is also topologically transitive; this can be deduced from [5], or may be established directly as follows: given nonempty open sets U, V ⊆ T2 , choose i, j so that U ∩ Hi and V ∩ Vj are both nonempty. Then there exist k, l such that (φik U ) ∩ (ψjl V ) is nonempty. Now recall that the quotient of T2 under the map (x, y) → (−x, −y) is homeomorphic to the 2-sphere S 2 . The quotient map π : T2 → S 2 is a 2-fold cover ramified over 4 points. (As a geometry, the quotient S 2 is an orifold commonly called a pillow; see [7, Chapter 13.1.4].) Clearly the G-action on T2 induces a faithful G-action on S 2 such that the map π is a semi-conjugacy. It follows that since the G-action on T2 is chaotic, the G-action on S 2 is also chaotic. The point z = π(± 12 , ± 12 ) is a fixed point of this action; removing this point we obtain a faithful chaotic action of G on S 2 \{z} ∼ = R2 . By construction, the elements of this action are compactly supported. This provides the required example. 3. Counterexample in higher dimensions We now provide a counterexample in Rn for n > 2. The idea is to first build a slightly more complicated 2-dimensional example, and then employ the skew product construction as in [3]. Set H1 = {(x, y) ∈ T2 : 0 |y| 1/4}, V1 = {(x, y) ∈ T2 : 0 |x| 1/4} and for each i > 1, consider the following sets (see Fig. 2): Hi = (x, y) ∈ T2 : 3/8 − 2−i−1 |y| 3/8 − 2−i−2 ,
G. Cairns, T.D. Pham / Topology and its Applications 155 (2007) 161–164
163
Fig. 2.
Vi = (x, y) ∈ T2 : 3/8 − 2−i−1 |x| 3/8 − 2−i−2 . Consider the set 3 3 . N := (x, y): |x| or |y| 8 8 We define homeomorphisms φi , ψi of N for each i ∈ N, as follows. For the linear automorphisms φ and ψ of T2 i+1 i+1 defined above, set φi = φ 2 on Hi and set φi to be the identity map elsewhere on N ; similarly, set ψi = ψ 2 on Vi and set ψi to be the identity map elsewhere. Notice that φi (resp. ψi ) is the identity on the boundary of Hi (resp. Vi ). Consider the group G = φi , ψi : i ∈ N. The action of G on N is obviously faithful and by the same argument as we used above, it is chaotic on N . Notice that by construction, this action is trivial on the boundary of N . Now we construct a chaotic action on the closed ball B n of dimension n > 2. Consider the skew product, for each g ∈ G, f (g) : N × Tn−2 → N × Tn−2 , (x1 , x2 , . . . , xn ) → g(x1 , x2 ), x3 + h3 (x), . . . , xn + hn (x) , where the maps hj : N → S 1 are defined by 2j if|x | 1/4, 1 hj (x) = (4x1 ) 0 otherwise, for x = (x1 , x2 ) and j ∈ N. Notice that for all g ∈ G, the map f (g) is the identity on the boundary of N × Tn−2 . We will show that this action of G on N × Tn−2 is chaotic. In order to directly apply the results of [3], consider the set 1 1 . M := (x, y): |x| or |y| 4 4 As shown in the proof of [3, Lemma 3.2], f (φ1 ◦ ψ1 ) is a chaotic homeomorphism of M × Tn−2 . Let U and V be two arbitrary nonempty open subsets in N × Tn−2 . Assume that U = U1 × U and V = V1 × V , where U1 and V1 are nonempty open subsets in N , and U , V are nonempty open subsets in Tn−2 . If U1 and V1 both intersect M then since f (φ1 ◦ ψ1 ) is topologically transitive on M × Tn−2 , there is g ∈ G such that f (g)(U ) ∩ V = ∅.
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If U1 or V1 do not intersect M, then since G acts chaotically on N , there exist g1 , g2 ∈ G such that g1 (U1 ) ∩ M = ∅ and g2 (V1 ) ∩ M = ∅. So from above, there is g ∈ G such that f (g)(f (g1 )(U )) ∩ (f (g2 )(V )) = ∅. So we have f (g2−1 ◦ g ◦ g1 )(U ) ∩ V = ∅. Hence the action of G on N × Tn−2 is topologically transitive. Since the points in N × Tn−2 with rational coordinates are periodic, the set of periodic points is dense in N × Tn−2 . Therefore, G is chaotic on N × Tn−2 . The quotient of N under the map (x, y) → (−x, −y) is homeomorphic to the closed disc D 2 . So dividing N ×Tn−2 by the map σ : (x1 , x2 , x3 , . . . , xn ) → (−x1 , −x2 , x3 , . . . , xn ) one obtains a chaotic action of G on the solid torus D 2 × Tn−2 which is again chaotic and trivial on the boundary. Now consider the construction of the closed ball B n from D 2 × Tn−2 as in the proof of [3, Lemma 3.2]. Since B n is constructed from D 2 × Tn−2 by a succession of identifications on the boundary of the space and g is the identity on the boundary of D 2 × Tn−2 , the action of G on D 2 × Tn−2 is semi-conjugate to a chaotic action on the quotient space B n [3, Lemma 3.2]. Moreover, by construction, each map in the action is compactly supported and is the identity on the boundary of the ball. Removing the boundary, we get the required action on Rn . Remark 1. Let D denote the direct sum of countably infinitely many copies of the infinite cyclic group. The group G of the above construction is isomorphic to the free product G = D ∗ D. References [1] [2] [3] [4]
J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (4) (1992) 332–334. G. Cairns, G. Davis, D. Elton, A. Kolganova, P. Perversi, Chaotic group actions, Enseign. Math. (2) 41 (1–2) (1995) 123–133. G. Cairns, A. Kolganova, Chaotic actions of free groups, Nonlinearity 9 (4) (1996) 1015–1021. G. Cairns, A. Kolganova, A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math. 37 (2) (2007) 371–397. [5] R.L. Devaney, Linked twist mappings are almost Anosov, Global theory of dynamical systems, in: Proc. Internat. Conf., Northwestern Univ., Evanston, Ill, 1979, in: Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 121–145. [6] A.C. Naolekar, P. Sankaran, Chaotic group actions on manifolds, Topology Appl. 107 (3) (2000) 233–243. [7] W.P. Thurston, The geometry and topology of three-manifolds Lecture notes; available at http://www.msri.org/communications/books/gt3m, 1980.
An example of a chaotic group action on Euclidean space by compactly supported homeomorphisms Grant Cairns ∗ , Thanh Duong Pham Department of Mathematics, La Trobe University, Melbourne, Australia 3086 Received 7 May 2007; received in revised form 13 October 2007; accepted 13 October 2007
Abstract We give an example of a group action on Euclidean space for which each map in the action is compactly supported, and such that the action is chaotic in the sense that it is topologically transitive and the set of points with finite orbit is dense. This resolves a conjecture of Naolekar and Sankaran. © 2007 Elsevier B.V. All rights reserved. Keywords: Chaos; Group action
1. Introduction We say that an action by homeomorphisms of a group G on a topological space X is chaotic if the following two conditions are satisfied: (i) the set of points with finite G-orbit is dense in X, (ii) the action is topologically transitive in the sense that for all nonempty open sets U, V ⊆ X, there exists g ∈ G for which gU ∩ V is nonempty. This is the natural generalization to group actions of Devaney’s definition of chaos for functions [1] and has been studied in [2–4]. In [6], Naolekar and Sankaran conjectured that on Euclidean space, there are no chaotic group actions via compactly supported homeomorphisms. They established this in several cases, including the cases of finitely generated groups and solvable groups. We give counterexamples to this conjecture on Rn , for all n 2. The n = 2 counterexample is quite easy to describe so we will treat this case before turning to the general case. 2. Counterexample in the plane Consider the torus T2 = R2 /Z2 , which for convenience we identify with the square with vertices (± 12 , ± 12 ) and opposing edges identified in the usual manner. For each i ∈ N, consider the following sets (see Fig. 1): Hi = (x, y) ∈ T2 : 2−1 − 2−i |y| 2−1 − 2−i−1 , Vi = (x, y) ∈ T2 : 2−1 − 2−i |x| 2−1 − 2−i−1 . * Corresponding author.
E-mail addresses: [email protected] (G. Cairns), [email protected] (T.D. Pham). 0166-8641/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2007.10.002
162
G. Cairns, T.D. Pham / Topology and its Applications 155 (2007) 161–164
Fig. 1.
Let φ and ψ be the linear automorphisms of T2 defined by the matrices
1 2 01
and
1 0 21
respectively. It is well known i
[3]. For i ∈ N, set φi = φ 2 on Hi and set φi that the group Γ = φ, ψ generated by φ and ψ acts chaotically on i to be the identity map elsewhere; similarly, set ψi = ψ 2 on Vi and set ψi to be the identity map elsewhere. Notice that φi (resp. ψi ) is the identity on the boundary of Hi (resp., Vi ); for example, x 1 2i+1 x x = = (mod 1). φi 2−1 − 2−i−1 2−1 − 2−i−1 0 1 2−1 − 2−i−1 T2
It follows that the maps φi and ψi are homeomorphisms. Consider the group G = φi , ψi : i ∈ N generated by all the φi and ψi . Since each generator of G is the restriction of an element of Γ , every point with finite Γ -orbit also has finite G-orbit. Thus the points with finite G-orbit form a dense subset of T2 . The action of G on T2 is also topologically transitive; this can be deduced from [5], or may be established directly as follows: given nonempty open sets U, V ⊆ T2 , choose i, j so that U ∩ Hi and V ∩ Vj are both nonempty. Then there exist k, l such that (φik U ) ∩ (ψjl V ) is nonempty. Now recall that the quotient of T2 under the map (x, y) → (−x, −y) is homeomorphic to the 2-sphere S 2 . The quotient map π : T2 → S 2 is a 2-fold cover ramified over 4 points. (As a geometry, the quotient S 2 is an orifold commonly called a pillow; see [7, Chapter 13.1.4].) Clearly the G-action on T2 induces a faithful G-action on S 2 such that the map π is a semi-conjugacy. It follows that since the G-action on T2 is chaotic, the G-action on S 2 is also chaotic. The point z = π(± 12 , ± 12 ) is a fixed point of this action; removing this point we obtain a faithful chaotic action of G on S 2 \{z} ∼ = R2 . By construction, the elements of this action are compactly supported. This provides the required example. 3. Counterexample in higher dimensions We now provide a counterexample in Rn for n > 2. The idea is to first build a slightly more complicated 2-dimensional example, and then employ the skew product construction as in [3]. Set H1 = {(x, y) ∈ T2 : 0 |y| 1/4}, V1 = {(x, y) ∈ T2 : 0 |x| 1/4} and for each i > 1, consider the following sets (see Fig. 2): Hi = (x, y) ∈ T2 : 3/8 − 2−i−1 |y| 3/8 − 2−i−2 ,
G. Cairns, T.D. Pham / Topology and its Applications 155 (2007) 161–164
163
Fig. 2.
Vi = (x, y) ∈ T2 : 3/8 − 2−i−1 |x| 3/8 − 2−i−2 . Consider the set 3 3 . N := (x, y): |x| or |y| 8 8 We define homeomorphisms φi , ψi of N for each i ∈ N, as follows. For the linear automorphisms φ and ψ of T2 i+1 i+1 defined above, set φi = φ 2 on Hi and set φi to be the identity map elsewhere on N ; similarly, set ψi = ψ 2 on Vi and set ψi to be the identity map elsewhere. Notice that φi (resp. ψi ) is the identity on the boundary of Hi (resp. Vi ). Consider the group G = φi , ψi : i ∈ N. The action of G on N is obviously faithful and by the same argument as we used above, it is chaotic on N . Notice that by construction, this action is trivial on the boundary of N . Now we construct a chaotic action on the closed ball B n of dimension n > 2. Consider the skew product, for each g ∈ G, f (g) : N × Tn−2 → N × Tn−2 , (x1 , x2 , . . . , xn ) → g(x1 , x2 ), x3 + h3 (x), . . . , xn + hn (x) , where the maps hj : N → S 1 are defined by 2j if|x | 1/4, 1 hj (x) = (4x1 ) 0 otherwise, for x = (x1 , x2 ) and j ∈ N. Notice that for all g ∈ G, the map f (g) is the identity on the boundary of N × Tn−2 . We will show that this action of G on N × Tn−2 is chaotic. In order to directly apply the results of [3], consider the set 1 1 . M := (x, y): |x| or |y| 4 4 As shown in the proof of [3, Lemma 3.2], f (φ1 ◦ ψ1 ) is a chaotic homeomorphism of M × Tn−2 . Let U and V be two arbitrary nonempty open subsets in N × Tn−2 . Assume that U = U1 × U and V = V1 × V , where U1 and V1 are nonempty open subsets in N , and U , V are nonempty open subsets in Tn−2 . If U1 and V1 both intersect M then since f (φ1 ◦ ψ1 ) is topologically transitive on M × Tn−2 , there is g ∈ G such that f (g)(U ) ∩ V = ∅.
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G. Cairns, T.D. Pham / Topology and its Applications 155 (2007) 161–164
If U1 or V1 do not intersect M, then since G acts chaotically on N , there exist g1 , g2 ∈ G such that g1 (U1 ) ∩ M = ∅ and g2 (V1 ) ∩ M = ∅. So from above, there is g ∈ G such that f (g)(f (g1 )(U )) ∩ (f (g2 )(V )) = ∅. So we have f (g2−1 ◦ g ◦ g1 )(U ) ∩ V = ∅. Hence the action of G on N × Tn−2 is topologically transitive. Since the points in N × Tn−2 with rational coordinates are periodic, the set of periodic points is dense in N × Tn−2 . Therefore, G is chaotic on N × Tn−2 . The quotient of N under the map (x, y) → (−x, −y) is homeomorphic to the closed disc D 2 . So dividing N ×Tn−2 by the map σ : (x1 , x2 , x3 , . . . , xn ) → (−x1 , −x2 , x3 , . . . , xn ) one obtains a chaotic action of G on the solid torus D 2 × Tn−2 which is again chaotic and trivial on the boundary. Now consider the construction of the closed ball B n from D 2 × Tn−2 as in the proof of [3, Lemma 3.2]. Since B n is constructed from D 2 × Tn−2 by a succession of identifications on the boundary of the space and g is the identity on the boundary of D 2 × Tn−2 , the action of G on D 2 × Tn−2 is semi-conjugate to a chaotic action on the quotient space B n [3, Lemma 3.2]. Moreover, by construction, each map in the action is compactly supported and is the identity on the boundary of the ball. Removing the boundary, we get the required action on Rn . Remark 1. Let D denote the direct sum of countably infinitely many copies of the infinite cyclic group. The group G of the above construction is isomorphic to the free product G = D ∗ D. References [1] [2] [3] [4]
J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (4) (1992) 332–334. G. Cairns, G. Davis, D. Elton, A. Kolganova, P. Perversi, Chaotic group actions, Enseign. Math. (2) 41 (1–2) (1995) 123–133. G. Cairns, A. Kolganova, Chaotic actions of free groups, Nonlinearity 9 (4) (1996) 1015–1021. G. Cairns, A. Kolganova, A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math. 37 (2) (2007) 371–397. [5] R.L. Devaney, Linked twist mappings are almost Anosov, Global theory of dynamical systems, in: Proc. Internat. Conf., Northwestern Univ., Evanston, Ill, 1979, in: Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 121–145. [6] A.C. Naolekar, P. Sankaran, Chaotic group actions on manifolds, Topology Appl. 107 (3) (2000) 233–243. [7] W.P. Thurston, The geometry and topology of three-manifolds Lecture notes; available at http://www.msri.org/communications/books/gt3m, 1980.