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ADVANCES IN MATHEMATICS

Vol.39, No.2 April, 2010

One-dimensional Topological Dynamical Systems of Discrete Group Actions SHI Enhui (Department of Mathematics, Suzhou University, Suzhou, Jiangsu, 215006, P. R. China)

Abstract: In this survey it is given a brief account of recent research achievements in the study of 1-dimensional topological dynamical systems of discrete group actions. The content contains: expansiveness, ping pong game and geometric entropy; invariant sets and invariant measures; topological k-transitivity; sensitivity and Devaney’s chaos; bounded Euler classes and classifications. Some open questions are reported. Key words: topological dynamical system; discrete group; continuum; 1-manifold; conjugate classification MR(2000) Subject Classification: 37B05; 57S25 / CLC number: O189.1 Document code: A Article ID: 1000-0917(2010)02-0129-15

0 Introduction The theory of classical dynamical systems deals with properties of the actions of additive groups Z and R that are asymptotic in character. Recently people have made great progress in the study of dynamical systems of general group actions. Some new branches have been established, such as ergodic theory of amenable group actions, rigidity theory of higher rank semisimple Lie groups and lattices, homogeneous dynamical systems, and algebraic systems of Zd -actions, etc.. These achievements not only extend the theory of dynamical systems itself, but also have many important applications in areas such as number theory, geometry and statistical physics[20] . First let us recall some basic notions. Let X be a topological space, Homeo(X) the homeomorphism group of X, and G a discrete group. We call a group homomorphism φ : G → Homeo(X) a continuous action (or action, for brevity) of G on X. If φ is injective, then φ is said to be faithful. For x ∈ X, the set Orb(x) = {φ(g)(x) : g ∈ G} is said to be the orbit of x. Two actions φ1 and φ2 of G on X1 and X2 are topologically conjugate if there is a homeomorphism h from X1 to X2 such that for every g ∈ G, one has φ2 (g) = hφ1 (g)h−1 . For convenience, we often use g(x) instead of φ(g)(x). Let φ : G → Homeo(X) be an action of group G on one-dimensional space X. We would like to understand the relationships between: • algebraic properties of G (such as commutativity, nilpotency, solvability, amenability, lattices in semisimple Lie groups, etc.), Received date: 2009-04-18. Foundation item: Supported by the National Natural Science Foundation of China(No. 10801103) and by the Natural Sciences Fund for Colleges and Universities in Jiangsu Province (No. 08KJB110010). E-mail: [email protected]

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• The topology of X (such as the arc, the circle, the line, dendrites, and other onedimensional continua), • Dynamical properties of the action φ (such as expansiveness, sensitivity, topological transitivity, minimality, mixing, finite orbits, chaos, topological entropy, etc.). This study program is influenced by the ideas of Zimmer[71] . We are mainly interested in the following two questions: Question 0.1 Given a space X, a group G, and a dynamical property P , can G act on X with the property P ? Question 0.2 Given a space X and a group G, classify all the possible actions of G on X up to topological conjugations. Now let us recall some basic notions in group theory and continuum theory. A connected compact metric space is called a continuum. A locally connected continuum is called a Peano continuum. If X is homeomorphic to the closed interval [0, 1] (resp. the unit circle S1 ), then X is called an arc (resp. a circle). A Peano continuum having no circle is called a dendrite. If for any two points x, y in a continuum X, there is a unique arc [x, y] ⊂ X from x to y, then X is said to be uniquely arcwise connected. Clearly a dendrite is uniquely arcwise connected. A continuum is called planar if it can be embedded in the plane. For the other notions such as Siepinski curve, arc-like continuum, tree-like continuum, etc., one may refer to [42]. Suppose that G is a group with identity e. Let a, b ∈ G. The commutator [a, b] is defined by [a, b] = a−1 b−1 ab. For any two subsets A and B of G, define [A, B] to be the subgroup generated by the set {[a, b] : a ∈ A, b ∈ B}. Let G0 = G and Gi+1 = [Gi , G], for i = 0, 1, 2, · · · . Thus we get a sequence of normal subgroups of G : G0 = G  G1  G2  · · · . If there is some natural number n such that Gn = {e}, then G is called nilpotent. Also we can define another sequence of normal subgroups G0 = G  G1  G2  · · · by letting G0 = G and Gi+1 = [Gi , Gi ] for i = 0, 1, 2, · · · . If there is some n such that Gn = {e} then G is called solvable. The minimal n such that Gn = {e} is called the derived length of G. A solvable group G is called polycyclic, if for some k, G has i a sequence of normal subgroups G = N0  N1  · · ·  Nk = {e} such that each NNi+1 is cyclic. Ni When each of the quotients Ni+1 is infinite cyclic, G is said to be poly-infinite-cyclic. One may refer to [51] for the details of these notions. For the notations such as amenability, lattices in simple Lie groups, cohomology of groups etc. will be introduced in the context and the related references will be given. The related notions in dynamical systems will be introduced in the context. One may consult[69] for the detailed introduction of notions and recent achievements in topological dynamical systems and refer to [68, 70] for many important results in the classical 1-dimensional dynamics. Of course the content in this survey is far from completeness and it only reflects the author’s own interests. Though we are mainly interested in continuous actions of groups on onedimensional spaces, some results about smooth actions or actions on higher dimensional spaces are introduced for completeness.



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1 Expansiveness, Ping Pong Game and Geometric Entropy Let X be a compact metric space with metric d. Let φ : G → Homeo(X) be an action of group G on X. The action φ is said to be expansive if there is a positive constant c > 0 such that for any two different points x, y ∈ X, there is some g ∈ G such that d(g(x), g(y)) > c and such c is called an expansivity constant for φ. A homeomorphism is said to be expansive if the cyclic group generated by it is expansive. Which space can admit an expansive homeomorphism (i.e. expansive Z-action) is an important question in topological dynamical systems. It is well known that the Cantor set, 2-adic solenoid, and 2-torus admit expansive homeomorphisms[66] . O’Brien and Reddy proved that each compact orientable surface of positive genus admits an expansive homeomorphism[45]. On the other hand, it is shown that the arc and the circle admit no expansive homeomorphism[5, 24] . Kawamura proved that no Peano continuum having a free arc admits an expansive homeomorphism[32] . Kato proved that one-dimensional compact ANRs, Peano continua in the plane, arc-like continua, and dendroid admit no expansive homeomorphism[25−28] . Hiraide proved that there exists no expansive homeomorphism on the 2-sphere S 2 (see [21]). Recently Mouron proved that there exist no expansive homeomorphisms on tree-like continua[41] , and Kato and Mouron proved that hereditarily indecomposable compacta do not admit expansive homeomorphisms[30]. In 1999, Ward proposed the following question. Question 1.1[31]

Can the unit circle S1 admit an expansive nilpotent group action?

By a skillful induction process, Shi and Zhou proved: Theorem 1.2[56]

There is no expansive Zn -action on S1 .

By means of the integrals of some functions specially defined on graphs, Mai and Shi proved further: Theorem 1.3[35]

There is no expansive commutative group action on S1 .

Recently, in an unpublished paper, Hurder solved Ward’s question using the ping pong game technique[22] . In fact he gets more. Before the statement of his results, let us recall some definitions. Consider a finitely generated group G. Let S = {σ0 , σ1 , · · · , σk } be a symmetric generating set of G, where σ0 is the identity. The symmetric hypothesis means that σi ∈ S implies σi−1 ∈ S. An element g ∈ G has word length g  n if there exist indices i1 , i2 , · · · , in such that g = σi1 σi2 · · · σin . The word length g is the least integer n such that g  n. The growth rate of G is the number gr(G; S) = lim supn→∞ n−1 log|G(n)|, where |G(n)| is the number of all elements of G with word length≤ n. Suppose that φ is an action of G on a compact metric space (X, d). Given ε > 0 and an integer n > 0, we say that x, y ∈ X are (n, ε)-separated if there exists g ∈ G such that  g  n and d(g(x), g(y)) > ε. A finite subset E is said to be (n, ε)-separated if for any two different points x, y ∈ E, x and y are (n, ε)-separated. Let S(φ, ε, n) denote the largest cardinality of any (n, ε)-separated subset of X. Now define h(φ, ε) = lim sup n→∞

log S(φ, ε, n)  0. n

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The geometric entropy of φ is the limit h(φ) = limε→0 h(φ, ε). The above definition of geometric entropy of a group action was introduced by Ghys, Langevin and Walczak[15] . When the acting group G is a cyclic group generated by a homeomorphism f , then the geometric entropy of G is just the topological entropy of f . Ping-pong game is an important technique in determining the existence of free subgroups or free sub-semigroups of some group. This concept dates from the work of Blaschke, Klein, Schottky and Poincar´e. Let φ be an action of G on X. A pair of maps {g1 : K → K1 , g2 : K → K2 } is called a ping-pong game for φ if K ⊂ X is a closed set, K1 , K2 ⊂ K are closed disjoint subsets and g1 , g2 ∈ G satisfy g1 (K) ⊂ K1 , g2 (K) ⊂ K2 . It is known and easy to show that if an action φ : G → Homeo(X) has a ping pong game, then G must have positive geometric entropy and contain a free subsemigroup. Hurder proved the the following Theorem 1.4 and 1.5 in [22]. Theorem 1.4[22] If the action φ : G → Homeo(S1 ) is expansive, then φ has a ping pong game. In particular, φ has positive geometric entropy and G contains a free subsemigroup. It is well known that no nilpotent group contains a free sub-semigroup. So the following theorem is a direct corollary of Theorem 1.4. Theorem 1.5[22] There is no expansive nilpotent group action on S1 . Existence of expansive group actions on other continua has also been studied by some authors. Shi et al proved the following: Theorem 1.6[55] There are no expansive nilpotent group actions on Peano continua having no θ-curves. Remark 1.7 In fact, the authors only established the above theorem for Zd -actions in [55]. But using the same method and Theorem 1.9, it is easy to see that Theorem 1.6 holds. The following theorem is due to Mai and Shi. Theorem 1.8[36] There are no expansive commutative group actions on Peano continua having free dendrites. Using the ping pong game technique, Shi and Wang generalized Theorem 1.4, Theorem 1.5 and Theorem 1.8 to the following Theorems 1.9 and Theorem 1.10. Theorem 1.9[53] Each expansive nilpotent group action on a Peano continuum having a free dendrite must have a ping pong game and positive geometric entropy. Theorem 1.10[53] There are no expansive nilpotent group actions on Peano continua having free dendrites. Remark 1.11 (a) Theorem 1.10 does not hold for solvable group actions. In fact Shi and Zhou constructed an expansive solvable group action on [0, 1] (see [59]). (b) Shi and Zhou constructed an expansive Z2 -action on an infinite-dimensional compact metric space in [56]. Thus the well known result “there are no expansive homeomorphisms on infinite-dimensional compact metric space” due to M˜an´e does not hold for the case of Z2 -actions[37] . The following question is open. Question 1.12 Do Peano continua in the plane, arc-like continua, tree-like continua and the 2-sphere S 2 admit expansive nilpotent group actions? A continuum X is called a regular curve if for each x ∈ X and each open neighborhood U



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of x, there is an open neighborhood V of x such that V ⊆ U and the boundary set BdX (V ) of V is a finite set. It is known that every local dendrite is a regular curve[34] , while there are many regular curves which are not local dendrites, such as the Sierpi´ nski’s triangle. It is well known that a homeomorphism on the closed interval I = [0, 1] or on the unit circle S1 must have zero topological entropy[62] . Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero[52]. Recently Wang et al proved the following Theorem 1.13 and Theorem 1.14. Theorem 1.13[65] Suppose that X is a regular curve and G is a finitely generated group. Let φ : G → Homeo(X) be an action. Then the geometric entropy h(φ) is less than or equal to the growth rate gr(G) of G. Corollary 1.14[65] Suppose that G is a finitely generated group of polynomial growth. Then every G-action on a regular curve must have zero geometric entropy. In particular, every finitely generated nilpotent group action on a regular curve must have zero geometric entropy. There does exist a one-dimensional continuum which admits a homeomorphism with positive topological entropy. For example, Kennedy proved the following theorem which answers a question of Barge. Theorem 1.15[33]

There is a pseudoarc homeomorphism with positive topological entropy.

2 Invariant Sets and Invariant Measures Let φ : G → Homeo(X) be an action of group G on a compact metric space X. If for every point x ∈ X the orbit Orb(x) is dense in X, then φ is said to be minimal. This is equivalent to saying that φ has no proper G-invariant closed subset of X. A closed subset Y of X is said to be a minimal set, if the restriction to Y of the action φ is minimal. If a minimal set Y consists of finite points, then Y is said to be a finite orbit, that is, the finite set Y is an orbit of some point x. If a minimal set Y consists of just one point x, then x is said to be a fixed point of φ. A Borel measure μ on X is said to be G-invariant if for any Borel subset A of X, we have μ(g(A)) = μ(A) for all g ∈ G. The following question is interesting: Question 2.1 When does a group action on a continuum have a fixed point or a finite orbit? The following theorem is due to Isbell. Theorem 2.2[23]

Each commutative group action on a dendrite must have a fixed point.

Bing asked whether every homeomorphism on a uniquely arcwise connected continuum has a fixed point[3] . Mohler solved Bing’s question by proving the following theorem. Theorem 2.3[40] a fixed point.

Every homeomorphism on a 1-arcwise connected continuum must have

Shi and Sun generalized the results of Isbell and Mohler by proving the following theorem. Theorem 2.4[24] Every nilpotent group action on a uniquely arcwise connected continuum must have a fixed point. Recently Shi and Zhou obtained further the following theorem:

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Theorem 2.5[61] Every solvable group action on a uniquely arcwise connected continuum either has a fixed point or has a 2-periodic point. Let G be a Lie group and G be its Lie algebra. The real rank of G is the maximal dimension of an abelian subalgebra U such that for every a ∈ U the linear operator ad(a) : G → G is diagonalizable over R. For instance, the real rank of SL(n, R) is n − 1. A lattice in a Lie group G is a discrete subgroup Γ such that the quotient G/Γ has finite measure with respect to a right invariant Haar measure. A conjecture of Zimmer states that if Γ is a lattice in a simple Lie group G with real rank at least 2 and φ : Γ → Homeo(S1 ) is an action of Γ on the unite circle, then φ(Γ) is a finite group. This conjecture was solved under some additional assumptions[17, 67, 12] . The following theorem is due to Ghys. Theorem 2.6[17] Every continuous action of a higher rank lattice on a circle must have a finite orbit. It is well known that minimal sets always exist for group actions on compact metric spaces. Naturally one may ask the following question: Question 2.7 Let φ : G → Homeo(X) be an action of G on a compact metric space X. Then what can we say about the topology of minimal sets K and about the dynamics of φ restricted to K. The following proposition can be seen in [18]. Proposition 2.8 Let φ : G → Homeo+ (S1 ) be an action of the group G on the unite circle 1 S . Then there are three mutually exclusive possibilities. Case 1 There is a finite orbit. Case 2 φ is minimal. Case 3 There is a minimal set K which is homeomorphic to a Cantor set. This set K is unique, contained in the closure of any orbit. Recall that a group action φ : G → Homeo(X) is said to be equicontinuous, if for any ε > 0, there is a δ > 0 such that d(φ(g)(x), φ(g)(y)) < ε for all g ∈ G, whenever d(x, y) < δ. An action φ : G → Homeo(S1 ) is said to be of C 1 if for every g ∈ G, φ(g) is a C 1 -diffeomorphism on S1 . The following proposition is taken from [22]. Proposition 2.9 Let φ : G → Homeo+ (S1 ) be a minimal action. Then φ is either equicontinuous, or expansive. Remark 2.10 In Proposition 2.9, if φ is equicontinuous, then it is well known that φ is conjugate to a minimal group action generated by rotations on S1 . In particular φ has an invariant probability measure on S1 . For C 1 -action, Hurder proved the following theorem. Theorem 2.11[22] Let φ : G → Homeo+ (S1 ) be a C 1 -action with minimal set K. Then either there is a G-invariant probability measure supported on K, or there is a ping pong game for φ and φ has positive geometry entropy. Remark 2.12 To fit the purpose of this survey, the statement of Theorem 2.11 is simplified here. From Theorem 2.11, we know if φ has no invariant measure on S1 , then φ(G) must have



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a nonabelian free subsemigroup. It was conjectured by Ghys that if φ : G → Homeo(S1 ) is a minimal action, then φ has a G-invariant measure or G has a non-abelian free subgroup. This conjecture was solved by Hurder for analytic actions in [22] and was solved by Margulis in the general case. Theorem 2.13[38] Let φ : G → Homeo(S1 ) be a minimal action. Then φ has a G-invariant measure or G has a non-abelian free subgroup. A topological group G is amenable if there is an invariant mean on B(G), the bounded right uniformly continuous functions on G. When G is discrete, there is an intrinsic characterization given by Føler: G is amenable if and only if for any ε > 0 and any finite subset K of G, there is a finite set A of G such that |gAA| < ε, for all g ∈ K, where | | denotes cardinal number. |A| It is well known that every solvable group is amenable and every amenable group action on a compact metric space must have an invariant measure. For details about amenability, one may consult[46] . Plante considered the existence of invariant measures of some amenable group actions on the noncompact space R and obtained the following theorem: Theorem 2.14[47] If G is a finitely generated nilpotent group acting on the line, then there is a G-invariant Borel measure on R which is finite on compact sets. Solvable group actions on R need not have invariant measures, e.g., subgroups of the affine group which contain nontrivial translations and nontrivial dilations. So Plante considered the existence of quasi-invariant measures for solvable group actions on R. Let G be a group acting on R. A measure μ is called quasi-invariant for G if for each g ∈ G there is a positive constant c(g) such that g∗ μ = c(g)μ. Let F ⊂ R be a G-invariant set. If there is a symmetric generating set G1 ⊂ G such that G1 (x) is bounded for every x ∈ F , then G is said to be boundedly generated on F . G is boundedly generated if it is boundedly generated on some nonempty set F . Plante proved the following theorem. Theorem 2.15[48] Let G be a group acting on R. If G is solvable with normal series 1 = G0 ⊂ G1 ⊂ · · · ⊂ Gn = G such that each Gi is boundedly generated. Then there exists a nontrivial Borel measure on R which is G-quasi-invariant and finite on compact sets. If differentiability is assumed, stronger results are obtained: Theorem 2.16[48] An arbitrary abelian group of C 2 diffeomorphisms of the line has an invariant measure and an arbitrary solvable real analytic group acting on the line has a quasiinvariant-measure. By a Kleinian group we mean a group Γ acting freely and properly discontinuously on hyperbolic 3-space, H3 . The limit set Λ(Γ) of Γ consists of the points x ∈ ∂H3 such that there is a sequence γi ∈ Γ and some y ∈ H3 with limi→∞ γi (y) = x. Clearly Λ(Γ) is a Γ-invariant closed subset of ∂H3 . It is well known that the action of Γ on Λ(Γ) is minimal[50] . In some cases, Λ(Γ) is known to be a dendrite and its Hausdorff dimension has been calculated by some authors[4, 39] . Motivated by the above process, minimal group actions on dendrites are studied by Shi, Wang and Zhou and the following theorem is proved. Theorem 2.17[57]

If a group G acts on a nondegenerate dendrite X minimally, then X

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admits no G-invariant finite measure. In particular, G cannot be amenable.

3 Topological k-transitivity Let φ : G → Homeo(X) be an action of group G on topological space X. The action φ is said to be topologically transitive, if for any two nonempty open subsets U and V of X, there is some g ∈ G such that g(U ) ∩ V = ∅. If there is some point x ∈ X such that the orbit Gx is dense in X then φ is said to be point transitive and such x is called a transitive point. It is well known that when G is countable and X is a Polish space these two notions are the same and in fact the collection of transitive points form a dense Gδ set in X. For an integer k ≥ 1, φ is said to be topologically k-transitive, if for any two families of nonempty open subsets U1 , U2 , · · · , Uk and V1 , V2 , · · · , Vk of X, there is some g ∈ G such that g(Ui ) ∩ Vi = ∅ for each i = 1, 2, · · · , k. Topological 2-transitivity is usually called weak mixing. The following well known theorem is due to Furstenberg. Theorem 3.1[13, 19] For commutative group actions, topological 2-transitivity implies topological k-transitivity for each k ≥ 2. Naturally, one may ask the following question. Question 3.2 For nilpotent group actions, does topological 2-transitivity imply topological k-transitivity for each k ≥ 2? Though the question is still open now, Wang et al gave the following counterexample for solvable group actions. Example 3.3[64] Let T , S, Mα and Mβ be defined by T (x) = x + 1, S(x) = −x, Mα (x) = αx and Mβ (x) = βx for all x ∈ R respectively, where α, β > 1, and log(α) and log(β) are rationally independent. Then the group G = T, S, Mα , Mβ is solvable and the action of G on R is 2-transitive but not 3-transitive. Clearly if φ : G → Homeo+ (R) is an orientation preserving group action of G on R, then φ cannot be 2-transitive. So, to study the topological transitivity of orientation preserving group actions on R, the following definition was introduced in [63]. First we introduce an ordering  in the collection of all open intervals contained in R. For any two open intervals (a, b) and (c, d) in R, we say that (a, b)  (c, d) if a ≤ c. Definition 3.4[63] Let φ : G → Homeo+ (R) be an orientation preserving group action of G on R. We say φ is pseudo-k-transitive if for any two families of open intervals (a1 , b1 )  (a2 , b2 )  · · ·  (ak , bk ) and (c1 , d1 )  (c2 , d2 )  · · ·  (ck , dk ) there is some g ∈ G such that g((ai , bi )) ∩ (ci , di ) = ∅ for all i = 1, 2, · · · , k. Question 3.5 (a) Which solvable group does have a faithful topologically transitive action on the line? (b) What can one say about actions with higher transitivity? Wang et al proved the following Theorem 3.6 and Theorem 3.8. Theorem 3.6[63] Every noncyclic poly-infinite-cyclic group has a faithful topologically transitive orientation preserving action on the real line. Remark 3.7 Ghys proved that if G is a countable group which is solvable with torsion free abelian quotients, then G can act faithfully on the real line by orientation preserving homeomorphisms[18]. Farb and Franks proved that every finitely generated torsion-free nilpotent



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group has a faithful C 1 action on the line[11] . Theorem 3.8[63] (a) Finitely generated nilpotent groups have no pseudo-2-transitive action on the line. (b) Polycyclic solvable groups have no pseudo-3-transitive action on the line. (c) Suppose G is a solvable group of derived length n. Then G has at most pseudo-(4n − 1)-transitive action on the line. The following questions are still open. Question 3.9 Is it true that every noncyclic group which has a faithful orientation preserving action on R must have a topologically transitive action on R? Question 3.10 Does there exist a pseudo-3-transitive solvable group action on R? Wang et al also studied the topological transitivity of group actions on dendrites and the following results are obtained. Theorem 3.11[64] Each weakly mixing group action on a dendrite must have a ping-pong game. Moreover, if the acting group is finitely generated, then the action has positive geometric entropy. In particular, there is no weakly mixing nilpotent group action on a dendrite. Proposition 3.12[64] There are no topologically 4-transitive group actions on dendrites. Question 3.13 Does there exist a topologically 3-transitive group action on a dendrite?

4 Sensitivity and Devaney’s Chaos Let G be a group acting on a topological space X. This action is said to be chaotic if it is topologically transitive and all finite orbits are dense in X. This notion was first introduced by Cairns et al. in [7] which is a generalization of Devaney’s chaos for maps[10] . The action is said to be sensitive if there is a constant c > 0 such that for any nonempty open subset U of X, there is a g ∈ G such that diam(g(U )) > c. Using the ideas in [2], it is easy to see that Devaney’s chaos implies sensitivity for group actions. Question 4.1 Which space can admit a chaotic or a sensitive G-action, for a given group G with some specified algebraic structures? Recall that a group is said to be residually finite if for every nonidentity element g of G, there is a normal subgroup , not containing g, of finite index in G. For the algebraic structure of a group which has a chaotic action is characterized by Cairns ect.. Theorem 4.2[7] A group G possesses a faithful chaotic action on some Hausdorff space if and only if G is residually finite. The following Theorem 4.3 and Theorem 4.4 are obtained by Cairns etc.. Theorem 4.3[7] The circle does not admit a chaotic action of any group. Theorem 4.4[7] Every compact surface(with or without boundary) admits a chaotic Zaction. Cairns and Kolganova obtained the following theorem. Theorem 4.5[8] Every compact triangulable manifold of dimension greater than 1 admits a faithful chaotic action of every countably generated free group. Remark 4.6 Chaotic actions of certain finitely generated infinite abelian groups on evendimensional spheres, and of finite index subgroups of SLn (Z) on tori are constructed in [43].

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Chaotic homeomorphisms or group actions on continua are also studied by some authors. The following theorem is due to Aarts and Oversteegen. Theorem 4.7[1] The Sierpinski curve admits a chaotic Z-action. Kennedy proved the following theorem. Theorem 4.8[33] If P is a pseoudoarc, then P admits a chaotic Z-action. Wang et al proved the following theorem. Theorem 4.9[64] Dendrites admit no chaotic group actions. For sensitive group actions, Shi et al obtained: Theorem 4.10[58] There are no sensitive commutative group actions on dendrites. Remark 4.11 Recently, the author have shown that Theorem 4.10 also holds for Peano continua having free dendrites in an unpublished paper. Recall that a continuum is Suslinian, if it contains at most countable mutually disjoint nondegenerate subcontinua. Kato[29] asked: Suppose that a continuum X admits a sensitive homeomorphism, is X not Suslinian? Mai and the author answered this question by proving the following theorem. Theorem 4.12[35] There is a Suslin continuum which admits a sensitive homeomorphism. The following two questions are open. Question 4.13[7] Is there a faithful chaotic action of Z2 on the 2-torus T2 or the 2-sphere 2 S ? Question 4.14 Does there exist a sensitive nilpotent group action on a Peano continuum having a free dendrite?

5 Bounded Euler Classes and Classifications Rotation number is the main invariant of homeomorphisms of the circle which was introduced by Poincar´e[48]. The following well known classification theorem is due to Poincar´e. Theorem 5.1 Let f be an element of Homeo+ (S 1 ). Then f has a periodic orbit if and only if the rotation number ρ(f ) is rational, i.e. belongs to Q/Z. If the rational number ρ(f ) is irrational, then f is semi-conjugate to the rotation on the circle of angle ρ(f ) ∈ R/Z. This semi-conjugacy is actually a conjugacy if the orbits of f are dense. For orientation preserving group actions on S 1 , Ghys established a classification theorem in [16] which is an analogy of the Poincar´e’s classification theorem, using bounded Euler class (an invariant of group actions on S 1 ). Let us first recall some notions in algebra. Let Γ be a group and A be an abelian group. A k-cochain of Γ with values in A is a map c : Γk+1 → A which is homogeneous, i.e. such that c(γγ0 , γγ1 , · · · , γγk ) = c(γ0 , γ1 , · · · , γk ) identically. The set of these cochains is an abelian group denoted by C k (Γ, A). The coboundary map dk from C k (Γ, A) to C k+1 (Γ, A) is defined by dk c(γ0 , · · · , γk+1 ) =

k  (−1)i c(γ0 , · · · , γˆi , · · · , γk ). i=0

Clearly dk+1 ◦ dk = 0 and the cohomology group H k (Γ, A) is defined to be the quotient of cocycles (i.e. the kernel of dk ) by coboundaries (i.e. the image of dk−1 ).



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It is easy to see that a homogeneous map c : Γk+1 → A can be written in a unique way in the −1 form c(γ0 , · · · , γk ) = c¯(γ0−1 γ1 , γ1−1 γ2 , · · · , γk−1 γk ) for a unique function c¯ : Γk → A. Conversely, given a map c¯ there is a unique homogeneous map c satisfying this relation. Such a c¯ is called the inhomogeneous cochain associated to c. i ˜ p → Γ → 1. This means that Now consider a central extension of Γ by A : 0 → A → Γ ˜ Γ contains a subgroup isomorphic to A contained in its center and that the quotient by this ˜ and consider subgroup is isomorphic to Γ. Choose a set theoretical section s from Γ to Γ c¯(γ1 , γ2 ) = s(γ1 γ2 )−1 s(γ1 )s(γ2 ). It is easy to check that this element projects on the identity element of Γ under p and hence is an element of image of i and can be identified with an element of A. Thus we obtain a map c¯ : Γ2 → A and an associated homogeneous cochain c : Γ3 → A. One can check that c is a cocycle. The cohomology class of c in H 2 (Γ, A) is called the Euler class of the extension under consideration and it does not depend on the choice of a section.  + (S 1 ) be the group of homeomorphisms of R which commute with integral transLet Homeo  + (S 1 ) can naturally induced an orientation lations. It is well known that each element of Homeo preserving homeomorphism on S 1 and each element of Homeo+ (S 1 ) can also be lifted to an  + (S 1 ). Thus we have the following center extension: element of Homeo p  + (S 1 ) → Homeo+ (S 1 ) → 1. 0 → Z → Homeo

(∗)

˜ = {(γ, f˜) ∈ Γ × Let φ be a homomorphism from some group Γ to Homeo+ (S 1 ). Define Γ p ˜→  + (S 1 ) : φ(γ) = p(f˜)}. Thus we get a central extension of Γ by Z: 0 → Z → Γ Homeo Γ→1  where p is the canonical projection on Γ. This extension leads to an Euler class which is called the Euler class of the homomorphism φ and denoted by φ∗ (eu) ∈ H 2 (Γ, Z). It is obviously a dynamical invariant in the sense that two conjugate homomorphisms φ1 and φ2 have the same Euler class in H 2 (Γ, Z). But this is indeed a very poor invariant, since it cannot even detect the rotation number by the fact that H 2 (Z, Z) = 0. To fill up this deficiency, we need to use bounded Euler classes. Consider again an abstract group Γ and let A = Z or R. Then define a bounded k-cochain as a bounded homogeneous map from Γk+1 to A. This is a sub A-module of C k (Γ, A) denoted by Cbk (Γ, A). It is clear that the coboundary dk of a bounded k-cochain is a bounded k + 1cochain. Thus we can define the cohomology of this new differential complex, that is called the bounded cohomology of Γ with coefficients in A and denoted by Hbk (Γ, A). Let us look at  + (S 1 ) as the central extension (∗) again. Now we select a section σ : Homeo+ (S 1 )→Homeo follows: for each f ∈ Homeo+ (S 1 ), let (the unique) σ(f ) ∈ p−1 (f ) be such that σ(f )(0) ∈ [0, 1]. Then the associated inhomogeneous cocycle c¯ is: c¯(f1 , f2 ) = σ(f1 f2 )−1 σ(f1 )σ(f2 ). It is easy to see that the associated 2-cocycle c is bounded and integral and thus we obtain an element of Hb2 (Homeo+ (S1 ), Z) that we call the bounded Euler class. If φ is a homomorphism from Γ to Homeo+ (S 1 ), then we can pull back this bounded Euler class to obtain an element in Hb2 (Γ, Z) which is denoted by φ∗ (eu) and is called the bounded Euler class of φ. Let φ : Γ → Homeo+ (S 1 ) be a group action. If φ is not minimal, then from Proposition 2.8, there are two possibilities: φ can have a finite orbit or can have a minimal set K homeomorphic to the Cantor set. In the first case, suppose φ has a finite orbit consisting of k elements. Then,



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every element of Γ must permute these k points cyclically so that we get a homomorphism r : Γ → Z/kZ. We call this r the cyclic structure of the finite orbits. Conversely, consider a homomorphism r : Γ → Z/kZ and the corresponding action on the circle by rotations of order k. The bounded Euler class of this action is an element of Hb2 (Γ, Z) : we call these elements the rational elements in Hb2 (Γ, Z). In the second case, there is a canonical way of collapsing the connected components of S 1 \ K to construct another homomorphism φ¯ which is minimal. Ghys proved the following classification theorem for group actions on S 1 . Theorem 5.2[16] Let φ1 , φ2 be two homomorphisms from a group Γ to Homeo+ (S 1 ). Assume that the bounded Euler classes φ∗1 (eu) = φ∗2 (eu) = c ∈ Hb2 (Γ, Z). Then we have (1) If c is a rational class, then φ1 (Γ) and φ2 (Γ) have finite orbits with the same cyclic structure. (2) If c is not rational, then the associated minimal homomorphisms φ¯1 and φ¯2 are conjugate. Conversely, if φ1 (Γ) and φ2 (Γ) have finite orbits of the same cyclic structure or if they have no finite orbit and their associated minimal homomorphisms are conjugate (by an orientation preserving homeomorphism), then they have the same bounded Euler class. Remark 5.3 There are also several remarkable classification theorems of group actions on S 1 , for example, the classification of convergence group due to Tukia, Casson-Jungreis, Gabai[9, 14] and the classification of solvable group actions on S 1 under some assumptions on smoothness[6, 44] . But these contents are beyond the goal of this survey, so we don’t talk about them here. Now we introduce a classification theorem for topologically transitive Zd -actions on the line R obtained by Shi and Zhou. Suppose φ : Zn → Homeo+ (R) is topologically transitive. If for any subgroup F of Zn with coset index |Zn : F | = ∞, the restriction to F of φ, φ|F : F → Homeo+ (R) is not topologically transitive, then φ is said to be tightly transitive. For every irrational number α ∈ (0, 1) and every positive integer n ≥ 2, we will construct a tightly transitive Zn -action φα,n : Zn → Homeo+ (R) by induction as follows. When k = 2, let f (x) = x + 1, g(x) = x + α, for all x ∈ R. Then f and g generate a minimal Z -action φα,2 : Z2 → Homeo+ (R). 2

Suppose that φα,k has been constructed for 2 ≤ k ≤ n − 1. Then we construct φα,n as follows. Let  : R → (0, 1) be the homeomorphism defined by (x) = π1 (arctanx + π2 ), for x ∈ R. For each a ∈ R, define La : R → R by La (x) = x+a for all x ∈ R. Define φ˜ : Zn−1 → Homeo+ (R) by, for each g ∈ Zn−1 ,  ˜ φ(g)(x) =

φα,n−1 (g) −1 (x − i) + i, for x ∈ (i, i + 1), i ∈ Z, x,

for x ∈ Z.

˜ ˜ Clearly φ(g) and L1 are commutative for each g ∈ Zn−1 . So {φ(g) : g ∈ Zn−1 } and L1 generate n a Z -action φα,n . From the above construction, it is easy to see that φα,n is tightly transitive with countably many non-transitive points. The following two theorems are due to Shi and Zhou.



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Theorem 5.4[60] For any n ≥ 2 and any irrational numbers 0 < α, β < 1, α = β, φα,n and φβ,n are conjugate by an orientation preserving homeomorphism if and only if there are 1 +n1 α integers m1 , n1 , m2 , n2 with |m1 n2 − n1 m2 | = 1 such that m m2 +n2 α = β. [60] n Theorem 5.5 Let φ : Z → Homeo+ (R) be a tightly transitive group action of Zn on R with countably many non-transitive points. Then φ is conjugate to φα,n by an orientation preserving homeomorphism for some irrational number α ∈ (0, 1). From Theorem 5.4 and Theorem 5.5, the topologically conjugate classifications of tightly transitive Zn -actions on the line with countably many non-transitive points is completed and all the classes are parameterized by the set of orbits of irrational numbers under the fractional linear actions of SL(2, Z) on R. Acknowledgment The author is deeply grateful to Professor Ye Xiangdong from whom the author learned much of topological dynamics and ergodic theory.

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