ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS ALEX CLARK AND ROBBERT FOKKINK Abstract. We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition for rigidity that unifies previous rigidity results and we settle an old question of Walters. In the second part we consider the rigidity of hyperbolic systems. An algebraic dynamical system (X, S) has a phase space X that is a compact abelian group and a transformation semigroup S acts on X by affine transformations. In this paper X is always connected and metrisable, S is a discrete commutative semigroup with unity (a monoid) and the action of S preserves the Haar measure. A continuous map f : X Y between phase spaces of S-algebraic dynamical systems is equivariant if f(s x) = s f(x) for all s S. An algebraic dynamical system (Y, S) is topologically rigid if all equivariant maps into Y are affine; i.e., if f : X Y is equivariant for any X, then f(x) = y + A(x) for some y Y and a homomorphism A between X and Y. Usually, the notion is restricted to a subclass of all S-algebraic dynamical systems. For instance, if (Y, S) is ergodic, then all (X, S) are assumed to be ergodic as well. Let Z[S] be the semigroup ring that consists of all formal sums of elements of S with the natural multiplication. If S acts by endomorphisms then X has the structure of a Z[S]- module. An algebraic dynamical system does not necessarily have this structure since S acts by affine transformations, but this can be adjusted. If T is affine then T (x) = a + A(x) for some homomorphism A and we say that A is the linear part of T. If S acts on X by affine transformations, then we can give X the structure of a Z[S]-module by replacing the affine transformations x s x by their linear parts. By Pontryagin duality this induces a Z[S]-module structure on the character group ˆX. The linear space C(X, R) of real-valued continuous maps inherits an S-action that is defined by f(x) f(s x). These are endomorphisms, hence C(X, R) is a Z[S]-module. The submodule C 0 (X, R) consists of all functions such that fdµ = 0 for the normalized Haar measure µ. In the first part of the paper we show that the rigidity of Y depends on the Z[S]-module structure of the C 0 (X, R), where X ranges over the subclass of S-dynamical systems. More specifically, we obtain the following algebraic characterization of rigidity that unifies previous results of [1, 7, 2, 3]. Theorem 1. Let µ be the Haar measure on X and let C 0 (X, R) C(X, R) be the submodule of all functions for which fdµ = 0. All equivariant maps from X to Y are affine if and only if the only Z[S]-module homomorphism between Ŷ and C 0(X, R) is the trivial homomorphism. We are indebted to [3], which implicitly contains this theorem for the case that S is equal to Z d and acts by endomorphisms. The techniques we use can be viewed as an extension of the techniques used in [2]. 1991 Mathematics Subject Classification. Primary 54 H15; Secondary 37 B10, 11 R04. Key words and phrases. rigidity, hyperbolic automorphism, affine map. 1
2 ALEX CLARK AND ROBBERT FOKKINK It is not surprising that many algebraic dynamical systems are rigid: if S acts transitively on X, which it does if the action is ergodic (with respect to Haar measure), then an equivariant f is determined by a single value. So it is natural to study rigidity under a condition less restrictive than equivariance. In the second part of this paper we introduce such a condition for the subcategory of hyperbolic dynamical systems. In this case S is equal to Z and it acts on X by endomorphisms. With WX s = {x X : lim n n x = 0} and WX u = {x X : lim n n x = 0} denoting the stable and unstable groups of X, we say a continuous map f : X Y between hyperbolic algebraic systems respects the hyperbolic structure if f(0) = 0 and f(wx s ) W Y s and f(w X u ) W Y u. Theorem 2. Suppose that X is a hyperbolic algebraic system such that the path components of the stable and the unstable groups are dense. Then every f : X Y that respects the hyperbolic structure is a homomorphism. 1. Rigidity of algebraic dynamical systems Let T denote the circle group, so the character group ˆX is equal to Hom(X, T) with the compact-open topology. Lin [5] observed that Hom( ˆX, R) is very similar to the tangent space T p M of a manifold M. Definition 3. The tangent space L(X) of a compact connected abelian group X is the group of functionals Hom( ˆX, R) with the compact-open topology. The exponential map from R to T induces a homomorphism exp: L(X) X onto the path-component of the unit element 0 X. Let A be an endomorphism on X and let  be the dual action on ˆX. Then ϕ ϕ  is an operator on L(X) that has the property exp L(A) = A exp, so L(A) is a lift of A to L(X). It follows that the Z[S]-module structure on X carries over to the tangent space. Theorem 4 (Lin). The homomorphism exp: L(X) X has the homotopy lifting property. In other words, a null-homotopic f : Y X factors through a map F : Y L(X) such that f = exp F. If Y is a connected topological group and f is a homomorphism, then F is a homomorphism and F (Y ) is a compact subgroup of a linear space. So F is trivial and we find that a null-homotopic homomorphism from Y to X is trivial. It follows that any continuous map from Y to X is homotopic to at most one homomorphism. Gorin and Scheffer [4, 6] proved that in the base-point preserving category, every homotopy class contains a homomorphism: Theorem 5 (Gorin-Scheffer). Let X, Y be compact and connected abelian groups. Then each continuous f : X Y for which f(0) = 0 is homotopic to a unique homomorphism. The theorem of Gorin and Scheffer generalizes a theorem by Van Kampen that is used in [3] and [7]. A vector valued function f : X L(Y ) can be integrated with respect to Haar measure. We denote this integral by fdµ. It is an element of L(Y ). Proposition 6. Let A be an endomorphism on Y and let L(A) be its lifted operator on L(Y ). For any continuous f : X L(Y ) we have that L(A)( fdµ) = L(A) fdµ. Proof. By our assumptions f(x) is a functional on Ŷ for each x X. The integral fdµ is the functional that maps a character χ ˆX to x X f(x) (χ) dµ. The lifted operator L(A) acts on a functional ϖ by L(A) = ϕ Â. Hence L(A)( fdµ) maps χ to x X f(x) (χ A) = L(A) fdµ.
ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS 3 If S is equal to N then the action on X is induced by a single affine transformation T. We denote the algebraic dynamical system by (X, T ) in this case. The following result is related to a Theorem 1 of [2] and Lemma 1 of [7]. Theorem 7. Let (X, T 1 ) and (Y, T 2 ) be equivariant algebraic dynamical systems. Let A 2 be the linear part of T 2. Then every equivariant map from X to Y is affine if and only if each f : X L(Y ) such that f T 1 = L(A 2 ) f is constant. Proof. We say that a map f : X L(Y ) is equivariant if f T 1 = L(A 2 ) f. It is obvious that the composition of such an equivariant map f and exp induces an equivariant map between (X, T 1 ) and (Y, A 2 ). If h : X Y is an equivariance, then so is x h(x) + exp f(x). If f is not constant, then neither is exp f. Hence if every equivariant map between X and Y is affine, then every equivariant map between X and L(Y ) is constant. Assume that all equivariant maps between X and L(Y ) are constant. We have to show that an equivariant map g : X Y is affine. By the Gorin-Scheffer theorem there exists a homomorphism E : X Y that is homotopic to g g(0). Since g T 1 T 2 g is constant, E T 1 T 2 E is homotopic to a constant map. But E T 1 T 2 E is an affine map, so it must be a constant map x c for some c Y. Define h(x) = g(x) E(x) g(0). Note that h is a null-homotopic map. Now compute h(t 1 (x)) = g(t 1 (x)) E(T 1 (x)) g(0) = T 2 (g(x)) T 2 (E(x)) c g(0) = A 2 (g(x) E(x)) c g(0) = A 2 (g(x) E(x) g(0)) c + A 2 (g(0)) g(0) = A 2 (h(x)) c + A(g(0)) g(0) So h T 1 A 2 h is constant. Now h is null-homotopic, so by Lin s Theorem h = exp h for some h: X L(Y ) and by the equation above ht 1 L(A 2 ) h is constant, say equal to c. By the invariance of the Haar measure hdµ = h(t1 (x))dµ. Since h T 1 L(A 2 ) h is equal to c, by the proposition above we find that hdµ L(A2 )( hdµ) = c. It follows that the map f(x) = h(x) hdµ is equivariant from X to L(Y ), hence it is constant. This implies that g E is constant, so g is affine. Corollary 8. Let X and Y be algebraic dynamical systems. Then every equivariant map from X to Y is affine if and only if each S-equivariant map between X and L(Y ) is constant. Proof. For any s S apply Theorem 7 to the affine transformations x s x on X and y s y on Y. The following example provides a negative answer to a question of Walters [7, p. 100] that had apparently remained unsettled: if (Y, T ) is rigid and if T is ergodic, is it true that Ŷ is torsion? Example 9. Let σ be the two-sided shift on the infinite torus T Z. Then τ = 2σ is rigid in the sense that any τ-equivariant map f : T Z T Z is affine. Proof. The tangent space of T Z is equal to R Z and the induced operator L(τ) is equal to (r n ) (2r n+1 ). In particular, L(τ) is bijective. If f : T Z R Z is τ-equivariant, then f(x) = 0 for any x such that τ n (x) = 0 for some n N. It is not hard to check that the x with this property form a dense subgroup of T Z. So f is constant and by Theorem 7, τ is rigid. Theorem 10. All equivariant maps from X to Y are affine if and only if the only Z[S]- module homomorphism between Ŷ and C 0(X, R) is the trivial homomorphism.
4 ALEX CLARK AND ROBBERT FOKKINK Proof. First observe that C 0 (X, R) is a quotient of C(X, R) under the projection f f fdm so an equivariant map between Ŷ and C(X, R) projects onto an equivariant map between Ŷ and C 0(X, R). This projection is trivial if and only if the image of Ŷ in C(X, R) consists of constant maps. It suffices to prove the theorem for systems with a single affine transformation (X, T 1 ) and (Y, T 2 ). The general case follows just like Corollary 8 follows from Theorem 7. Since L(Y ) is equal to Hom(Ŷ, R) an equivariant map f : X L(Y ) induces a map f : X Ŷ R by (x, χ) f(x)(χ). It is continuous in the first coordinate, additive in the second coordinate, and has the property that f(x, χ A 2 ) = f(t 1 (x), χ). For each fixed character χ the real-valued map x f(x, χ) is continuous. Denote this map by f χ. Then χ f χ is an equivariance between Ŷ and C(X, R). Conversely, any such equivariance induces an equivariant f : X L(Y ) defined by f(x)(χ) = f χ (x). The maps f χ are constant for all χ if and only if f is constant. By Theorem 7 equivariant maps between X and Y are affine if and only if the image of equivariant maps between Ŷ and C(X, R) are in the subset of constant maps. The result now follows by the projection of C(X, R) to C 0 (X, R). We show how the rigidity results of Walters [7] and Bhattacharya and Ward [3] relate to Theorem 10. Theorem 11 (Bhattacharya-Ward). Suppose S is Z d, and consider the subcategory of S- actions that are mixing. If Ŷ is a Noetherian module, then Y is rigid if and only if Ŷ is a torsion module. Proof. The proof depends on some non-trivial observations on the algebraic structure of C 0 (X, R) and Ŷ. In Lemma 4.2 of [3] it is shown that C 0(X, R) is torsion-free if X is mixing. So if Ŷ is torsion then the trivial homomorphism is the only homomorphism between Ŷ and C 0 (X, R). If Ŷ is not a torsion module, then Ŷ / Tor(Ŷ ) is a non-trivial torsion-free module. By Lemma 2.2 of [3] it embeds as a submodule of free module, since Ŷ is Noetherian. Hence Ŷ / Tor(Ŷ ) maps nontrivially to C 0 (X, R) for any nontrivial X. Walters also considers the case that Ŷ is a torsion module and that S is equal to N. His rigidity result is easiest to state by extension of the base, replacing Ŷ by R Z Ŷ. Since the action on Y is generated by a single affine transformation T, R Z Ŷ is a vector space on which A, the linear part of T, acts. Recall that f : X C is an eigenfunction of (X, T ) if f(t x) = λf(x) for some λ C. Theorem 12 (Walters). If Ŷ is torsion, then each equivariant map from (X, T 1) to (Y, T 2 ) is affine if there exists no eigenfunction f C 0 (X, R) at an eigenvalue λ of A 2 on R Z Ŷ. We note that Walters avoids the use of the eigenfunction in his original statement. He prefers the equivalent condition that there exists no non-constant δ ˆX such that δ A p 1 = δ and δ T p 1 1 (0) = λ p for all such p. Proof. Suppose that there exists a non-trivial equivariant map g : R Z Ŷ C 0 (X, R). Then some element y R Z Ŷ satisfies g(y) 0. Since Ŷ is torsion, the module M generated by y is finite dimensional. By the Jordan decomposition theorem g(e λ ) is nonzero for some generalized eigenspace E λ M, which has a basis v i for i = 1,..., k such that A 2 (v i ) = λv i + v i 1 (with v 0 = 0). If i is the least index such that g(v i ) 0 then g(v i )
ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS 5 is an eigenfunction of X at an eigenvalue of A 2. Hence if no such eigenfunction exists, then (Y, T 2 ) is rigid. C(X, R) is an S-module even if S acts on X by arbitrary continuous transformations. So the algebraic condition of Theorem 10 applies to a general S-action on X, suggesting that a more general rigidity result is possible. According to the Gorin-Scheffer theorem, an affine transformation represents a homotopy class once we fix a base-point. So if all equivariant maps are affine, then this implies that no two equivariant maps are homotopic, once we fix a base-point. To fix our ideas, we consider the base-point preserving category with the unit element as the base-point. Let C p (X, R) be the submodule of real-valued maps such that f(0) = 0. We say that Y is homotopically rigid if if no two base-point preserving equivariant maps into Y are homotopic. Theorem 13. An S-algebraic dynamical system Y is homotopically rigid with respect to a category of S-dynamical systems X if and only if the only Z[S]-module homomorphism between Ŷ and C p(x, R) is the trivial homomorphism. Proof. Suppose that f and g are two homotopic equivariant maps from X to Y. Then f g is equivariant too and since it is homotopic to the trivial map, it can be lifted to an equivariant map h: X L(Y ). In the proof of Theorem 10 we found that such an equivariance is equivalent to a module homomorphism Ŷ C(X, R) given by χ (x h(x)(χ)). Now we are dealing with pointed spaces and f and g preserve the base-point, so h(0)(χ) = 0 for all χ. It follows that the module homomorphism has image in C p (X, R). Hence, the only module homomorphism between Ŷ and C p(x, R) is the trivial one if and only if each homotopy class contains at most one equivariant map. 2. Rigidity of hyperbolic dynamical systems In the previous section we saw that many affine transformations are rigid in the sense that conjugating maps are affine. Conjugation is a restrictive notion for equivalence that is sometimes weakened to obtain other classifications. Conjugacy of flows, for instance, is far more restrictive than (orbit) topological equivalence of flows. Conjugation preserves some dynamical properties that topological equivalence does not. In this section we show that many algebraic dynamical systems remain rigid even if we replace conjugation by a relation that resembles topological equivalence for flows. We consider groups as pointed spaces (X, 0), and in what follows any map f : X Y is assumed to preserve the base-point: f(0) = 0. We only consider algebraic systems in this section for which S = Z so we are dealing with an automorphism A that acts on a compact connected abelian group. Recall that WX s = {x X : lim n A n (x) = 0} and WX u = {x X : lim n A n (x) = 0} denote the stable and unstable groups. Now let W X s and W X u be the stable and unstable manifolds of the lifted operator L(A) on the tangent space L(X). Notice that the image of W X s under the exponential map is equal to the path component of 0 in WX s since the image of L(X) is the path component of 0. Definition 14. We say that an algebraic dynamical system (X, A) is hyperbolic if the tangent space decomposes as L(X) = W s X W u X. For example, an expansive automorphism on a compact and connected group is hyperbolic. There are examples of non-expansive hyperbolic automorphisms, in particular examples on infinite-dimensional groups.
6 ALEX CLARK AND ROBBERT FOKKINK Note that a continous base-point preserving map f : X Y lifts to L(f): L(X) L(Y ). To see this, note that f exp: L(X) Y is null-homotopic. Also note that the lift is unique by connectedness of L(X). If X and Y are hyperbolic, then we say that f respects the hyperbolic structure, if L(f)( W X s ) W Y s and L(f)( W X u ) W Y u. Definition 15. We say that a hyperbolic system (X, A) is rigid if any f : X Y that respects the hyperbolic structure is a homomorphism. Lemma 16. Any f : X Y that respects the hyperbolic structure is homotopic to a homomorphism that respects the hyperbolic structure. Proof. According to the Theorem of Gorin-Scheffer there exists a unique h that is homotopic to f. Let H t be a homotopy between f h and the constant map. By Theorem 4 there exists a covering homotopy H t : X L(Y ) of H t. By the compactness of X, there is a uniform bound on the diameter of the arcs { H t (x): 0 t 1}. Now H t lifts to a homotopy L(H t ) between L(f) and L(h) and L(H t ) is equal to H t exp. So we find that L(h)( W X s {0}) is a linear subspace of L(Y ) that is within a bounded distance of the linear subspace W Y s {0} since the arcs {L(H t )(x): 0 t 1} are bounded. The linearity implies that the distance is zero, so the image of L(h) is contained in W Y s {0}. By the same argument h({0} W X u ) is contained in {0} W Y u. Theorem 17. Let (X, A) be a hyperbolic algebraic dynamical system and let W s (X,0) and W u (X,0) be the path-components of 0 in W s X and W u X, respectively. If both W s (X,0) and W u (X,0) are dense in X, then (X, A) is rigid. Proof. Suppose that f : X Y preserves the hyperbolic structure. By the previous lemma, f is homotopic to a homomorphism h with h(w s (X,0) ) W s (Y,0) and h(w u (X,0) ) W u (Y,0). Let D : (X, 0) (L(Y ), 0) be the lift of h f. By connectivity D(W(X,0) s ) W Y s {0}. By density D(X) W Y s {0}. Similarly D(X) {0} W Y u. Hence D(X) = 0 and f is equal to h. Expansive toral automorphisms, for example, satisfy the conditions of the theorem. More generally, if an expansive automorphism on a compact connected group has stable and unstable group with non-trivial path components, then it is rigid. The shift on the dyadic solenoid, on the other hand, has a totally disconnected stable group. Even more so, it is not difficult to construct a homeomorphism h on the dyadic solenoid that preserves both the stable and the unstable group (so it preserves more than just the hyperbolic structure), such that h is not a homomorphism. 3. Acknowledgement A part of the paper was written when the first author visited Delft University on NWO grant B 61-563. References [1] D. Z. Arov, Topological similitude of automorphisms and translations of compact commutative groups, Uspehi Mat. Nauk 18 1963 no. 5 (113), 133 138. [2] S. Bhattacharya, Orbit equivalence and topological conjugacy of affine actions on compact abelian groups, Monatsh. Math. 129 (2000), no. 2, 89 96. [3] S. Bhattacharya, T. Ward, Finite entropy characterizes topological rigidity on connected groups, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 365 373. [4] E.A. Gorin, A function-algebra variant of the Bohr-van Kampen theorem, (Russian) Mat. Sb. (N.S.) 82 (124) 1970 260 272.
ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS 7 [5] V. Ja. Lin, Semi-invariant integration with values in a group, and some applications of it, (Russian) Mat. Sb. (N.S.) 82 (124) 1970 233 259. [6] W. Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562 567. [7] P. Walters, Topological conjugacy of affine transformations of compact abelian groups, Trans. Amer. Math. Soc. 140 1969 95 107. [8] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. Faculty of Mathematics, University of North Texas, Denton, Texas, USA Delft University, Faculty of Electrical Engineering, Mathematics and Information Technology, P.O.Box 5031, 2600 GA Delft, Netherlands