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Let rn(ǫ, f) be the minimum cardinality of an (n, ǫ)-spanning set and sn(ǫ, f) be the maximum cardinality of an (n, ǫ)-separated set. Then we have the following relation ǫ r (ǫ, f) ≤ s (ǫ, f) ≤ r ( , f) n n n 2 and this implies the following equivalent definition of the topological entropy: Definition 1.2. Let f : X → X be a continuous map. The topological entropy of f is defined as 1 1 h(f) = lim lim sup log sn(ǫ, f) = lim lim sup log rn(ǫ, f). ǫ→0+ n→∞ n ǫ→0+ n→∞ n SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 3 In [10], Shub stated the following conjecture. Entropy conjecture. Let f be a continuous map on a compact manifold M to itself. Then
h(f) ≥ log r(f∗), where f∗ : H∗(M, R) → H∗(M, R) denotes the homomorphism induced by f on the total homology of M, i.e., dim M R R (1.1) H∗(M, )= M Hi(M, ) i=0
n 1 and r(f∗) denotes the spectral radius of f∗, i.e., r(f∗) = lim (kf∗ k n ), which coincides n→∞ with the maximum of the moduli of the eigenvalues of f∗. The entropy conjectures means that f∗ must capture some but not necessarily all of the mixing f does. Since the expansion (1.1) is clearly invariant under f∗, we have
r(f∗) = max r(f∗i) 1≤i≤dim M where f∗i denotes the restriction of f∗ to the space Hi(M; R). Hence the entropy con- jecture is equivalent to the system of inequalities
h(f) ≥ log r(f∗i), i =1, ··· , dim M. In fact, the entropy conjecture only holds for some special cases. In [5], Manning showed that Theorem 1.3. Let M be a compact differentiable manifold without boundary and f : M → M a continuous map. Then
h(f) ≥ log r(f∗1) where f∗1 : H1(M; R) → H1(M; R) is the induced map on the first homology group. Corollary 1.4. If M has dimension ≤ 3 and f is a homeomorphism, then h(f) ≥ log r(f∗) where f∗ is the map induced on the homology of all dimensions. In [8], the authors propose to imitate for homeomorphisms the “Markov approxi- mation” procedure for diffeomorphisms mentioned in [11] and construct a dense set of homeomorphisms where the entropy conjecture holds. Theorem 1.5. If dim M ≥ 5, the entropy conjecture holds for all homeomorphisms belonging to an open dense set of the space of all homeomorphisms of M with the C0- topology. Next, due to Nitecki in [7], this holds for diffeomorphisms with a hyperbolic structure. Thin would imply that any sufficiently small perturbation in the C0-topology of such a homeomorphism could not decrease the topological entropy. For arbitrary homeomorphisms by [9], however, the entropy conjecture does not hold. It is also shown in [5] that it is not necessary to suppose that M is a manifold. It suffices that the following conditions be met: 4 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU (1) For any ǫ > 0, there exists a δ > 0 such that any two points x and y for which d(x, y) < δ can be joined by a path of diameter < ǫ. (2) There exists an ǫ0 > 0 such that any loop of diameter < ǫ0 is contractible in M. Inspired by the relaxation of the restrictions Manning did on the space on which f acts, we attempt to generalize the existing results regarding smooth manifolds to a metric scale.
2. Scaled homology and cohomology Definition 2.1. Given ǫ> 0, a map f : X → Y is said to be ǫ-continuous if there exists δ = δ(ǫ) > 0 such that
sup{dY (f(x1), f(x2)) : dX (x1, x2) < δ} < ǫ. n Rn+1 n Let ∆ = {(t0, ..., tn) ∈ | Pi=0 ti = 1 and ti ≥ 0, ∀i}, the standard n-simplex. An ǫ-continuous map σ : ∆n → X is said to be an ǫ-singular n-simplex. Moreover, ǫ we define the n-dimensional ǫ-singular chain group, denoted by Cn(X), to be the free ǫ ǫ abelian group generated by all ǫ-singular n-simplices of X, and ∂n : Cn(X) → Cn−1(X) to be the boundary operator defined as the one in classic singular homology theory, i.e., n i ∂nσ = X(−1) σ|[v0, ··· , vˆi, ··· , vn]. i=0 ǫ Then we call (C∗(X), ∂) an ǫ-singular chain complex. ǫ ǫ Definition 2.2. Let (C∗(X), ∂) be an ǫ-singular chain complex. Define Zn(X) = ǫ Ker(∂n) to be the subgroup of cycles and Bn(X) = Im(∂n+1) to be the subgroup of ǫ ǫ ǫ th boundaries. Furthermore, Hn(X) = Zn(X)/Bn(X) is said to be the n ǫ-homology group of X. ǫ Z Proposition 2.3. If (X,d) is a metric space with diameter < ǫ, i.e., then H0(X) = ǫ and Hn(X)=0 for all n> 0. Proof. As in the case of singular homology theory, it is easy to see that if X is a point ǫ ǫ ∼ Z {pt}, then Hn(X) = 0 for n > 0 and H0(X) = since there is a unique ǫ-singular n-simplex σn for all n. Now let f be a constant map sending all of X to x0 ∈ X. Since diamX < ǫ, f is ǫ-homotopic to the identity map id : X → X, i.e., there exists F : X × I → Y given by F (x, t)= ft(x) that is ǫ-continuous such that f0 = f, f1 = id. n n ǫ ǫ Let ∆ × 0 = [v0, ··· , vn] and ∆ × 1 = [w0, ··· ,wn] and let P : Cn(X) → Cn+1(X) be such that i P (σ)= X(−1) F · (σ × id)|[v0, ··· , vi,wi, ··· ,wn] i for σ : ∆n → X, an ǫ-continuous map. It is easy to check that P is a chain homotopy between the chain maps f♯ and id♯ that are induced by f and id respectively, and f∗ : ǫ ǫ ǫ Z ǫ Hn(X) → Hn({pt}) are isomorphisms for all n. Therefore, H0(X)= and Hn(X)=0 for all n> 0. SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 5 Thus, a metric space (X,d) with ǫ-scale can be informally seen as a “locally con- tractible” space. Like in the singular homology theory, we can define the reduced ǫ- ǫ homology groups H∗(X) to be the homology groups of the augmented chain complex e / ǫ ∂2 / ǫ ∂1 / ǫ ε / Z / ··· C2(X) C1(X) C0(X) 0 ǫ ǫ Z ǫ where ε(Pi kiσi)= Pi ki. It can be easily check that H0(X)= H0(X)⊕ and Hn(X)= ǫ e Hn(X) for all n> 0. e ǫ ǫ ǫ ǫ ǫ For a subspace A ⊂ X, let Cn(X, A) = Cn(X)/Cn(A). Then ∂ : Cn(A) → Cn−1(A) ǫ ǫ 2 induces a quotient boundary map ∂ : Cn(X, A) → Cn−1(X, A) and we have ∂ =0. So ǫ we can define the relative ǫ-homology group Hn(X, A) = Ker ∂/ Im ∂. ǫ + An interesting object is the limit of Hn(X) as ǫ → 0 . Note that an n-dimensional µ-singular simplex is naturally an n-dimensional ǫ-singular simplex if 0 < µ ≤ ǫ. Then, let ϕµǫ∗ be the homomorphism induced by the natural inclusion chain map µ ǫ ǫ .(ϕµǫ :(Cn (X), ∂) ֒→ (Cn(X), ∂). In particular, ϕǫǫ∗ is the identity on Hn(X Since (R+, ≤) is a directed partially-ordered set, we obtain an inverse system ǫ ((Hn(X))ǫ∈R+ , (ϕµǫ∗)µ≤ǫ∈R+ ). Furthermore, we can take the inverse limit lc ǫ H (X) = lim ((H (X)) R+ , (ϕ ) R+ ) n ←−−− n ǫ∈ µǫ∗ µ≤ǫ∈ ǫ∈R+ −→ ǫ R+ = {[a] ∈ Y Hn(X):[aǫ]= ϕµǫ∗([aµ]) for all µ ≤ ǫ in }. ǫ∈R+ + Let {ǫm} be a non-increasing sequence of R converging to 0. Then we have
ǫ ǫm ∞ lim ((H (X)) R+ , (ϕ ) R+ ) ∼= lim((H (X)) , (ϕ ) N), ←−−− n ǫ∈ µǫ∗ µ≤ǫ∈ ←− n m=1 ǫkǫl∗ l≤k∈ ǫ∈R+ + since {ǫm} is a cofinal subset of the directed index set R . We call the inverse limit above the nth infinitesimal-scaled homology group of X or lc-homology group of X (the homology group under infinitesimal scale or the homology group of the “local contract- ification” of X). The lc-homology group of X is said to be stable if there exists ǫ > 0 such that for µ ∼ ǫ all 0 < ǫ2 ≤ ǫ1 ≤ ǫ, ϕǫ2,ǫ1∗ is an isomorphism, i.e., for any 0 <µ ≤ ǫ, Hn (X) = Hn(X). Later we will show that when X is a Riemannian manifold, its lc-homology group will be stable. ǫ Actually, here if we let ǫ → ∞, we get a direct system ((Hn(X))ǫ∈R+ , (ϕµǫ∗)µ≤ǫ∈R+ ). Then we can take the direct limit whose form will be very similar to the coarse homology theory introduced in [6]. To begin with, we check that lc-homology is a functor from category of topological spaces Metu to category of abelian groups Ab, where Metu denotes the category that has metric spaces as its objects and uniformly continous maps between metric spaces as its morphisms. Proposition 2.4. If f : X → Y is a uniformly continuous map, then f induces a lc lc lc homomorphism f∗ : Hn (X) → Hn (Y ), ∀n ≥ 0. 6 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU Proof. Let σ : ∆n → X be an ǫ-continuous map. Composing with f, we get a µ- ǫµ n continuous map f♯ (σ)= fσ : ∆ → Y , where µ is determined by f and σ. In fact, we can choose µ = inf{r > 0 : whenever dX (σ(v1), σ(v2)) < ǫ,dY (f(σ(v1)), f(σ(v2))) < r}. ǫµ ǫ µ ǫµ ǫµ Extend f♯ : Cn(X) → Cn (Y ) linearly by f♯ ( i niσi) = i nif♯ (σi) and we have ǫµ ǫµ ǫµ P P f♯ ∂ = ∂f♯ , which implies that f♯ takes cycles to cycles and takes boundaries to ǫµ ǫ µ boundaries. Hence f♯ induces a homomorphism f∗ : Hn(X) → Hn (Y ). Let {ǫn} be a monotonically non-increasing sequence such that limn→∞ ǫn = 0. For n each ǫk-continuous map ϕ : ∆ → X, composing with f, we get a corresponding µk- ǫkµk n continuous map f♯ (ϕ)= fϕ : ∆ → Y , where µk = inf{rk > 0 : if dX (σ(v1), σ(v2)) < ǫk,dY (f(σ(v1)), f(σ(v2))) < rk}. Since f is uniformly continuous, µn → 0 as ǫn → lc 0. Hence, after taking the inverse limit, we obtain the induced homomorphism f∗ : lc lc Hn (X) → Hn (Y ). Naturally, we can define the corresponding lc-cohomology group of X in the following ways. Let G be an abelian group. For the ǫ-singular chain complexes
/ ǫ ∂ / ǫ ∂ / ǫ / ··· Cn+1(X) Cn(X) Cn−1(X) ··· we apply the Hom(−,G) functor and obtain the corresponding ǫ-singular cochain com- plex o n+1 o δ n o δ n−1 o ··· Cǫ (X; G) Cǫ (X; G) Cǫ (X; G) ··· n ǫ ∗ n n+1 where Cǫ (X; G) = Hom(Cn(X); G) and δ = ∂ : Cǫ (X; G) → Cǫ (X; G) sending n n+1 ϕ ∈ Cǫ (X; G) to ϕ∂ ∈ Cǫ (X; G). Then we can define the ǫ-singular cohomology group n Hǫ (X; G) = Ker δ/ Im δ. ∗ n n Let iµǫ : Hǫ (X; G) → Hµ (X; G) be the homomorphism induced by the natural inclusion µ ǫ n ∗ iµǫ : Cn (X) ֒→ Cn(X) and we obtain a direct system ((Hǫ (X))ǫ∈R+ , (iµǫ)µ≤ǫ∈R+ ). Then we can take the direct limit n n ∗ n R+ R+ Hlc(X) =−−−→ lim ((Hǫ (X))ǫ∈ , (iµǫ)µ≤ǫ∈ )= M Hǫ (X)/S R+ ǫ∈ ǫ∈R+ ∗ n where S is generated by {qµiµǫ([ϕǫ]) − qǫ[ϕǫ]:[ϕǫ] ∈ Hǫ (X), qǫ is the embedding : n n .{(Hǫ (X) ֒→ L Hǫ (X ǫ∈R+ Next, we will check whether the lc-homology satisfies the Eilenberg-Steenrod axioms of homology theory. Obviously, the lc-homology theory satisfies the dimension axiom.
lc Theorem 2.5 (Dimension Axiom). Hn ({pt})=0 for all n> 0. Theorem 2.6 (Homotopy Axiom). Let f,g : X → Y be two uniformly continuous maps. If f and g are uniformly homotopic, i.e., there is a family of uniformly continuous maps ft : X → Y, t ∈ [0, 1] such that f0 = f, g0 = g and the associated map F : lc lc X × [0, 1] → Y given by F (x, t) = ft(x) is uniformly continuous, then f∗ = g∗ : lc lc Hn (X) → Hn (Y ), ∀n ≥ 0. SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 7 + Proof. Let {ǫn} be a non-increasing sequence of R with limn→∞ ǫn = 0. and we can take lc ǫ ∞ H (X) = lim((H m (X)) , (ϕ ) N) n ←− n m=1 ǫk,ǫl∗ l≤k∈ −→ ∞ ǫm N = {[a] ∈ Y Hn (X):[aǫl ]= ϕǫk,ǫl∗([aǫk ]) for all l ≤ k ∈ }. m=1 Then, we have −→ −−→ lc N f∗ ([a]) = {[f(a)] : [f(aǫl )] = ϕµk,µl∗([f(aǫk )]) for all l ≤ k ∈ } and −→ −−→ lc N g∗ ([a]) = {[g(a)] : [g(aǫl )] = ϕωk,ωl∗([g(aǫk )]) for all l ≤ k ∈ }, where an ǫ∗-continuous map composed with f and g will be a µ∗-continuous map and an ω∗-continuous map respectively. Since a µk-continuous map is naturally an ωk-continuous map if µk ≤ ωk, without loss of generality, we can assume µ∗ = ω∗. Next, given a homotopy F : X × I → Y from f to g with associated maps uniformly n continuous and an ǫk-singular simplex σ : ∆ → X, we can form the composition n F · (σ × id) : ∆ × I → X × I → Y with associated maps θk-continuous. As in the case of singular homology theory, we can construct a prism operator which cuts ∆n × I into a sum of (n + 1)-simplices in ∆n × I by n i P (σ)= X(−1) F · (σ × id)|[v0,··· ,vi,wi,··· ,wn] i=0
n n θk where ∆ × 0 = [v0, ··· , vn] and ∆ × 1 = [w0, ··· ,wn]. Then P (σ) ∈ Cn+1(Y ). Pick τk τk = max{µk, θk} and we have P (σ),g♯ − f♯ − P ∂ ∈ Cn+1(Y ). As we did in singular homology theory, ∂P (σ)= g♯(σ) − f♯(σ) − P ∂(σ). ǫk τk Hence, f and g induce the same homomorphism f∗ = g∗ : Hn (X) → Hn (Y ),k = lc lc lc lc 1, 2, ··· . By taking the inverse limit, f∗ = g∗ : Hn (X) → Hn (Y ), ∀n ≥ 0. lc lc lc Corollary 2.7. The maps f∗ : Hn (X) → Hn (Y ) induced by a uniform homotopy equivalence f : X → Y are isomorphisms for all n. In particular, if X is compact and lc contractible, then Hn (X)=0 for any n. e Remark 2.8. Actually, we cannot simply omit the restriction ”uniformly homotopic”. Indeed, let p ∈ S1 and consider a space S1 \{p}. If ǫ is sufficiently small, then we cannot 1 ǫ 1 1 ∼ ǫ 1 distinguish between Cn(S ) and Cn(S \{p}). Naturally, we have Hn(S ) = Hn(S \{p}) 1 (we will give a rigorous proof later in this section) since B(p, 2 ǫ) \{p} is contractible under the ”ǫ-scale” and can be regarded as a point. By taking inverse limit, we have lc 1 Z 1 H1 (S \{p})= , even though S \{p} is obviously contractible. The reason for this is that the homotopy map connecting the retraction of S1 \{p} to a point and the identity map is not uniformly continuous. Theorem 2.9 (Excision Axiom). Let Z and A be two subspaces of X such that the closure of Z is contained in the interior of A, i.e., Z¯ ⊆ A˚ and dist(Z,∂A¯ ) > 0. Then lc lc the inclusion map i : (X − Z, A − Z) → (X, A) induces an isomorphism i∗ : Hn (X − 8 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU lc ˚ ˚ Z, A−Z) → Hn (X, A) for all n. Equivalently, for subspaces A, B ⊆ X with X = A∪B, lc lc (the inclusion (B, A ∩ B) ֒→ (X, A) induces isomorphisms Hn (B, A ∩ B) → Hn (X, A for all n. Proof. For the cover {A, B} with X = A ∪ B, pick 0 <ǫ< dist(Z,∂A¯ ) and denote ǫ{A,B} ǫ Cn (X) by Cn(A + B), the formal sums of chains in A and chains in B. As we do for singular homology theory, by barycentric subdivision of chains, we can construct a ǫ ǫ ǫ ǫ chain map S : Cn(X) → Cn(X) and a chain homotopy T : Cn(X) → Cn+1(X), s.t. ∂T + T ∂ = id − S. In addition, for the cover A, B of X and any ǫ-simplex σ : ∆n → X, there exists m ∈ N, m ǫ{A,B} s.t. S σ ∈ Cn (X) since dist(A ∩ B) > ǫ. It will follow that S, T restrict to maps ǫ{A,B} i on Cn (X). Let Dm = P0≤i Theorem 2.10 (Refined Additivity Axiom). Let X = F Xλ. If there exists ǫ> 0 such λ∈Λ lc ∼ lc that for any k,λ ∈ Λ, dist(Xk,Xλ) > ǫ, then Hn (X) = L Hn (Xλ) for all n. λ∈Λ 1 Example 2.11. X = { ; k =0, 1, 2, ···}∪{0}. 2k k+1 ǫk Take ǫk = 1/2 . For a fixed ǫk, H0 (X) is the free abelian group generated k+1 ǫk by elements of X that are ≥ 1/2 and {0}, i.e., elements of H0 (X) are given by Zk+3 ǫk lc Z|X| (a0, a1, ··· , ak+2) ∈ . Hence the inverse limit of (H0 (X))n∈N is H0 (X)= , the direct product of Z, where |X| denotes the cardinality of X. (k) (k) Next, for ǫk, X can be written as X = X1 F X2 with 1 1 1 1 X(k) = {0, 1, , ··· , },X(k) = { , , ···} 1 2 2k 2 2k+1 2k+2 (k) (k) ǫk ∼ ǫk ǫk and for n ≥ 1, Hn (X) = Hn (X1 ) ⊕ Hn (X2 ). By dimension axiom, 1 Hǫk (X(k)) ∼= Hǫk ({0}) ⊕ Hǫk ({1}) ⊕···⊕ Hǫk ({ })=0. n n n n n 2k (k) ǫk (k) Since the diameter of X2 is less than ǫk, by proposition 2.3, Hn (X2 ) = 0. Therefore, ǫk lc Hn (X) = 0 holds for all ǫk and by taking the inverse limit, Hn (X)=0 for n ≥ 1. SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 9 Consider the commutative diagram / ǫ iǫ / ǫ jǫ / ǫ / 0 Cn(A) Cn(X) Cn(X, A) 0 ∂ ∂ ∂ 0 / Cǫ (A) / Cǫ (X) / Cǫ (X, A) / 0 n−1 iǫ n−1 jǫ n−1 where iǫ is inclusion and jǫ is the quotient map. By Snake Lemma, we obtain an induced long exact sequence in ǫ-homology: / ǫ iǫ∗ / ǫ jǫ∗ / ǫ ∂ǫ / ǫ / ··· Hn(A) Hn(X) Hn(X, A) Hn−1(A) ··· Then we have the following proposition. lc ∼ lc ¯ Proposition 2.12. Let A be a subspace of X. Then we have Hn (A) = Hn (A) for all n. ǫ ∼ ǫ ¯ Proof. First we claim that for any ǫ > 0, Hn(A) = Hn(A). Indeed, by the long exact sequence in ǫ-homology / ǫ ¯ ∂ǫ / ǫ iǫ∗ / ǫ ¯ jǫ∗ / ǫ ¯ / ··· Hn+1(A, A) Hn(A) Hn(A) Hn(A, A) ··· ǫ ¯ it suffices to show that Hn(A, A) = 0 for all n. ǫ ¯ ǫ ¯ ǫ k Let [α] ∈ Hn(A, A) where α ∈ Cn(A) and ∂α ∈ Cn−1(A). Write α = Pi=1 niαi where n αi : ∆ → A¯ is ǫ-continuous. For each αi, ∃δi > 0, s.t. ωi = sup{d(αi(x1),αi(x2)) : d(x1, x2) < δi} < ǫ. By definition, k n ∂α = X X niαi|[v0, ··· , vˆj, ··· , vn] i=1 j=0 n which is a formal sum of ǫ-continuous maps. Hence the restrictions of αi : ∆ → A¯ to the faces of ∆n consist of ǫ-continuous maps whose images are contained in either A¯ or A, and in the formal sum of ∂α, all of those ǫ-continuous maps whose images intersect ∂A are cancelled. n Then, we construct βi : ∆ × I → A¯ by n n βi(X λlvl, t) ≡ αi(X λlvl) l=0 l=0 l6=j l6=j n where l=0 λl = 1 for j =0, 1, ··· , n and Pl6=j n n βi(X λlvl, 0) = αi(X λlvl) l=0 l=0 n where l=0 λl = 1. P n Moreover, for other points (x, t) ∈ ∆ × I, if αi(x) ∈ A, then we let βi(x, t)= αi(x). On the other hand, if αi(x) ∈/ A, we do a perturbation for αi(x) under the scale of 10 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU (ǫ − ωi)/3 and we obtain αi(x). Then we let βi(x, t) = αi(x). Clearly, βi is still ǫ- n continuous and since ∆ × eI is a simplicial complex, βi cane be seen as a formal sum of k ǫ ¯ ǫ ǫ-singular n + 1-simplices. Let β = Pi=1 niβi ∈ Cn+1(A). Then ∂β − α ∈ Cn(A). So ǫ ¯ ǫ ∼ ǫ ¯ [α] = 0 in Hn(A, A) and Hn(A) = Hn(A) for ∀ǫ> 0. lc ∼ lc ¯ Finally, by taking the inverse limit, Hn (A) = Hn (A) for all n. But unfortunately, the inverse limit functor preserving left exactness is still not exact in our case. i Example 2.13. Let A be the set of positive integers and let X be the set of all n + n where i, n are integers with 0 ≤ i < n, i.e., 1 1 2 1 1 3 1 X = {1, 2, 2 2 , 3, 3 3 , 3 3 , 4, 4 4 , 4 2 , 4 4 , 5, 5 5 , ···} Consider the short exact sequence / / ǫ / / 0 Ker iǫ∗ H0(A) Im iǫ∗ 0 ǫ where H0(A) is the same for all 0 <ǫ< 1, which is the direct sum of countably Z ǫ infinitely many copies of and H0(X) is a finitely generated abelian group. Without ǫn ǫn loss of generality, take ǫn =1/n. Then elements of iǫn∗(H0 (A)) ⊂ H0 (X) are given by a ǫn sequence of integers (a0, a1, ··· , an), i.e., iǫn∗(H0 (A)) is generated by only n generators. Let ([a] ) ∈ lim Im(i ), which can be written as ǫn ←− ǫn ((a1, a2), (a1, a2, a3), ··· , (a1, a2, ··· , am), ··· ) and can have countably infinite length (the number of nonzero coefficients). But the element of lim Hǫn (A) can be characterized by the element of the direct sum of countably ←− 0 infinitely many copies of Z, so its length can only be finite. Therefore, there is no lc element of H0 (A) that can go to the elements with infinitely many nonzero coordinates lc in lim Im iǫn∗, i.e., H0 (A) → lim Im iǫn∗ is not surjective and we fail to show the exactness ←−ǫ ←− at H0(X). Furthermore, we can give a counterexample showing that the lc-homology doesn’t satisfy the exactness axiom. i Example 2.14. Let B be the set of all n + n where i, n are integers with 0 ≤ i < n, i.e., 1 1 2 1 1 3 1 B = {1, 2, 2 2 , 3, 3 3 , 3 3 , 4, 4 4 , 4 2 , 4 4 , 5, 5 5 , ···}. ǫn Take ǫn to be such that 1/n<ǫn ≤ 1/(n − 1). Clearly, H0 (B) is the free abelian group ǫ3 Z4 generated by the elements of B which are ≤ n. For instance, for n = 3, H0 (B) = , 1 2 ǫ3 which is freely generated by {[1], [2], [2.5], [3] = [3 3 ] = [3 3 ]}. Thus, elements of H0 (B) 4 are given by the set of tuples {(a1, a2, a5/2, a3) ∈ Z }. Add a “dummy variable” a0 = −(a1 + a2 + a5/2 + a3) and we obtain a sequence of five integers (a0, a1, a2, a5/2, a3) whose sum is 0. ǫn lc Z|B+| Take B+ = B `{0}. Then the inverse limit of (H0 )n∈N is H0 (B)= , the set of all infinite sequences of integers (a0, a1, a2, a5/2,... ) indexed by the elements of B+. Let A = B ×{0, 1} = B B, the disjoint union of two copies of B with distance lc lc ` lc > ǫn. Then H0 (A) = H0 (B) ⊕ H0 (B) whose elements are given by pairs of infinite sequences of integers (a0, a1, a2, a5/2,... ), (b0, b1, b2, b5/2,... ). SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 11 ǫ Take X = A ∪{1, 2, ···}× I. For any ǫ < 1, H1(X, A) is the free abelian group generated by the set of vertical line segments in X, which is the direct sum of countable Z ǫ ∼ lc lc copies of . Therefore, H1(X, A) = H1 (X, A) and elements in H1 (X, A) can only have finite nonzero coefficients. lc lc Note that the homomorphism H0 (A) → H0 (X) adds together the coefficients an, bn for all integers n, so the kernel consists of all pairs of infinite sequence (a∗, b∗) such that ak + bk = 0 for all integers k and ax = bx = 0 for all x ∈ B that are not integers. So the lc lc Z kernel of H0 (A) → H0 (X) is a product of infinitely many copies of , i.e., it can have lc countably infinite nonzero coefficients that cannot be the image of H1 (X, A). Therefore, the sequence / lc / lc / lc ··· H1 (X, A) H0 (A) H0 (X) lc is not exact at H0 (A). Remark 2.15. However, lc-cohomology satisfies the exactness axiom. Indeed, the lc- cohomology sequence / n / n / n+1 / n+1 / ··· Hlc(X) Hlc(A) Hlc (X, A) Hlc (X) ··· with any coefficients is exact since it is the direct limit of the exact sequences / n / n / n+1 / n+1 / ··· Hǫ (X) Hǫ (A) Hǫ (X, A) Hǫ (X) ··· and the direct limit serves as an exact functor. In particular, for our counterexample, the lc-cohomologies are all countably generated since they are countable direct limits of countably generated groups. Therefore, the Universal Coefficient Theorem does not hold for lc-cohomology theory in general. To end up this section, we will show the equivalence of lc-homology and singular homology in one special case. Theorem 2.16. Let (X,d) be a compact n-dimensional manifold endowed with a Rie- lc mannian structure. Then the homomorphisms Hn(X) → Hn (X) are isomorphisms for all n. Proof. Since X is a Riemannian manifold, by Theorem 5.1 in [2], every point has a geodesically convex neighborhood. So for any x ∈ X, there exists ǫx > 0 such that ∼ Rn B(x, ǫx) = and X = Sx∈X B(x, ǫx). By compactnesss, there exists N > 0 such that X N B x , ǫ δ > can be expressed as Si=1 ( i xi ). Let 0 be a Lebesgue number for this cover, i.e., for ∀x ∈ X, B(x, δ) ⊂ B(xk, ǫxk ) for some 1 ≤ k ≤ N. Hence each point of X has a δ-ball homeomorphic to Rn. Now for µ ≤ ǫ < δ, consider the short exact sequence / µ / ǫ / ǫ µ / 0 Cn (X) Cn(X) Cn(X)/Cn (X) 0 and by Snake Lemma, we have the following long exact sequence / ǫ µ / µ / ǫ / ǫ µ / ··· Hn+1(X ,X ) Hn (X) Hn(X) Hn(X ,X ) ··· 12 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU ǫ µ where Hn(X ,X ) denotes the homology group of ǫ µ ǫ µ ∂ : Cn(X)/Cn (X) → Cn−1(X)/Cn−1(X). ǫ µ Then we claim that Hn(X ,X ) = 0 for all n. ǫ µ ǫ µ k Let [α] ∈ Hn(X ,X ) where α ∈ Cn(X) and ∂α ∈ Cn−1(X). Write α = i=1 niσi n P with σi : ∆ → X an ǫ-continuous map. By definition, k n ∂α = X X niσi|[v0, ··· , vˆj, ··· , vn] i=1 j=0 n which is a formal sum of µ-continuous maps. Hence the restrictions of σi : ∆ → X to the faces of ∆n consist of µ-continuous maps and ǫ-continuous maps and in the formal sum of ∂α, all of those ǫ-continuous maps are cancelled. For σi, for any x ∈ Im(σi) ⊂ X, B(x, δ) forms an open cover of its image Sx∈Im(σi) ǫ ǫ n in X. Define S : Cn(X) → Cn(X) by sending each ǫ-singular n-simplex σ : ∆ → X to n n σ♯S∆ where S∆ is the signed sum of the n-simplices in the barycentric subdivision of ∆n and Sσ is the corresponding signed sum of the restrictions of σ to the n-simplices of the barycentric subdivision of ∆n. Since µ ≤ ǫ < δ, we can guarantee that there exists m > 0, s.t. Smσ = N k σ with k = ±1, Im(σ ) ⊂ B(x , δ) for some x ∈ Im(σ ), i Pl=1 l il l il l l i where σ : ∆n → X and N ∆n = ∆n. il l Sl=1 l Rn Next, for each σil :[v0, v1, ··· , vn] → X, let ϕl : B(xl, δ) → be the coordinate map n Rn and let fil = ϕl ◦ σil : ∆l → . Then we do the following construction, n Rn Fil : ∆l × I → n n n (X λjvj, t) 7→ (1 − t)fl(X λjvj)+ t X λjfl(vj). j=0 j=0 j=0 X (v2, 1) B(xl, δ) σil (v0, 1) (v1, 1) 2 ∆l × I ϕl (v2, 0) 2 (Pj=0 λj vj , 0) (v0, 0) (v1, 0) fl = ϕl ◦ σil SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 13 −1 n Let βil = ϕl ◦Fil : ∆l ×I → X that can be seen as a homotopy equivalence between n an ǫ-continuous map and a continuous map. Since ∆l × I is a simplicial complex, βil can be regarded as a formal sum of ǫ-singular n + 1-simplices. ǫ µ k N Finally, given [α] ∈ Hn(X ,X ), we can take β = i=1 ni l=1 βil to be such that P P µ ∂β is a formal sum of α as well as some µ-continuous maps, i.e., α − ∂β ∈ Cn (X). ǫ µ µ ∼ ǫ Therefore, Hn(X ,X ) = 0 and Hn (X) = Hn(X) for 0 <µ ≤ ǫ < δ. Actually, from the ǫ ∼ argument above, we can even see that Hn(X) = Hn(X) for 0 <ǫ<δ. Therefore, by lc ∼ universal property of the inverse limit, Hn (X) = Hn(X) for all n. n lc n ∼ Z lc n Example 2.17. Let p be a point in S , then Hn (S \{p}) = and Hi (S \{p})=0 for i =6 n. Indeed, Sn \{p} = Sn and Sn is a compact Riemannian manifold, so lc n ∼ lc n ∼ n n Hi (S \{p}) = Hi (S ) = Hi (S ). 3. Topological entropy and lc-homology In this section, we give a generalization of the entropy conjecture and consequently establish a connection between topological entropy and lc-homology. Let (X,d) be a compact metric space and f : X → X be a continuous map. For the lc lc lc lc induced homomorphism f∗i : Hi (X) → Hi (X), denote the spectral radius of f∗i by lc r(f∗i), i.e., 1 lc lc n n r(f∗i) = lim k(f∗i) k n→∞ lc n lc n where k(f∗i) k is the operator norm of the linear operator (f∗i) . lc Theorem 3.1. h(f) ≥ log r(f∗1). Proof. Let Hlc(X) = lim((Hǫn (X))∞ , (ϕ ) ) and fix ǫ > 0. Then any loop of 1 ←− 1 n=0 ǫkǫl∗ l≤k 0 diameter < ǫ0 can be seen as contractible under ǫ0-scale. Choose µ < ǫ0 such that 1 whenever d(x, y) <µ, d(f(x), f(y)) < ǫ0. Then take δ < 3 µ and clearly there exists an ǫ0-continuous path from x to y whenever d(x, y) < 3δ. Let P be a δ-net in X and define ǫ0 R C1 (P, 3δ; ) to be the free abelian group generated by {[x, y]: x, y ∈ P,d(x, y) < 3δ}, where [x, y] denotes some fixed ǫ0-path from x to y of diameter <µ, with real coefficients. ǫ0 R Similarly, let C0 (P ; ) be the free abelian group generated by P with real coefficients ǫ0 R ǫ0 R and we can define ∂ǫ0 : C1 (P, 3δ; ) → C0 (P ; ) by ∂ǫ0 [x, y]= y − x. For any ǫ < 3δ, consider the ǫ -coordinate of lim((Hǫn (X))∞ , (ϕ ) ) that is 1 0 ←− 1 n=0 ǫkǫl∗ l≤k ǫ1 ǫ1ǫ0 R ǫ0 R contained in ϕǫ1ǫ0∗(H1 (X)). Let H1 (P, 3δ; ) be the subgroup of H1 (P, 3δ; ) whose ǫ1 R ǫ1 R representatives are in C1 (P, 3δ; ) and it is naturally isomorphic to ϕǫ1ǫ0∗(H1 (X; )). ǫ1 R ǫ1 R Indeed, every ǫ0-homology class in ϕǫ1ǫ0∗(H1 (X; )) has a representative in C1 (P, 3δ; ) ǫ1 R obtained by breaking down ǫ1-paths in C1 (X; ) into combinations of short ǫ1-paths ǫ1 R joining points of P , since ǫ1 < 3δ. Then we can define a norm k k on C1 (P, 3δ; ) by k X aiσik = X |ai| i i ′ ǫ1ǫ0 R and a norm k k on H1 (P, 3δ; ) by k[u]k′ = inf kσk [σ]=[u] 14 BINGZHEHOU,KIYOSHIIGUSA,ANDZIHAOLIU ǫ1ǫ0 R (since H1 (P, 3δ; ) is finite dimensional, all norms on it are equivalent). ǫ1ǫ0 R Take ǫ1 to be sufficiently small. For a non-zero class [u] ∈ H1 (P, 3δ; ), take n ǫ1 R ′ σ = Pi=1 aiσi ∈ C1 (P, 3δ; ) to be its cycle with kσk ≤ 2k[u]k . Let Qk be a minimal (k, δ)-spanning set for f. For each i, we do the following construction with σi : I → X. Define 2 k−1 k−1 Fk = id ×f × f ×···× f : X 7→ X × f(X) ×···× f (X) k and then the points of Fk(Qk) have δ-neighbourhoods in X (endowed with d∞ metric that takes the largest of the distances in each of the k factors) that cover Fk(X). For the sequence of these neighbourhoods through which Fk(σi(I)) passes, we pick a sequence x1, x2, ··· , xb ∈ Qk k of some length b such that each Fk(xp) is in a δ-neighbourhood of Fk(σi(I)) ⊂ X and j j j j j j d(f (xp−1), f (xp)) < 3δ for 1 τi = [σi(0), x1] ∗ [x1, x2] ∗ [x2, x3] ···∗ [xb−1, xb] ∗ [xb, σi(1)] where ∗ denotes the composition of paths. It is possible that the length b > |Qk|(the cardinality of Qk). If xj = xl for some j =6 l, then τi contains a loop from xj to xl = xj and this loop is contained in an ǫ1 + δ neighbourhood of σiI which itself has diameter < ǫ1. So the diameter of the loop is < ǫ0. Hence we could suppress xj+1, ··· , xl and assume that b ≤|Qk|. m (m) Since σi is ǫ0-continuous, without loss of generality, we can assume that f σi is ǫ0 - (k−1) (k−2) (0) (k−1) continuous with ǫ0 ≥ ǫ0 ≥ ··· ≥ ǫ0 = ǫ0. We claim that under ǫ0 -scale, k−1 k−1 f σi is homologous (∼) to f τi ∼ vi, where k−1 k−1 k−1 k−1 k−1 k−1 vi = [f (σi(0)), f (x1)] ∗···∗ [f (xb−1), f (xb)] ∗ [f (xb), f (σi(1))] (1) Indeed, f[xj−1, xj] ∼ [f(xj−1), f(xj)] under ǫ0 -scale since f does not extend the path 2 2 (2) [xj−1, xj] so much. Similarly, f[f(xj−1), f(xj)] ∼ [f (xj−1), f (xj)] under ǫ0 -scale etc. k−1 k−1 k−1 (k−1) and so f [xj−1, xj] ∼ [f (xj−1), f (xj)] under ǫ0 -scale and k−1 k−1 k−1 f σ ∼ X aif σi ∼ X aif τi ∼ X aivi = v. Let p : X → P be a map such that d(x, px) < δ for ∀x ∈ X. Then we replace α by the (k−1) (k−1) ǫ0 -singularly homologous cycle α obtained by replacing each ǫ0 -singular simplex − ǫ(k 1) k−1 k−1 k−1 k−1 0 R [f (x), f (y)] in v by [pf (x), pf (y)] which is a generator of C1 (P, 3δ; ) since d(pf k−1(x), pf k−1(y)) < 3δ. k−1 ǫn ∞ Since fǫ0∗1 is the induced homomorphism on the ǫ0-coordinate of lim((H1 (X))n=0) − ←− ǫ(k 1) 0 R ′ to H1 (X; ), by the choice of k k , we have k−1 ′ kfǫ0∗1 ([u])k ≤kαk ≤ (1 + |Qk|) X |ai| ǫ1ǫ0 R whereas on H1 (P, 3δ; ), 1 1 k[u]k′ ≥ kσk = |a |. 2 2 X i SCALED HOMOLOGY AND TOPOLOGICAL ENTROPY 15 Therefore kf k−1([u])k′ ǫ0∗1 < 2(1 + |Q |) k[u]k′ k ǫ0 R k−1 ′ and this holds for any non-zero [u] ∈ H1 (X; ) and all k. Hence kfǫ0∗1 k < 2(1+ |Qk|). 1 ǫ k ′ k n ∞ But r(fǫ0∗1) = lim kfǫ ∗1k , so for the ǫ0-coordinate of lim((H1 (X))n=0), k→∞ 0 ←− 1 1 log r(f ) = lim log kf k k′ ≤ lim sup log 2(1 + |Q |) ǫ0∗1 k ǫ0∗1 k k+1 1 = lim sup log |Q | = h(f, δ) ≤ h(f). k k Similarly, this will hold for each coordinate of the given inverse limit. Hence, lc log r(f∗1) ≤ h(f). References [1] R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114(1965) 309–319. [2] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag GTM 82, 1982. [3] R. Bowen, Topological entropy and axiom A, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. [4] A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge 2002. [5] A. 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Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109–132. [12] M. Shub and R. Williams, Entropy and stability, Topology 14 (1975), 329–338. Bingzhe Hou, School of Mathematics, Jilin university, Changchun, P.R.China, 130012 Email address: [email protected] Kiyoshi Igusa, Department of Mathematics, Brandeis university, Waltham, MA, 02453 Email address: [email protected] Zihao Liu, Department of Mathematics, Brandeis university, Waltham, MA, 02453 Email address: [email protected]