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Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Extending our results of [17], we conﬁrm that Entropy Conjecture holds for every continuous self-map of a compact K ( π, 1) manifold with the fundamental group π torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by the logarithm of the spectral radius of an associated ”linearization matrix” with integer entries. From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.


Introduction
Let M be a compact manifold and f : M → M a continuous self-map of M. The topological entropy h top (f ), denoted shortly by h(f ), is defined as lim ǫ→0 lim sup n→∞ 1/n log sup #Q, supremum over all Q being (ǫ, n)-separated. Q is called (ǫ, n)-separated, if for every two distinct points x, y ∈ Q, max j=0,...,n d(f j (x), f j (y)) ≥ ǫ. Here d is a metric on M consistent with the topology; in fact h(f ) does not depend on the metric (cf. [28]).
Entropy Conjecture, denoted shortly as EC, says that the topological entropy of f is larger or equal to the logarithm of the spectral radius of the linear operator induced by f on the linear spaces of cohomology of M with real coefficients. It was posed by M. Shub in seventies who asked what suppositions on f and M imply EC.
We prove the following Theorem A Assume that a compact manifold M is a K(π, 1)-space with the fundamental group π being torsion free and virtually nilpotent. Then EC holds for every continuous self-map f of M. In particular EC holds for every continuous self-map of any compact infra-nilmanifold.
• A group π is called virtually nilpotent if it contains a finite index nilpotent subgroup Γ. We can assume that Γ is a normal subgroup of π.
(Indeed, for any pair of groups K ⊂ L, where K has a finite index in L, one has a homomorphism ρ : L → Sym(L/K) into the symmetry group of the quotient space. Then ker ρ is a normal subgroup of L, it has finite index, and is contained in K.) • One can replace the assumption that π is virtually nilpotent by the assumption that π has polynomial growth, [9].
• Theorem A is a step towards proving a conjecture by A. Katok [11] saying that EC holds for every continuous map if the universal cover of M is homeomorphic to an Euclidean space R d .
• One can even ask whether EC holds for every continuous self-map of a K(π, 1)) compact manifold (or a finite CW-complex).

Affine maps
We refer to the following theorem by A. Malcev and L. Auslander about the existence of a model [7] p.76: Assume that π is finitely generated torsion free virtually nilpotent group. Then it contains a finite index normal subgroup Γ which can be embedded as a lattice, i.e. a discrete co-compact subgroup, in a connected, simply connected nilpotent Lie group G. The embedding can be extended to the embedding of π in the group Aff(G) of affine mappings of G, so that π ∩ G = Γ. More precisely, if C ⊂ Aut(G) denotes the maximal compact subgroup of the group of automorphisms of G, then π ⊂ G ⋉ C ⊂ G ⋉ Aut(G); this embedding of π is called an almost Bieberbach group.
It follows then from the definition that π acts on G properly discontinuously (First note that if α ∈ π has a fixed point z ∈ G, then α ℓ (z) = z and α ℓ ∈ Γ for ℓ = #(π/Γ). Then α ℓ = e the unity of π, hence α = e by the assumption that π is torsion free.) The quotient manifold IN = G/π is called an infra-nilmanifold. It is regularly finitely covered by the nilmanifold N = G/Γ, with the deck transformation group equal to H = π/Γ. Note that every compact manifold finitely covered by a nilmanifold, in particular every infra-nilmanifold, satisfies the assumptions of Theorem A. Indeed, G is homeomorphic to R d , where d = dim M (cf. [22]). If π were not torsion free, a cyclic subgroup generated by an element {g} ≃ Z p of a prime order would act freely on R d . The latter is impossible, as follows from the Smith theory (cf. [1]).
The image of the embedding of π into G ⋉ C ⊂ Aut(G) will be denoted by π IN . It is a deck transformation group of the cover p IN : G → G/π IN and, distinguishing an arbitrary z ∈ G, it can be identified in a standard way with the fundamental group π 1 (IN, p IN (z)) of IN.
Consider now any M being K(π, 1) as in Theorem A. The group π acts properly discontinuously on the universal cover spaceM and it can be identified with π M , which is the deck transformation group of the universal cover p M :M → M.
(Similarly to N → IN, we have a regular finite coverM /Γ → M =M/π, with the deck transformation group H.) Every continuous f : M → M induces an endomorphism F = F f of π M , unique up to an inner authomorphism. To define this use f # : π 1 (M, z) → π 1 (M, f (z)) between the fundamental groups and standard identifications of these groups with π M . For more details see Section 3.
When we identify π with π IN , then we can consider F as an endomorphism of π = π IN . By K. B. Lee, [13, Th.1.1 and Cor 1.2], there exists an affine self- Hence, in view of (1), there exists a factor φ = φ f of Φ on IN under the action of π IN . In general one calls factors of affine Φ on G satisfying (1), affine maps on IN, in particular φ f is an affine map on IN.
Since both M and IN are K(π, 1)-spaces, they are homotopically equivalent, [26]. One can find a homotopy equivalence h : M → IN inducing our identification between π M and π IN . Then φ • h is homotopic to h • f , see [26,Section 8.1] and Section 3. Note, [13], ∧ k D f of D f . Of course it is equal to j:|λ j |>1 |λ j |, where λ j ∈ σ(D f ), provided at least one λ j has absolute value larger than 1. Otherwise it is equal to 1 (in ∧ 0 D f ).

Linearization matrices
One can assign to an endomorphism F = f # not only a linear map D f : R d → R d , G ≡ R d , but also an integer d × d matrix A [f ] called the linearization of the homotopy class [f ], because endomorphisms f # are in the one-one correspondence with homotopy classes [f ] of self-maps of a K(π, 1)-space. Sometimes we shall use the notation A f . An endomorphism F : π → π does not preserve the nilpotent subgroup Γ ¡ π in general. But Γ contains a subgroup Γ ′ ¡ π such that Γ ′ is nilpotent has finite index in π and is invariant for F (cf. [14] or Proposition 5). Since Γ ′ has finite index in Γ it is also a lattice in G. By (1) the endomorphism B : G → G is an extension of F : Γ ′ → Γ ′ , see Section 2.
Let G be a nilpotent connected simple-connected Lie group, Γ its lattice. By the definition the descending central series of is a homomorphism of G preserving a lattice Γ it preserves each subgroup Γ i thus it induces an endomorphism B i on each factor group Γ i /Γ i+1 , 0 ≤ i ≤ k. Clearly Γ i /Γ i+1 is abelian and torsion free of dimension d i . Therefore the action of which is uniquely defined up to a choice of basis, i.e. up to a conjugation by a unimodular matrix. Finally we put [12] and [10] for some extensions to [17], [10]).
In fact the matrices A can be defined directly, using F = f # : π → π, without constructing G. One can use the series of isolators

Conclusion
We are in the position to formulate a sharper version of Theorem A, namely Theorem B For every continuous self-map f of a compact manifold M which is a K(π, 1)-space with the group π torsion free and virtually nilpotent, In the case M is an infra-nilmanifold the equality holds for every affine map φ : M → M, a factor of an affine Φ satisfying (1), in particular for φ f . In consequence for every Maybe considering of M is not needed and it is sufficient to consider only IN. Since the topological entropy is an invariant of conjugation by a homeomorphism, this would follow from Borel conjecture, which states that the fundamental group π of a manifold being K(π, 1)-space determines M up to homeomorphism, This has been confirmed by Farrell and Jones [5] for a class of groups that contains the almost-Bieberbach groups, except in dimension 3.
Formulating Theorems A and B we have followed a suggestion by M. Shub [24], to assume a discrete group point of view. Having given an endomorphism F = f # : π → π of a finitely generated torsion free virtually nilpotent group, we associate to it a linear operator D f , or an integer d × d matrix A f . As suggested in [24] the logarithm of spectral radius of sp (∧ * D f ), or sp (∧ * A f ), is "a kind of volume growth" of f # . In Theorem A it is replaced by the spectral radius of the map induced on real cohomologies of the group π.
We shall present two proofs of Theorems A and B.
The first one, in Section 2, concludes Theorems A and B from analogous theorems in [17], with f : N → N a continuous map of a compact nilmanifold. However this proof of Theorems A and B does not work in dimension 3.
The second proof, in Section 3, holds for f on M and uses only a homotopy equivalence between M and IN. It directly repeats the arguments of [17].
An important observation is that A [f ] is an integer matrix. This allows, in Section 4, to prove absolute estimates from below for sp (∧ * A f ), where f is an expanding map of a compact manifold (without boundary) or an Anosov diffeomorphism of a compact infra-nilmanifold. The latter uses number theory results estimating the Mahler measure of an integer polynomial.
The authors would like to express their thanks to K. Dekimpe, E. Dobrowolski, T. Farrell, A. Katok and M. Shub for helpful conversation.

Entropy Conjecture on infra-nilmanifolds
The proof of Theorems A and B we present in this section will hold for M finitely covered by a nilmanifold and follows from two standard facts and the main theorem of [17] in which the topological entropy of a continuous map of nilmanifold is estimated by the corresponding quantities. We begin with the following Proof: It is elementary and is given in [6]. Briefly: The p-preimage of an (n, ǫ) − fseparated set in M is (n, ǫ) −f-separated (in a metricd onM is being the lift of a metric d on M chosen to define the entropy), hence h(f ) ≤ h(f ) . (In fact only the continuity of p was substantial in this proof). Conversely, let Q be an (n, ǫ) −f -separated set inM consisting of points in a ball B(z, ǫ/2). Let δ > 0 be a constant such that p is injective on every ball inM of radius δ. We prove that the set p(Q) is (n, ǫ) − f -separated. Indeed, take ǫ < δ and suppose that for x, y ∈ Q we have d(f j (p(x)), f j (p(y)) < ǫ for all j = 0, 1, ..., n. Let j 0 ≥ 0 be the smallest j ≤ n such that d(f j (x),f j (y) ≥ ǫ. Then d(f j (x),f j (y) ≥ δ − ǫ (i.e. projections by p are close to each other but the points are in different components of preimages of a small ball under the cover map). This is not possible for j = 0 by Q ⊂ B(z, ǫ/2). If it happens for another j it means that thef image of two points within the distance < ǫ have distance ≥ δ − ǫ, what for ǫ small enough contradicts the uniform continuity off . P Definition 4 Let Γ ⊳ π be a normal nilpotent subgroup of finite index in π and let let s be an endomorphism of π. We say that a group Γ ′ ⊂ Γ is s -admissible if Proposition 5 For a nilpotent group Γ normal and of finite index in a group π there exists a group Γ ′ ⊂ Γ, Γ ′ ⊳ π, admissible for every endomorphism s of π. (sometimes such Γ ′ is called a fully characteristic subgroup).
Proof: Repeat verbatim the argument of Lemma 3.1 of [14] and define Γ ′ = group generated by {γ k : γ ∈ π}, where k is the order of H = Γ/Γ ′ . This is a subgroup preserved by every endomorphism of π, in particular π is normal.
Next we define a group Γ(k) := group generated by {x k } : x ∈ Γ .
Of course Γ(k) ⊂ Γ ′ . It is enough to show that Γ(k) is of a finite index in Γ. Apply an argument used in [14]: Since Γ is nilpotent, it is polycyclic, [22]. But for any polycyclic group Γ the corresponding group Γ(k) has a finite index, cf. [22,Lemma 4.1]. In particular adapting the argument of Lemma 4.1 of [22] one shows its assertion for a nilpotent group, by an induction over the length of nilpotency. P Note that the number k used to define the group Γ(k) is not unique, e.g. we can take any its multiple getting a smaller group with the required property. To get a larger group Γ ′ than that of [14] we can use k equal to the LCM{#h; h ∈ H} the order of element, instead of k = #H the order of H. Proof: The assertion follows from Proposition 5 for π the fundamental group of M, N = G/Γ for Γ a subgroup of π and s = f # . We can assume that Γ (hence Γ ′ ) is normal in π, see Introduction. We defineÑ := G/Γ ′ . A liftf exists, since the homomorphism f # : π → π preserves Γ ′ identified withp # π(Ñ), see [26]. P Together with Corollary 3 and EC for nilmanifolds, [17], this proves EC for all continuous self maps of M finitely covered by nilmanifolds, in particular for all M being infranilmanifolds.
Proof: of Theorem B (for M finitely covered by a nilmanifold G/Γ).
As above we find an admissible Γ ′ ⊂ Γ. The quality (1) for α = g ∈ Γ ′ takes the form As the left hand side expression is equal to fact B in (b, B) was found in [13] just as an extension of f # |Γ ′ to G.) B is a lift to G/Γ ′ of φ f homotopic to f . Theorem B follows from Corollary 6 (B isf there), from Proposition 1, and from Theorem B for self-maps f of nilmanifolds, [17]. P

Another proof of EC
Now we provide another proof of Theorems A and B without additional assumptions by showing that a modification of the proof for the nilmanifolds given in [17] works. Let us remind the notation. M is a compact manifold, being K(π, 1) for π a virtually nilpotent torsion free group π. G is a connected simply connected nilpotent Lie group and IN = G/π where π is embedded in Aff(G) as π IN , acting discontinuously on G so that IN is an infra-nilmanifold, see Existence of a Model Theorem in Introduction. We have the universal covers p M :M → M and p IN : G → IN.
Remark that we use the right action, thus IN = G/π, instead for π\G used in [13]. Then the action of an affine map (d, D), d ∈ G, D ∈ Endo(G), is given as (d, D)x = (Dx)d.
We assume that all metrics under consideration are induced by Riemannian metrics. We need the following Proof: By compactness the lift τ Γ of τ IN to G/Γ, where Γ = G ∩ π, is equivalent to ρ Γ , the projection of ρ to G/Γ. Therefore the lifts to G are also equivalent. P Let us stop for a while on the homotopy equivalence between M and IN making some explanations from Introduction more precise. By construction we get  Let x n , n = 0, 1, 2, ... be anf trajectory. Hence w n =h(x n ) is a ξ 1 − Φ-trajectory in the metric τ G , hence, by Lemma 7, a ξ 2 − Φ-trajectory in ρ, the right invariant metric on G. Finally , the latter equality by the right invariance of ρ. Hence w n is a ξ 2 + ρ(b, e) trajectory for B ′ .
Note that the spectra of the derivatives (linearizations) of DB(e) and DB ′ (e) coincide as these operators are conjugate. Now we define a mapping Θ from (w n ) to a B ′ -trajectory in G u the unstable subgroup for B ′ by proceeding as in [17]: First we define w n → π u (w n ), the "projection" to G u , i.e. we write w n = g cs · g u where g cs ∈ G cs the central stable subgroup and π u (w n ) := g u ∈ G u . Next Θ(w n ) is defined as the unique B ′ -trajectory in G u subexponentially "shadowing" π u (w n ). Finally, we define θ(x) := Θ •h(x).
For an arbitrary ǫ > 0, for ((1 + ǫ) j , n) − B ′ -separated points in G u , j = 0, ..., n, (contained in a small disc), i.e. such that for some j their j-th images under B ′ are within the distance at least (1 + ǫ) j , we choose points w in their Θ preimages (also in a small disc) and next points x inh-preimages, in a small disc. This is a crucial point which uses the fact thath is onto, since |deg h| = 1, compare Remark 4.8 in [17].
If two points p M (x), p M (y) are (ǫ, n) − f -close (i.e. not separated), then so are x, y. Henceh(x) andh(y) are (ξ 4 , n)-close (with respect to Φ, hence B ′ ) in ρ for a constant ξ 4 . Hence their Θ images are ((1 + ǫ) j , n) − B ′ -close, a contradiction. P Note that we did not use an admissible group constructed in Proposition 5

Remark 9
The statement of Theorem A, in a weaker form for the flat manifolds, was posed as a question by Szczepański in his article [27]. Earlier, a very special case of entropy conjecture an for affine map of a compact affine manifold was proved by D. Fried and M. Shub in [8].

Absolute estimates of entropy
Famous Lehmer conjecture in number theory states that there exists a constant C, called Lehmer constant, such that for every integer polynomial w(x) = a 0 x d +a 1 x d−1 + + · · · +a d , not being a product of cyclotomic polynomials (all zeros being roots of 1) and x k , for the Mahler M(w) measure of w have where the product is taken over all zeros of w(x).
There are estimates of the Mahler measure which depend on the degree of an irreducible polynomial (the degree of an algebraic number). Using an estimate given by Voutier in 1996 (cf. [29]), which is the best known valid for every d > 1, not only asymptotically, we get the following Proof: Let w(x) = w 1 (x) · w 2 (x) · · · w k (x), d j = deg w j , d 1 ≤ d 2 ≤ · · · ≤ d k , be a decomposition of the characteristic polynomial of the linearization matrix A f into irreducible terms. If h(φ f ) > 0 then by Theorem B at least one eigenvalue of A f has the absolute value larger than 1. Hence, by Theorem B, using the property the sequence τ (n) is decreasing with respect to n, P For other estimates of Mahler measure see for example [4]. In particular from Smyth's theorem [25] (which is a partial answer to the Lehmer conjecture) it follows where τ 0 is the real root of polynomial τ 3 − τ − 1. P One can check that the latter τ 0 is greater that 1.32471795. In particular, τ 0 does not depend neither on w(x) nor on its degree d. Theorem 11 is a statement about a homotopy property of f . A special case is when A f is a hyperbolic matrix invertible over integers, i.e. φ f is an Anosov automorphism, and d, the dimension of M, is odd. Then obviously the characteristic polynomial of A f is non-reciprocal, hence Theorem 11 applies and we obtain h(f ) ≥ 1.32471795. This is in fact an easy case whose proof does not need the use of Theorem B. Namely one can refer to Franks' theorem [7,Theorem 2.2], saying that such a map f is semiconjugate to φ f , i.e. there exists a continuous map θ : M → M such that θ • f = φ f • θ. This θ is found to be homotopic to identity, hence "onto". Therefore h(f ) ≥ h(φ f ), see Proposition 1. It is easy to check that if f is an Anosov diffeomorphism then A f is a hyperbolic invertible matrix.

Other remarks
• The "projection -shadowing" construction of Θ in the proof of Theorem B in Section 3 and in [17] can be considered as a strengthening of Franks' theorem to the case central direction exists.
• It is sufficient to assume A f is a hyperbolic endomorphism, i.e. without eigenvalues of absolute value 1, and without zero eigenvalues, to apply Franks' theorem. Then φ f is an Anosov endomorphism and the semiconjugacy holds between the inverse limits, cf. [23] and [20] .
• In the expanding case, i.e. if all the eigenvalues of A f have absolute values larger than 1, the product is at least 2. Therefore h(f ) ≥ log 2. In this case, instead of Theorem B, one can refer to Shub's theorem [23] saying that f is semiconjugate to φ f .
• Finally, if f itself is metric expanding on a compact orientable manifold (i.e. it expands all the distances between points close to each other, at least by a constant factor larger than 1) or at least if f is forward expansive, i.e. ∃δ > 0 such that ∀x = y ∃n ≥ 0 with d(f n (x), f n (y)) ≥ δ (as this implies expanding in an appropriate metric, see [21, Section 3.6]), then for its degree d(f ) one has immediately h(f ) ≥ log |d(f )| ≥ log 2, see [28].
Note that f expanding (in a metric induced by a Riemannian metric) can happen only on infra-nilmanifolds, [9].
• In general h(f ) ≥ log |d(f )| for all f being C 1 , see [18]. However the assumption that f is C 1 is essential in absence of the expanding property, namely there are easy examples of continuous, but not smooth maps f for which h(f ) < log | deg(f )|.