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The first type of dynamical systems we consider are expanding maps. We give the definition, some examples and an overview of the known results about their dynamical properties and the manifolds supporting such an expanding map.

What is an expanding map?

The definition of an expanding map generalizes Example 1.1.

Definition 3.16. Let M be a closed manifold with Riemannian metric k · k.

A differentiable map f : M → M is called expanding if there exist constants

c > 0, λ > 1 such that for every tangent vector v ∈ T M we have that

kDfn_{(v)k ≥ cλ}n_kvk

for every n ∈ N.

If k · k1 and k · k2are two Riemannian metrics on a closed manifold M , there

exist constants 0 < c1< c2such that

c1kvk1≤ kvk2≤ c2kvk1 ∀v ∈ T M.

This implies that the definition of an expanding map does not depend on the choice of Riemannian metric on the manifold M by varying the value of the constant c in the definition. Thus expanding maps can be considered as a topological type of self-map, an idea we pursue in Section 3.5.

In fact, we can always find a Riemannian metric on M such that we can choose

c = 1 in Definition 3.16.

Theorem 3.17. Let f : M → M be an expanding map of a closed manifold M. Then there always exist a Riemannian metric k · k on M and a λ > 1 such that

kDfn_{(v)k ≥ λ}n_kvk

As explained in Example 1.1, the easiest examples of an expanding map are given by the self-covers of the circle. We generalize this example to the class of infra-nilmanifolds.

Example 3.18. Let ¯α : Γ\G → Γ\G be an affine infra-nilmanifold

endomorphism. If ¯α only has eigenvalues of absolute value > 1, then ¯α

is an expanding map and ¯α is called an expanding affine infra-nilmanifold

endomorphism. We give some concrete examples of expanding affine infra- nilmanifold endomorphism. (i) Let A =      λ1 0 . . . 0 0 λ2 . . . 0 .. . ... . .. ... 0 0 . . . λn      ∈ GL(n, Q)

be a diagonal matrix with λi∈ Z for every i. If |λi| > 1 for every i, then

A induces an expanding toral endomorphism on Tn_{. In particular, every}

torus Tn _{has an expanding map.}

(ii) Consider the nilmanifold M = H3(Z)\H3(R). Let a, b ∈ Z and take

α : H3(R) → H3(R) the automorphism defined as

α     1 x z 0 1 y 0 0 1    =   1 ax abz 0 1 by 0 0 1  .

Since α(H3(Z)) ⊆ H3(Z), the automorphism α induces an nilmanifold

endomorphism ¯α : M → M . If |a| > 1 and |b| > 1, then ¯α is an expanding

nilmanifold endomorphism.

The map α also induces an expanding nilmanifold endomorphism on every nilmanifold Nk\H3(R).

In essentially the same way as Example 1.1 it can be shown that every expanding affine infra-nilmanifold endomorphism ¯α is chaotic.

It is harder to construct infra-nilmanifolds not admitting an expanding map. Such examples will be given in Chapter 6.

Properties

The dynamical properties of expanding maps are already described in the first paper [96] studying them. In this paper M. Shub shows that every manifold

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admitting an expanding diffeomorphism is diffeomorphic to Rn for some n. By studying the lifts of expanding maps to the universal cover, he proofs the following theorem.

Theorem 3.19 (Shub, 1969). Every expanding map of a closed Riemannian manifold has chaotic behavior.

This generalizes Example 1.1 above.

In the same paper, he also shows that if two expanding maps on the same manifold are homotopic, they are topologically conjugate. This implies the following result.

Theorem 3.20 (Shub, 1969). Every expanding map of a closed Riemannian manifold is structurally stable.

The only examples of expanding maps we gave above were expanding affine infra-nilmanifold endomorphisms. The following result by M. Gromov in [54] shows that these are in fact the only possible ones.

Theorem 3.21 (Gromov, 1981). Every expanding map on a closed manifold is topologically conjugate to an expanding affine infra-nilmanifold endomorphism.

In fact, the major contribution of M. Gromov is showing that groups of polynomial growth are virtually nilpotent.

Theorem 3.22 (Gromov, 1981, Polynomial growth theorem). Let G be a finitely generated group. The group G is virtually nilpotent if and only if G has polynomial growth.

One of the results in [50] is that the fundamental group of a closed manifold admitting an expanding map has polynomial growth. The fundamental group of a closed manifold is always finitely generated and torsion-free and therefore Theorem 3.22 implies that the fundamental group of such a manifold is isomorphic to an almost-Bieberbach group. The results of [63, 97] then show that Theorem 3.21 holds.

By Theorem 3.21, the problem of determining the infra-nilmanifolds admitting an expanding map is equivalent to determining the infra-nilmanifolds which have an expanding affine infra-nilmanifold endomorphism. An interesting question is then if we can give an algebraic way of describing the infra-nilmanifolds with an expanding infra-nilmanifold endomorphism.

Research question 3. Give an algebraic description of the infra-

nilmanifolds supporting an expanding map.

Part II gives a complete answer to Research Question 3.

Expanding maps on infra-nilmanifolds

Although Theorem 3.21 already dates from 1981, there were not many results about the existence of expanding maps on infra-nilmanifolds before this thesis. We give a short overview.

A first type of results is finding properties of the infra-nilmanifolds which support an expanding map. In the paper [35] the authors show that the existence of an expanding map on an infra-nilmanifold Γ\G implies the existence of a positive grading on the Lie algebra g corresponding to G.

Other results construct expanding maps on certain classes of infra-nilmanifolds. In [76] it is shown that every infra-nilmanifold modeled on a 2-step nilpotent Lie group admits an expanding map. This result was later generalized to the situation of homogeneous Lie algebras in [36]. A homogeneous Lie algebra is a Lie algebra g with a positive grading

g =

k

M

i=1

gi

such that g is generated as Lie algebra by the subspace g1.

Theorem 8.21, which is the main result of this dissertation about expanding maps, implies all these results.

There is another type of questions which follow naturally from Theorem 3.21. Let M be a manifold which is homeomorphic to an infra-nilmanifold supporting an expanding map, is it true that also M admits an expanding map? In the papers [47, 49] the authors construct examples of manifolds admitting an expanding map and which are homeomorphic but not diffeomorphic to a torus Tn. The main idea is to start from a torus Tn and glue an exotic sphere to it such that the resulting manifold still admits an expanding map. The manifold constructed in this way are homeomorphic to the torus Tn _{you started from}

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