Eventually Periodic Points of Infra-Nil Endomorphisms
© Ku Yong Ha et al. 2010
Received: 14 August 2009
Accepted: 19 February 2010
Published: 28 February 2010
Hyperbolic toral automorphisms provide important examples of chaotic dynamical systems. Generalizing automorphisms on tori, we study (infra-)nil endomorphisms defined on (infra-)nilmanifolds. In particular, we show that every infra-nil endomorphism has dense eventually periodic points.
Let be an nonsingular integer matrix. Then induces a map on the -torus . If is hyperbolic, we say that is a hyperbolic toral endomorphism. If, in addition, , then is called a hyperbolic toral automorphism.
A hyperbolic toral automorphism provides an important example of a chaotic dynamical system. We review the most fundamental property about hyperbolic toral automorphisms, together with some definitions which are necessary to describe this property. See  for details.
A continuous surjection of a topological space is said to be topologically transitive if, for any pair of nonempty open sets and in , there exists such that . Intuitively, a topologically transitive map has points which eventually move under iteration from one arbitrary small neighborhood to any other. The continuous map of the metric space is said to have sensitive dependence on initial conditions if there exists such that, for any and any neighborhood of , there exist and such that . Intuitively, a map possesses sensitive dependence on initial conditions if there exist points arbitrarily close to which eventually separate from by at least under iteration of .
Proposition 1.1 (see [1, Theorem ]).
Anosov diffeomorphisms play an important role in dynamics. In , Smale raised the problem of classifying the closed manifolds (up to homeomorphism) which admit an Anosov diffeomorphism. Franks  and Manning  proved that every Anosov diffeomorphism on an infra-nilmanifold is topologically conjugate to a hyperbolic infra-nil automorphism. In , Gromov proved that every expanding map on a closed manifold is topologically conjugated to an expanding map on an infra-nilmanifold.
We will consider infra-nil endomorphisms in this paper. These include Anosov diffeomorphisms and expanding maps on infra-nilmanifolds up to topological conjugacy. The purpose of this paper is to show that the infra-nil endomorphisms have dense eventually periodic points. In the case of infra-nil automorphisms, this is already known (cf. [4, Lemma ]).
2. Toral Endomorphisms
Now we show that every toral endomorphism has dense periodic points. This generalizes [1, Proposition ] in which it is shown that every toral automorphism has dense periodic points.
Let be a toral endomorphism of the torus induced by the automorphism and let be a nonempty set of prime numbers. Then every point with coordinates in is an eventually periodic point of . Moreover, if for all , then every point with coordinates in is a periodic point of .
3. Nil Endomorphisms
We use to indicate the Lie algebra of . This Lie algebra has the same dimension and nilpotency class as . Moreover, in the case of connected, simply connected nilpotent Lie groups it is known that the exponential map is a diffeomorphism. We denote its inverse by . If is another connected, simply connected nilpotent Lie group, with Lie algebra , then we have the following properties.
If is a preferred basis of , then can be regarded as a basis for the vector space . We call the basis of preferred. In particular, if is a lattice of , then every preferred basis of becomes a preferred basis for the vector space .
We first generalize the concept of toral automorphisms to that of nil endomorphisms and show that every nil endomorphism has eventually dense periodic points.
Thus we see that the differential of any automorphism on has determinant and eigenvalues and . Thus if , then has an eigenvalue of modulus 1. Therefore, there are no hyperbolic nil automorphisms on any nilmanifold . (There are examples of hyperbolic nil, nontoral, automorphisms. In fact, we can find such examples from many literatures. For example, we refer to [2, 14–18].)
Among interesting properties of this identification, we recall the following ([7, Theorem 2.1.(3)]): for any homomorphism , there exists a polynomial function with rational coefficients such that for all . Moreover, any homomorphism of extends to a homomorphism of by using the same polynomial.
which is of the form (3.22) claimed above.
Now we fix a canonical basis of . A point of the nilmanifold is said to have rational coordinates or simply has rational coordinates if for some . First we show that if and with , then for some . We recall the following ([7, Theorem .( )]): there exists a polynomial function with rational coefficients satisfying for all . The group product on is defined using this polynomial . Now, suppose that and with . Then for some . Since , for some . Hence we have . Since , and is a polynomial function with rational coordinates, we must have . This proves our assertion. Therefore the points of with rational coordinates are well defined. Consequently for a subring of with , the points of with coordinates in are well defined.
It is known that every (infra-)nil automorphism has dense periodic points (see the proof of [4, Lemma ]). Now we will generalize this to the case of (infra-)nil endomorphisms. The proof below is exactly the same as that of Lemma 2.2, except that the coefficients involved are different and hence Lemma 3.6 is essential.
Let be a nil endomorphism of the nilmanifold . Let be a ring obtained from by adding finitely many primes , that is, . Then every point with coordinates in is an eventually periodic point of . Moreover, if for all , then every point with coordinates in is a periodic point of .
The canonical basis of induces the canonical bases and of and , respectively, where stands for the image of in under the natural surjection . Hence the points in with rational coordinates are well defined. Furthermore the points in with rational coordinates are also well-defined.
Every nil endomorphism has dense periodic points.
4. Infra-Nil Endomorphisms
Let be a connected, simply connected nilpotent Lie group and let be a maximal compact subgroup of . A discrete and cocompact subgroup of is called an almost crystallographic group. Moreover, if is torsion-free, then is called an almost Bieberbach group and the quotient space an infra-nilmanifold. In particular, if , then is a nilmanifold. Recall from  that is the maximal normal nilpotent subgroup of with finite quotient group , called the holonomy group of .
Since for all , we have for all . Hence is the unique extension of the homomorphism of the lattice of . If is an isomorphism, then is also an isomorphism. Conversely, assume that is an isomorphism. Using the fact that is torsion-free, we can show that is injective. This fact implies that is also injective on the finite group and hence must be an isomorphism. Therefore, is an isomorphism. (The converse was suggested by a referee.) If is an infra-nil automorphism, then being implies by Lemma 3.1 that is an isomorphism and thus is a nil automorphism, and vice versa. Note also that is an infra-nil endomorphism if and only if is a nil endomorphism.
The authors would like to thank the referees for pointing out some errors and making careful corrections to a few expressions in the original version of the paper. The authors also would like to thank both referees for suggesting the apt title. The first author was partially supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-206-C00004), and the third author was supported in part by KOSEF Grant funded by the Korean Government (MOST) (no. R01-2007-000-10097-0).
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