Groups Having Elements Conjugate to Their Squares and Connections with Dynamical Systems

In recent years, dynamical systems which are conjugate to their squares have been studied in ergodic theory. In this paper we study the consequences of groups having elements which are conjugate to their squares and consider examples arising from topological dynamics and more general dynamical systems.


Introduction
Our intention is to introduce the reader to a number of topics in dynamical systems related by the group theoretic notion of conjugacy of an element to its square.In recent years there has been some interest in dynamical systems that are conjugate to their composition squares (see [1,2] and [3]).Although these ideas appeared in an ergodic theory setting, they can just as well be considered in the areas of topological and chaotic dynamics, and this is our point of view.This work arose during the teaching of a Senior Seminar (capstone) course given on chaotic dynamics.Some of the topics were introduced with the aim of having students give presentations in related areas using articles from various journals.The intention was to motivate students to learn more about chaotic and topological dynamics, ergodic theory and other aspects of dynamical systems (see [4] and [5]).We start in Section 2 with some group theoretic consequences of having elements conjugate to their square.The rest of the paper is concerned with examples.These include homeomorphisms of the interval [0,1] , the circle 1 S , and the 3-adic adding machine.We also take a look at transitive homeomorphisms having topological discrete spectrum and which are conjugate to their squares.

General Groups with Elements Conjugate to Their Squares
Let G be a group and G a  .We consider the consequences of a having the property that it is conjugate to its square, i.e., there exists G g  such that .Clearly  is one--to--one, and it is a homomorphism since , so that This shows that s is a square root of a which is conjugate to a .We summarize this, and give some additional properties of  as follows: Proposition 1 Suppose that G a  satisfies k a ka for some G k  .Define a map or not, which are not conjugate to a ).
Conversely suppose that are distinct square roots and the result follows.
(d) If . In particular,  is a group automorphism.
(e) Note that Q is conjugate to its square.Elements of G can be written as products of the form will be of the form (consider the cases where 0  n and 0 < n separately).

2
Q is abelian for suppose that m q  and consider In addition, 2 Q has the following properties (based on [1,6]): denotes the semidirect product of H and  where the multiplication is given by , so that r p = and q m = .
H is a subgroup of  with respect to addition and Clearly  is one-to-one and onto H . (c) If  , and we can check that , , then we can write and in a similar way , and


where we interpret ak k a 1 1/2 =  and similarly for other fractional powers.We see that  is well defined and onto.It is easy to check that  is one-to-one, and

 
, for some n odd, is not finite.
3 Let us consider an example related to the horocycle flow in ergodic theory (see [7]).Set , the centralizer of t g is the abelian group: In addition, we can check that t g is conjugate to its square , 1 0 The automorphism ) ( ) ( : ).This type of example can be generalized to SL ) , (  n , e.g., when , again conjugate to its square with abelian centralizer. Remark.Actions of the groups in Examples (1) and ( 2) above have importance in measurable dynamics (see [1] and [6] where the existence of weakly mixing rankone transformations conjugate to their square is demonstrated.This answered an open question in ergodic theory, see [2,3]).

Groups of Homeomorphisms
Denote by [0,1]  the set of homeomorphisms of the unit interval [0,1] , a group when given the operation of composition of functions.There are two possibilities for , possibly with other fixed points, but no period 2-points or points of greater period), or f is orientation reversing with a unique fixed point and no additional period 2-points or points of a greater period).We shall show that for , and we shall study properties of the conjugating map.The result below shows that any orientation preserving homeomorphism of [0,1] has a square root (in fact infinitely many).We illustrate a standard method of showing conjugacy using the notion of fundamental domain (see [8,9] and [10] for related results).
be orientation preserving, with fixed points are not equal, and f has infinitely many distinct square roots , each conjugate to f .Proof.(a) To keep the proof simple, we prove this for the case where there are only two fixed points, 0 and 1.The general case is similar. Either (since otherwise f will have additional fixed points).Suppose the latter holds (the former is treated similarly).In this case, if (0,1) and define a map  linearly on the interval ] ), ).These are called fundamental domains for f and 2 f .Roughly speaking, if we know the values ) (x and f is strictly increasing and continuous) as follows: Given any (0,1) , there is exactly one and we set ( ) = ( ( ( ))).


, then  is a homeomorphism with the property that , but f has no period 2-points, so ,  cannot have periodic points that are non-fixed in (0,1) , we must have , there is nothing more to prove, so suppose that 0 . In this case there is a unique fixed point, so either cannot happen as this would imply the existence of other fixed points.This concludes the proof of (b).
(c) Again we treat the case where f has only two fixed points as the general case is similar.Let ) (  and we proceed in an analogous way).Set , and generally Applying the Intermediate Value Theorem, we see that there exists ] ), ( [ An argument similar to that in (a) can be used to show that every square root of f is conjugate to f .Suppose now that there exists is an infinite set.The method of part (a) can be used to construct where a and b are chosen arbitrarily in (0,1) (again, for simplicity, assume that the only fixed points f has are 0 and 1 and that and continue the definition of h as in part (a).Because of the way h is defined on I we cannot have , thus giving infinitely many distinct square roots of f .Our hypotheses exclude from consideration, but the identity map has many square roots in [0,1]  .These are the involutions of [0,1]  and are easily constructed.Kuczma [10] has given a general treatment of the square roots of members of [0,1]  .We have used different methods with an aim of keeping the technicalities to a minimum.For example, if , then f has infinitely many ). Clearly orientation reversing homeomorphisms of [0,1] cannot be conjugate to their square.This is also seen from the following proposition.Denote by Fix ) (T the set of fixed points of a homeomorphism T .
Proposition 4 Let S and T be homeomorphisms of a compact metric space X .If S T ST

=
and T has finitely many fixed points, then In particular, T has no period 2-points.
, and let A # denote the number of elements of A .Clearly . Now A is a finite set, so that B is also a finite set and we must have , since S is a homeomorphism.

Groups with the Weak Closure Property
Suppose now that G is a complete metric topological group.We say that G a  has the weak closure property if the closure of } : In this case there is a dichotomy: the centralizer , for some sequence i n , so . We have shown that . Furthermore, we are assuming that a has the weak closure property, so so we must have equality throughout.It follows that 2 a has the weak closure property and , which may be regarded as an automorphism of the group , so s is conjugate to its square.Also, there exists a unique , so we deduce, on taking limits, that . This says that 1 s and 2 s are conjugate ( 1 2 = rs s r ), via the conjugation

The Topological Discrete Spectrum Theorem
The main examples having the weak closure property that we consider are those with (topological) discrete spectrum.Let X X T  : be a homeomorphism of a compact metric space X .Denote by ) ( X C the Banach space of complex valued continuous functions Suppose now that T is a transitive map: there is a point (an invariant function), then g must be constant, because it is a continuous function constant on the orbit of 0 f constant on X .It can also be seen that the eigenspace corresponding to any one eigenvalue is one-dimensional, and any finite set of eigenfunctions having distinct eigenvalues is a linearly independent set.Furthermore, the set of all eigenvalues of T is a countable subgroup of the circle group on a compact metric space X is said to have (topological) discrete spectrum if the eigenfunctions of T span ) ( X C (i.e., the smallest closed linear subspace containing the eigenfunctions is ) ( X C ).One of the main problems of topological dynamics is the question of when two dynamical systems (homeomorphisms on compact metric spaces) are conjugate.This is answered quite satisfactorily in the case of maps having discrete spectrum: The Discrete Spectrum Theorem of Halmos and von Neumann says: be transitive homeomorphisms defined on a compact metric space X and having discrete spectrum, then S and T are conjugate if and only if they have the same eigenvalue group  .
In addition, a transitive homeomorphism T having discrete spectrum is conjugate to a rotation T need not be conjugate, but will be conjugate if S is a group automorphism.For example, they are conjugate when where is dense in G , then R has the weak closure property.In addition, , the centralizer of R , is the set of all rotations on G .
Proof.For each , then there is a sequence of positive .
be arbitrary and choose k n so that G is a countable group (called the dual group of G ).The Pontryagin Duality Theorem says that the dual of  G (the second dual of G ) is topologically isomorphic to G (both a homeomorphism and a group isomorphism: see [13]).
Theorem 1 Let T be transitive with discrete spectrum and eigenvalue group  .T is conjugate to for some compact abelian group G and some G a  , where R has the weak closure property.It follows that the map ), this can be written as . We deduce that the map The eigenfunctions of R are the continuous characters , and the group of , where  G is the group of characters of G .We then see that the map is an automorphism which can be identified with which is therefore also an automorphism.
Conversely, if , so R and 2 R have the same eigenvalues, and are conjugate by the Discrete Spectrum Theorem.
Examples. 1. Set , a compact group when given the product topology and usual group operation.Let G be the subgroup of  defined by ( G is actually the inverse limit of the sequence , where all the arrows denote the power two homomorphism.This is written , and set , is an automorphism, then there is a transitive homeomorphism having discrete spectrum and eigenvalue group  , which is conjugate to its square.
2. Let = G the group of 3-adic integers, and the adding machine.We can think of G as , the group of integers modulo 3. The group operation on G can be defined as "carry to the right", so for example The adding machine is then defined by where 1= (1,0,0, )  and . T is transitive and since it is a group rotation, it has discrete spectrum.The eigenvalues are the n 3 th roots of unity. of the unit circle in the complex plane.Before stating his result we shall give some properties of circle homeomorphisms (see [8] or [4] for example).
Let    : F be a strictly increasing continuous function satisfying (respectively strictly decreasing with ( 1) ). F determines a homeomorphism of the circle , is a lift of f .Circle homeomorphisms and their rotation numbers have the following properties: 1) If f is orientation preserving with lift F , then for all exists and is independent of 4) An orientation preserving circle homeomorphism having points of period n can have points of no other period.
5) An orientation reversing circle homeomorphism has exactly 2 fixed points and can have any number of 2-cycles, but cannot have points of period greater than 2.
We have seen that irrational rotations of the circle cannot be conjugate to their squares, so the same is true for transitive circle homeomorphisms having irrational rotation number.The situation is different for circle homeomorphisms having fixed points.Proof.(a) Suppose f has a fixed point, then all other periodic points are fixed.Use these to partition the circle into subintervals and give an argument similar to that in Proposition 3(a) (see [9] or [14]).

Conversely if
, we may assume h is orientation preserving (otherwise look at 2 h ), so the above properties imply . An argument similar to that in Proposition 3(c) can be used to show that h must have an additional fixed point. If  .We deduce that h is a permutation of Fix ) ( f .If n is prime and the points are not periodic, then since a circle homeomorphism can only have points of one period, it must be an n -cycle for h .

Let
the Borel measurable subsets of [0,1] , and  a Borel measure on X .Denote by Aut ) ( X the group of all invertible measure preserving transformations of X X T  : (T will be one-to-one and onto, but possibly only after a set of measure zero is omitted).
These satisfy 1 ( ), ( ) . Aut ) ( X is a Polish group (but not a topological group).The 3-adic adding machine can be realized as a member of Aut ) ( X for =  Lebesgue measure on [0,1] in the following way. We define T as a rank-one (rational discrete spectrum) transformation whose eigenvalues are the n 3 th roots of unity, and constructed as follows: Starting with the unit interval [0,1) , subdivide into 3 equal subintervals and stack, placing [1/3,2/3) above [0,1/3) and [2/3,1) on top.Now define T by linearly mapping the bottom interval to the middle interval and mapping the middle interval to the top interval, but leaving T undefined on the top level.Continue this process inductively, so that at the n th stage, T is defined on the levels of a column consisting of n 3 equal subintervals.Again subdivide the column into 3 equal subcolumns and stack as before to extend the definition of T .Ultimately, T is defined almost everywhere on [  ), but no period 2-points.S has uncountably many periodic points of every order.

Concluding Remarks
1) In dynamical systems theory one studies the actions of groups on sets of homeomorphisms or on sets of measure preserving transformations.In the study of a single transformation we are looking at actions of the countable group  .The examples of this paper may be thought of as actions of the group } , the countable non-abelian group that was studied in Section 2. Let X be a compact space and ) ( X  its group of self-homeomorphisms.We can define a representation of G (an action) as a continuous homomorphism (see [4]). 3) Other examples of interest are the homeomorphisms of the Cantor set C , in particular for proving category/density type results.Every invertible measure preserving transformation may be modeled as a homeomorphism of C .The adding machine can be seen directly to be such an example.
4) It is natural to talk about conjugacy between a continuous transformation f defined on some metric space X and its square


, because such maps have period 2-points.However, conjugacies between continuous maps and their squares on higher dimensional spaces may lead to interesting dynamics.

2 )
see that actions of G on ) ( X  are determined by a pair of homeomorphisms S , All the examples we have considered have zero topological entropy (except for the last example whose entropy is infinite).This is because conjugate homeomorphisms have the same topological entropy,
Q is the conjugacy class of a , and every member of 2 2 The homeomorphism is said to be orientation preserving if F is strictly increasing and orientation reversing if F is strictly decreasing.The map F is said to be a lift of f .For example