Applied Mathematics, 2010, 1, 416-424
doi:10.4236/am.2010.15055 Published Online November 2010 (http://www.SciRP.org/journal/am)
Groups Having Elements Conjugate to Their Squares
and Connections with Dynamical Systems
Geoffrey R. Goodson
Department of Mathematics, Towson University, Towson, USA
Received July 15, 2010; revised September 23, 2010; accepted September 26, 2010
Abstract
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.
Keywords: Conjugacy to Square, Homeomorphism on a Compact Space
1. 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 consi-
dered 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 consequ-
ences 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 tran-
sitive homeomorphisms having topological discrete spec-
trum and which are conjugate to their squares.
2. General Groups with Elements Conjugate
to Their Squares
Let G be a group and Ga . We consider the con-
sequences of a having the property that it is conjugate
to its square, i.e., there exists Gg such that
.= 2gaga
Denote by )( aC the centralizer of a:
},=:{=)( gaagGgaC

and set
}.=:{=)( 2gagaGgaS 
If )(aSg

, we see that gaga nn 2 = for 

n
and n
n
ngaag2
= for all 
n. It follows that if G
is abelian, ea =, the identity of G.
Fix )(aSk

and define a map )()(: 2 aCaC 
by 1
=)(  ksks, for )(aCs

. We can see that

is
well defined since
),(=====)( 21211212 sakskakaskksakakskas  
so that )()( 2
aCs. Clearly  is one--to--one, and
it is a homomorphism since
).()(===)( 111 stkskktkktskts  

is onto, for if )( 2 aCr , set rkks 1 =, then
rs =)(

and )(aCs

, which shows that

is a
group isomorphism.
Since )()( 2
aCaC , there is an )(aCs  with
as=)(

, i.e., aksk=
1, or akks1
=, so that
.=== 1112 akakakakkks 
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 Ga  satisfies
kaka2
= for some Gk

. Define a map

G. R. GOODSON 417
).(,=)()()(: 12 aCsksksbyaCaC  
Then
(a)  is a well defined group isomorphism.
(b) akka 1
= is a square root of a in )(aC .
(c) )(=)(2 aCaC if and only if there is exactly one
square root of a in )(aC, which is conjugate to a.
(d) If )(aC is abelian, then  is independent of the
conjugation k, )(=)(2
aCaC , so  is a group auto-
morphism.
(e) If }:{=)( naaC n, then eam= for some
1>m odd.
Proof. (c) First assume )(=)(2 aCaC and 1
1
11 =akks 
and 2
1
22 =akks  are two square roots of a conjugate to
a. Then 2
21
21
21
1== kakkaka . It follows that
1
12
221
12 = kkaakk and )(=)(21 12 aCaCkk 
. In
particular
212
1
21
1
1
1
12
1
12 === ssakkakkkakakk  
(a may have other square roots either in )(aC or not,
which are not conjugate to a).
Conversely suppose that )(\)( 2aCaCb. We have
kaka2 =, so set bkk =
, then
,==== 222kabkakbabkaak 
since )(2
aCb. Therefore kak 1 )( is a square root
of a conjugate to a. If akkkak 11=)(  , then
akkkka 11 =  or baab = so )(aCb, a
contradiction. It follows that akk 1 and kak  1
)( are
distinct square roots and the result follows.
(d) If ii kaak 2
=, 1,2=i, we see that )(
1
1
2aCkk 
,
and since )(aC is abelian

11
21 21
=forall(),kksskks Ca
 
or 1
22
1
11 = skkskk , which says that  is independent
of the conjugation k.
Write 2
1
=as , then for )(aCr 
),(=====)( 11
2
1
1
2
1
1raakrkrakakkraakrkar  
so that )()(aCr  and since )()( 2 aCaC , it
follows that )(=)( 2 aCaC . In particular,  is a group
automorphism.
(e) Note that )(
1aCakk  , so that n
aakk =
1 for
some n, and this implies ean=
12 . (The same
proof can be used to show that if )(aC is an infinite,
finitely generated abelian group, then a cannot be
conjugate to its square).
Remark. If G is a topological group, then the homo-
morphism )()(:2 aCaC  is continuous.
Examples. 1. Set }=:,{= 2kakakaG  , a countably
infinite, finitely generated non-abelian group.
Let =
2
Q the subgroup of G generated by all
conjugations 1
gag , Gg

. 2
Q is the conjugacy
class of a, and every member of 2
Q is conjugate to its
square. Elements of G can be written as products of the
form nqm kak , 

qnm ,,, so that
)()(=
1mqnnqm kakakakgag  will be of the form
rpr kak , a member of )(aC (consider the cases where
0n and 0<n separately).
2
Q is abelian for suppose that mq  and consider
22
()()
=
==,
mp mqr q
mpqmrq
qm qm
mprqmqmprm
kak kak
kak ak
kaak kkak



 
and
22
()( )
=
==.
qr qmpm
qrmq pm
qm qm
qmqrpmmprm
kakkak
kak ak
kkaakkak




In addition, 2
Q has the following properties (based
on [1,6]):
Proposition 2 Let 2
Q be the subgroup of G
generated by the conjugations 1 gag , Gg .
(a) 2
Q is an abelian subgroup of )(aC, consisting
of elements of the form mnm kak , nm,.
(b) The map 
2
:Q

defined by
mmnm nkak 2=)( 
 is an isomorphism onto the
subgroup
H
of , },:2{= mnnH m of dyadic
rationals.
(c) The commutator subgroup of G is 2
Q and


2
/QG.
(d) 
2


HG where 
2
H denotes the semi-
direct product of
H
and  where the multiplication
is given by
.,,,),,2(=),)(,( mnHrhmnrhmrnhn
Proof. (b)

is well defined for if
)(=)( qrqmpm kakkak , mq, then
22
== =
qm qm
pqmrmqr qmqmr
akakakka


 , so that
r
p
= and qm =.
H
is a subgroup of  with respect to addition and
HQ 
2
:

is a homomorphism since if mq , then
2
(()( ))
=( )
=22=() ().
mp mqr q
qm
mprm
mq mpmqrq
kak kak
ka k
pr kakkak








Clearly

is one-to-one and onto
H
.
(c) If 22 QkQkg ji  , then jiQkji =
2
. In
addition, if Gg

then
g
is of the form

G. R. GOODSON 418
2
)(== Qkkakkkakg nmnqnnmnqm  . This shows that
2
=QkG i
i
 as a disjoint union. In addition, If
Ggg 
21 , we can write 11 =qkg i, 22 =qkgj for
some ji , and 221,Qqq, and we can check that
221 Qkgg ji
. It follows that 
2
/QG.
Let G be the commutator subgroup of G. If
Ggg
21 ,, then we can write 11 =qkgi, 22 =qkgj
for some ji, and 221,Qqq

. Then gkgg ji =
21
and in a similar way hkgg ji =
1
2
1
1, for some
2
,Qhg , so that 2
1
2
1
121 Qgggg 
 , and 2
QG
.
Also GQ 

2 since the commutator
1111
11 211 21
[, ]=
== ==,
kakkak k ka
kaka akka aa a


which gives
1111111
=[,] =[,()()]
g
aggk akggkggaggkgG



(d) Set 
21 =HG and define GG  1
:

by
nh kanh =),(
 where we interpret akka 11/2 = 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
,=),2(=)],)(,[( 2mnr
n
hn kamnrhmrnh
 
,===),(),( 22 mnr
n
hmnr
n
hmrnh kakkaakakamrnh 

so

is an isomorphism.
2 If we set }=,=:,{= 2ekkakakaG n
n , then
necessarily eam= for 12= 
n
m, so n G will be
finite, with order a multiple of m. The group
}=,=:,{= 2eakakakaH n
n , for some n odd, is not
finite.
3 Let us consider an example related to the horocycle
flow in ergodic theory (see [7]). Set
=(2,)=22matriceswith= 1,
ab
GSLadbc
cd





and G
t
gt








10
1
=, then for 0t, the centralizer of
t
g is the abelian group:
2
()=:,, =1.
0
t
ab
Cgab a
a










In addition, we can check that t
g is conjugate to its
square ,
10
21
=
2






t
gt via elements of G of the form
,
/20
=







c
dc
K where 2.=
2
c
The automorphism )()(: tt gCgC  is defined by
,
0
2
=
0















a
ba
a
ba
which is of course independent of the conjugation
K
(because )( t
gC is abelian).
This type of example can be generalized to
SL ),( n,
e.g., when 3=n, in place of t g, taking
2
1/2
01
001
tt
t





,
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 rank-
one transformations conjugate to their square is demon-
strated. This answered an open question in ergodic
theory, see [2,3]).
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
[0,1]


f: either f is orientation preserving (then
f is continuous, increasing, 0=(0)f and 1=(1)f,
possibly with other fixed points, but no period 2-points
or points of greater period), or f is orientation
reversing (so f is continuous, decreasing, 1=(0)f,
0=(1)f with a unique fixed point and no additional
period 2-points or points of a greater period). We shall
show that for [0,1]


f orientation preserving, f is
conjugate to its square ))((=)(
2xffxf , and we shall
study properties of the conjugating map. The result
below shows that any orientation preserving home-
omorphism 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).
Proposition 3 Let [0,1]


f be orientation pre-
serving, with fixed points
121
0= <<<<=1

nn cccc . Then
(a) f is conjugate to its square 2 f, i.e., there is a
map [0,1]


satisfying   2 =ff .
(b) If [0,1]


with   2 =ff , then either
ii cc =)(

, =1,,in, or 2=n and (0)= 1,

(1) = 0

.
(c) The conjugation

in (b) must have fixed points
i
d with 1 <<iiicdc for =1,2, ,1 in.
(d) The centralizers )( fC and )(2 fC are not
equal, and f has infinitely many distinct square roots
([0,1]

g with fg =
2), each conjugate to f.
Proof. (a) To keep the proof simple, we prove this for

G. R. GOODSON 419
the case where there are only two fixed points, 0 and 1.
The general case is similar.
Either xxf>)( for all 1<<0 x or xxf <)( for
all 1<<0 x (since otherwise f will have additional
fixed points). Suppose the latter holds (the former is
treated similarly). In this case, if (0,1)x then
0=)(
lim xf n n , so 0=x is an attracting fixed point
and 1=)(
lim xf n
n

 , or 1=x is a repelling fixed
point.
Fix (0,1), ba and define a map

linearly on the
interval ]),((= aafI onto the interval ]),((= 2bbfJ
(so that ba =)(

, )(=))(( 2bfaf  ). These are called
fundamental domains for f and 2
f. Roughly speak-
ing, if we know the values )(x

takes on
I
, then the
equation ))((=))(( 2xfxf

determines the values of
)(x

on the rest of [0,1].
Extend

to all of
)(=(0,1) =Ifn
n



(a disjoint union since xxf <)( and f is strictly
increasing and continuous) as follows: Given any
(0,1)x, there is exactly one n with
2
(), sothat(),
andweset( )=((( ))).
nn
nn
xfIfxI
x
ffx 



We can check that )(=))(( 2bfaf nn  for all 

n
and that

is well defined. If we let 0=(0)

,
1=(1)

, then

is a homeomorphism with the
property that )(=)( 2xfxf   for all [0,1]x.
(b) For =1,, in, ))((=))(( 2
iicfcf
 , so that
)(=))((
2
ii ccf
 , but f has no period 2-points, so
)(=))(( ii ccf


.
It follows that 1212 {,,, }={(),( ),,()
}

 
nn
cccccc ,
so that ji cc =)(

for some nj 

1.
Since [0,1]

,

cannot have periodic points
that are non-fixed in (0,1) , we must have ii cc =)(

for ni<<1 . If in addition 1=(1)0,=(0)


, there is
nothing more to prove, so suppose that
0=(1)1,=(0)


. In this case there is a unique fixed
point, so either 2=n or 3=n. It follows from (c)
(below) that 3=n 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
[0,1], 

f satisfy 0=(0)f, 1=(1)f and
))((=))(( 2xfxf  , xxf<)( for [0,1]x. Then for
any 1<<0 x, 0)( xfn as n.
Suppose that there exists 1<<<0 ba with ba=)(

(if not there must exist 1<<<0ab with ba =)(

and
we proceed in an analogous way). Set xxxg

)(=)(

,
then 0>=)(=)( abaaag 

.
Now )(=))((=))(( 22 bfafaf  , and generally
)(=))((2bfaf nn 
. We can find  kwith
abf k<)(, so that )(<))((=)(
2afbffbf kkkk.
Consequently,
0<)()(=)())((=))(( 2afbfafafafg kkkkk   .
Applying the Intermediate Value Theorem, we see that
there exists ]),([
*aafx k
 with 0=)( * xg , or
** =)( xx
 .
(d) Part (a) shows that f has a square root
 f
1.
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 )(\)(2fCfCh. Set



h=, then by Proposition 1(d) we see that  f
1
and
 f
1 are distinct square roots. It therefore
suffices to show that )(\)( 2fCfC is an infinite set.
The method of part (a) can be used to construct
)( 2 fCh with the property that bah =)( 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 xxf <)( for all (0,1)x). Set
)(=))(( 22bfafh and extend JIh : by linearity,
where ]),((= 2aafI, ]),((= 2bbfJ and continue the
definition of h as in part (a). Because of the way h is
defined on
I
we cannot have )(=))((=))(( bfahfafh
(unless ba=), so that )( fCh. By varying
(0,1),

ba we can define infinitely many distinct
)(\)( 2fCfCh , thus giving infinitely many distinct
square roots of f.
Our hypotheses exclude xxf=)( , [0,1]x 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
2
=)( xxf for [0,1]

x, then f has infinitely many
square roots in [0,1] (besides 2
=)( xxg ).
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 STST2 = and
T
has
finitely many fixed points, then
)).((=)(=)( 2TFixSTFixTFix
In particular,
T
has no period 2-points.
Proof. Let )(= TFixA , )(=2
TFixB , and let
A
#
denote the number of elements of
A
. Clearly BA .
If Bq

then

G. R. GOODSON 420
121 1 ()= ()()STq TSqSqAqSA


so SABA . Now
A
is a finite set, so that B is
also a finite set and we must have SABA== , since S
is a homeomorphism.
4. Groups with the Weak Closure Property
Suppose now that G is a complete metric topological
group. We say that Ga  has the weak closure
property if the closure of }:{ nan in G is equal to
)(aC .
In this case there is a dichotomy: the centralizer )(aC
of a is either trivial (}:{==)(  naaaC n), or
there is a sub--sequence n k of integers such that
ean
k as n, and the centralizer )(aC is
uncountable (see [11]).
Now if )(aC is trivial, with 2 a conjugate to a,
then we have seen that a must be of finite order, so we
shall assume throughout that )(aC is uncountable.
Let WC)(a denote the closure of the powers n
a,
n. It is clear that if a has the weak closure
property, then )(aC is an abelian group. As before
)()(: 2 aCaC is the isomorphism of Proposition 1.
Proposition 5 Suppose that a has the weak closure
property, and Gk satisfies kaka2
= (ean for
all 0n). Then
(a)  can be represented as an automorphism
)()(: aCaC , defined by 2 =)( ss.
(b) Every member of )(aC is conjugate to its square
via k and has a unique square root in )(aC .
(c) )(aC does not contain elements of even order.
Proof. (a) If )(aWCs , then i
n
iaslim
= , for
some sequence i n, so
2
2=
lim
=)(
lim
=)( saas i
n
i
i
n
i  ,
therefore )()( 2 aWCs . We have shown that
)())(( 2 aWCaWC  .
Now it is clear that )()(2aWCaWC  and
)()( 2 aCaC . Furthermore, we are assuming that a
has the weak closure property, so )(=)( aCaWC .
Hence
2
22
() ()
=()() =(())=(())(),
WC aWC a
Ca CaCaWCaWCa


so we must have equality throughout. It follows that 2
a
has the weak closure property and )(=)( 2aCaC .
Furthermore, 21 ==)( sksks 
 defines an isomor-
phism from )(aC to )( 2
aC , which may be regarded as
an automorphism of the group )(aC .
(b) If )(aCs , then 21==)(sksks 
, so
s
is
conjugate to its square. Also, there exists a unique
)(aCr

, such that srr ==)( 2
, i.e., every member
of )(aC has a unique square root in )(aC which is
conjugate to its square.
(c) Let }{\)(eaCr

be of finite even order,
er m=
2 say, then ererkkrmmm ==)(= 21 
, a
contradiction.
Most of our results are equally valid for groups with
elements a conjugate to 3 a or 4 a etc. The
following proposition is an exception to this claim:
Proposition 6 If a has the weak closure property,
then all conjugations between a and 2 a are con-
jugate.
Proof. Suppose that 1
s and 2 s are two conjugations
between a and 2 a, then )(
1
1
2aCss 
. Then since a
has the weak closure property
i
n
i
ass lim =
1
1
2

for some subsequence i n. We have 2
11
=
nn
ii
s
aas , for
all 1i, so we deduce, on taking limits, that
,)(=)( 1
2
1
1
21
1
21 ssssss 
or 11
21212 1
()=()
s
ssss s

. This says that 1 s and 2 s are
conjugate (12 =rssr ), via the conjugation 1
21
= ssr .
5. The Topological Discrete Spectrum
Theorem
The main examples having the weak closure property
that we consider are those with (topological) discrete
spectrum. Let XXT: be a homeomorphism of a
compact metric space
X
. Denote by )(XC the
Banach space of complex valued continuous functions
Xf: with the supremum norm. A non-zero
function )(XCf

is an eigenfunction of
T
with
corresponding eigenvalue 

if )(=)( xfTxf

for
Xx

.
Suppose now that
T
is a transitive map: there is a
point Xx

0 whose orbit }:{=)(00 nxTxO n is
dense in
X
. If there is )(XCg  with )(=)( xgTxg
for all Xx

(an invariant function), then
g
must be
constant, because it is a continuous function constant on
the orbit of 0
x (a dense set).
Let )(XCf

with )(=)( xfTxf

for all Xx

.
Then
|()|=|||()| |()|
sup
=|||()|| ()|=||| ()|,
sup supsup
xX
xX xXxX
fTxf xfTx
f
xfx fx



 


so 1|=|

and |=)(| xf 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

G. R. GOODSON 421
linearly independent set. Furthermore, the set of all
eigenvalues of
T
is a countable subgroup of the circle
group 1}|=:|{=
1zzS  (see Walters [6]).
A homeomorphism XXT : on a compact metric
space
X
is said to have (topological) discrete spectrum
if the eigenfunctions of
T
span )(XC (i.e., the
smallest closed linear subspace containing the eigen-
functions is )(XC ). 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 spec-
trum:
The Discrete Spectrum Theorem of Halmos and von
Neumann says:
Let XXTS :, 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 GGR:,
GgaggR,=)( on some compact abelian group G
(Ga  fixed with the property that }:{ nan is
dense in G ): see [5,12].
If G is a compact abelian group and =)(G
 the
set of self-homeomorphisms of G, define )(GT 

by bggT =)( for some Gb , a rotation on G. If we
set 2 =)( ggS then we see that STST2 = since
222
22222
()= ()=()=,
and() =()=.
STgS bgbgbg
TSgTgbg
T
and 2 T need not be conjugate, but will be
conjugate if S is a group automorphism. For example,
they are conjugate when
2/
=={ :=0,1,,1}
ik q
q
Gek q

 , (q odd),
GGT :, bggT =)( where qi
eb /2 =

and
2
=)(ggS .
However, if we set 1 =SG, the circle group, and
define a transitive map with discrete spectrum
11
:SST  by azzT =)( where 1 Sa is fixed with
1
n
a for all n, then 2 =)( zzS is onto but not
one--to--one and STST2
= (
2
T is said to be a factor
of
T
in this case). Clearly
T
and 2 T are not
conjugate since
T
has eigenvalue group
}:{=  nan and 2 T has eigenvalue group
}:{= 22  na n. The eigenfunctions of
T
are the
continuous characters of 1
S. These are the continuous
homomorphisms 11
:SS   , and are of the form
n
nzz =)(
 for each n.
T
is an example of what
is called an irrational rotation of the circle and is
conjugate to the map
[0,1)[0,1):

T,


xxT =)( (mod 1) where
2
=i ae


.
We now show that transitive rotations have the weak
closure property. In order to do this we need to specify
the topology on )(G
. It is known that on any compact
metrizable group G there is a metric

which is
rotation invariant: ),(=),(=),( ygxgyxgygx



for
all Ggyx

,, . Now we define a metric d on )(G

giving the compact- open topology:
).(,),,(
sup
),(
sup
=),( 11 GTSxTxSTxSxTSd
GxGx
 


With this topology, )(G
 is complete metric topolo-
gical group.
Lemma 1 Let G be a compact metric abelian group.
If GGR: is the rotation aggR =)( where
}:{ nan is dense in G, then R has the weak
closure property. In addition, )(RC, the centralizer of
R, is the set of all rotations on G.
Proof. For each Gb

, define a rotation GGTb:
by bggTb=)( , then there is a sequence of positive
integers k
n such that ba k
n in G, as

k. It
follows that b
k
nTR  in )(G
, as k.
It therefore suffices to show that the centralizer of R
is given by
}.:{=)( GbTRCb

Clearly )(}:{ RCGbTb

, so suppose that
)(RCT

then
=()=()()=(),
.
nn
TRRT Tag aTgTagaTg
gG


Let Gx

be arbitrary and choose k
n so that
xak
n, and set )(= eTb, then )(=)(eTaaTk n
k
n for
=1,2,k, implies that bxxT =)( for all Gx

, and
so }:{ GbTT b


.
Proposition 7 Let :TX X be transitive with
discrete spectrum and suppose that XXS : is a
homeomorphism satisfying STST2 =. Then
T
can be
represented as a rotation GGR :, aggR =)( on a
compact abelian group G, and S can be represented
as GGS :, 2
=)( bggS for some Gba ,.
Proof. The Discrete Spectrum Theorem tells us that
T
can be represented as a rotation aggR =)( on a
compact abelian group G, so that we can assume
2
22
()= ()
and( )=( )()=( ),
for all,
nn nn
SR gR Sg
SRgR SgSaga Sg
gG


Let Gx

and choose a sequence xa k
n as

k, and set )(= eSb , then we must have
2
=)( bxxS for all Gx

.
In the following proof we talk about the character

G. R. GOODSON 422
group

G of a (locally) compact abelian group G. This
is the set of all continuous characters 1 :SG   (i.e.,
continuous homomorphisms 1
:SG   ). If G is a
compact group,

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
2
T if and only if the map :

, 2
=)( 
is a
group automorphism.
Proof.
T
is conjugate to a rotation GGR:,
aggR=)( for some compact abelian group G and
some Ga , where R has the weak closure property.
It follows that the map )()(: RCRC  , 2
()=
s
s is
a group automorphism. Because }:{=)( GbTRCb

,
(where bggTb=)( ), this can be written as
2
2==)( b
bb TTT. We deduce that the map
,=)(,: 2 ggGG  
is a group automorphism.
The eigenfunctions of R are the continuous
characters 1
:SG   since
)()(=)(=))(( gaaggR




, and the group of
eigenvalues of R is

={( ):}aG   , where

G is the group of characters of G. We then see that
the map

22
:,()()=(())=( )=(),aaaa  
is an automorphism which can be identified with
:

, 2
=)( 
which is therefore also an
automorphism.
Conversely, if GGT : has eigenvalues

, a
countable subgroup of 1
S, for which the map
:

, 2
=)( 
is a group automorphism, then
2
=)(   , so R and 2 R have the same eigenvalues,
and are conjugate by the Discrete Spectrum Theorem.
Examples. 1. Set 11=SSi, 0i and 1
0=
=ii S

 , a
compact group when given the product topology and
usual group operation. Let G be the subgroup of

defined by
22 2
01201121
={( , ,,):=,=,,=,}.
nn
G
zzzzzz zzz


(G is actually the inverse limit of the sequence
11
SS , where all the arrows denote the power
two homomorphism. This is written 1
lim
=SG ).
Fix
 i
ez=
0 where 1 0
n
z for all n, and set
0012
=( , ,,)

zzz where 2
1
=nnzz , for = 0,1,2n.
Define a group rotation GGT : by




0
=)(T for G


. It can be shown that
T
is a
transitive homeomorphism having discrete spectrum, the
eigenvalue group being },:{= /2 nmeHm
in 
, and it is
easy to check that 2 =)(  S is an automorphism
which conjugates
T
to 2 T.
If we define HH :

, by 2 =)(  , then

is a
group automorphism and we see that the hypotheses of
Theorem 1 are satisfied.
Actually, it is a consequence of the Discrete Spectrum
Theorem that for any countable subgroup

of the
circle 1 S, there is a transitive homeomorphism
GGT : (

=G

a compact abelian group) having
discrete spectrum, and which has  as its eigenvalue
group. It follows that if

is any countable subgroup of
1
S for which



:

, 2 =)(  , is an auto-
morphism, 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
GGT : the adding machine. We can think of G as
31=
=

n G where {0,1,2}=
3
, the group of integers
modulo 3. The group operation on G can be defined as
“carry to the right”, so for example
(2,1, 0, 2,1,)(1, 2,1, 0,1,)=(0,1,1, 2, 2,).

 
The adding machine is then defined by
,1=)(

ggT
where 1= (1,0,0,)
 and 12 =( ,,)
g
gg G.
T
is
transitive and since it is a group rotation, it has discrete
spectrum. The eigenvalues are the n 3th roots of unity.
The map ggS 2=)( is a group automorphism of G
which conjugates
T
to 2 T. S is not transitive as sets
of the form




23 3
0,,,: i gg Gg are S inva-
riant. However, the only fixed point of S is the
identity of the group G. Since
T
has the weak closure
property, all other conjugations are conjugate to S.
3. One of the earliest results in dynamical systems is
due to Poincaré concerning homeomorphisms
11
:SSf  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
(1)=()1,Fx Fx

for all x
(respectively strictly decreasing with (1)

Fx
=()1

Fx for all 

x).
F
determines a
homeomorphism of the circle 11 :SSf  defined by
.=)( )(22 xiFixeef

In addition, every homeomorphism of 1 S arises in

G. R. GOODSON 423
this way. 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 2
22=)( ixix eef 
is
a circle homeomorphism with a single fixed point.
2
=)( xxF , 1<0 x and extended to all of  so that
1)(=1)( xFxF , 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 x,
)(=
)(
lim f
n
xF n
n


exists and is independent of
x
. Since the lift
F
is not
unique (any two differ by an integer), by choosing
1<(0)0 F, we may assume that 1<)(0f

 is
unique and we call )( f

the rotation number of f. It
satisfies )(=)(fnf n
 (mod 1) for n, and
)(=)( 1fhfh 
 for any other orientation preserving
circle homeomorphism h.
2) If 0=)( f

, then f has a fixed point. If )( f

is rational, then f has a periodic point and )( f

is
irrational if and only if f has no periodic points. There
are circle homeomorphisms having points of any given
period.
3) Poincaré showed: A transitive circle
homeomorphism having irrational rotation number is
conjugate to an irrational rotation of the circle.
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.
Theorem 2 Let 11 :SSf  be an orientation
preserving circle homeomorphism. Then
(a) f is conjugate to 2 f if and only if f has at
least one fixed point.
(b) If f has a single fixed point c and
hffh  2 = for some circle homeomorphism h, then
cch =)( , and h has at least one other fixed point. If
f has fixed points Fix12 ()={, ,, }
n
f
cc c, then h
permutes the set Fix)( f. If n is prime these points are
either fixed or constitute an n-cycle.
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 hffh  2 =, we may assume h is
orientation preserving (otherwise look at 2 h), so the
above properties imply
)(2=)(=)(=)( 221 ffhfhf 
, so that 0=)( f

,
and f must have a fixed point.
(b) If ccf =)( then ))((=))(( 2chfcfh , so
)(=))(( chchf as f cannot have any period 2-points.
Since c is unique, we must have cch=)( . An
argument similar to that in Proposition 3(c) can be used
to show that h must have an additional fixed point.
If f has n fixed points },,,,{ 21 n ccc  then as
above, )(=))(( ii chchf for ni ,1,2,=. 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.
4. Let [0,1]=X, =

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 XXT: (
T
will be one-to-one
and onto, but possibly only after a set of measure zero is
omitted). These satisfy 1 (),()TATA


,
))((=)(=))((1ATAAT  for all

A. 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
3th 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 nth 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 [0,1] . Denote by )(nBi the ith level
of the nth column (130 n i, 0n), then S can
now be defined inductively by mapping the level i
B to
the level i B2 (working modulo n 3 for 1n, where
[0,1)=
0
B). By following the orbit of )(nBxi

,
13<0  n i, we can see that STST2 = where
)(XAutS

.
5. Set ii X[0,1]=[0,1]= =



(where [0,1]=[0,1]i),

G. R. GOODSON 424
the Hilbert cube. Suppose we have a map
[0,1][0,1]:T conjugate to its square (for example
2
=)( xxT ) with kTkT 2
=. Set TkkT 11/2 = and
define XXT:
~
by
*
21012
*
1/41/22 4
21012
(,,,,,,)
=(,(),(),(), (), (),),
Txxxxx
TxTxTxTxTx




then if XXS: is the left shift map:
**
101012
(,, ,,)=(,,,,),

Sx xxxxx
(where the * represents the 0th coordinate), then we
can check that STTS 2
~
=
~
. In this case T
~ is a
homeomorphism having uncountably many fixed points
(e.g., if 2
=)( xxT), but no period 2-points. S has
uncountably many periodic points of every order.
6. 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 }=:,{= 2kakakaG , 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
)(:XGV , g VgV =)( . Suppose we set TVa=
and SVk=, then STVVV akka== and
STVVV k
aka
2
22== , so we see that actions of G on
)(X
 are determined by a pair of homeomorphisms S,
T
satisfying STST2 =.
2) 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,
and if )(Th denotes the entropy of a homeomorphism
T
, then )(2=)( 2ThTh (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 ho-
meomorphism 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 2 f. The logistic map
)(1=)( xxxf


, [0,1]x cannot be conjugate to
2

f for 3>

, because such maps have period 2-points.
However, conjugacies between continuous maps and
their squares on higher dimensional spaces may lead to
interesting dynamics.
7. References
[1] O. N. Ageev, “Spectral Rigidity of Group Actions:
Applications to the Case gr)=:,(2tstsst ,” Procee-
dings of the American Mathematical Society, Vol. 134,
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