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\topmatter

\title Irrational Rotations on Simply Connected Domains \endtitle

\author Beverly L. Brechner \endauthor

\address Department of Mathematics, University of Florida, 
Gainesville, Florida  32611-2082 \endaddress

\email brechner\@math.ufl.edu \endemail

\date February 10, 1993 \enddate

\keywords Denjoy homeomorphism, irrational rotation, Siegel disk
conjecture, simply connected domain \endkeywords

\subjclass 54F20, 54F50, 54G15, 54G20, 54H20 \endsubjclass

\abstract  We construct uncountably many simply connected, planar, bounded 
domains, whose closures are topologically distinct, and such that each of 
these domains admits a homeomorphism on its closure which induces a Denjoy 
homeomorphism on the prime end disk.  In particular, each of these homeomorphisms 
is topologically conjugate to the same irrational rotation on the interior of 
the unit disk, and is \lq\lq Denjoy-like" on the boundary.  The closed unit disk is 
one of the examples.  By using uncountably many different irrational rotations, 
we see that the closed unit disk admits uncountably many inequivalent 
homeomorphisms which are Denjoy on the boundary, and are (distinct) irrational 
rotations on the interior.

\endabstract

\thanks  These results were presented at the American Mathematical Society's 
Special Session on Function Spaces, Muncie, Indiana, Oct. 1989 
\endthanks


\endtopmatter

\document

\heading 1. Introduction \endheading  

This work was motivated by the well known Siegel Disk Problem, which asks
the following: Suppose $f$ is a rational function of degree at least two in the 
complex plane, and $M$ is a Siegel disk of this function; that is, $Int(M)$ is a 
bounded, simply connected domain on which $f$ is conjugate to an irrational 
rotation, and further, $Int(M)$ is maximal with respect to this property.  Must 
$Bd(M)$ be a simple closed curve?  

Some partial results, after relaxing the hypotheses, have been obtained by Herman 
[He], by Moeckel [Moe], and by Pommerenke and Rodin [PR].  

More recently, J.T. Rogers, Jr. in [R1,R2], and Mayer and Oversteegen in [MO],
have obtained some very interesting related results.  Rogers defines a \lq\lq local 
Siegel disk" and proves the following:   The boundary $Bd(G)$ of a bounded, irreducible, 
local Siegel disk satifies exactly one of the following two properties:
\roster
\item  If $\phi :Int(D) \longrightarrow G$ is the Riemann map from the interior of
the unit disk to $G$, then 
$\phi^{-1} :G \longrightarrow Int(D)$  must extend continuously, taking $Bd(G)$ to 
$Bd(D)$ = $S^1$, or 
\item  $Bd(G)$ is an indecomposable continuum. 
\endroster
Mayer and Oversteegen construct an 
example of an indecomposable cofrontier invariant under a homeomorphism of 
the plane such that the prime end number from both sides is the same, but the 
induced homeomorphism on the prime end circle from one side is conjugate to a rigid 
rotation, while the induced homeomorphism on the other side is conjugate to a 
Denjoy homeomorphism.  Both of these results are more topological 
in nature than the results mentioned in the previous paragraph.

In this paper, we modify the Siegel Disk Problem to a purely topological question, 
making the only requirement on $f$ be that $f$ is a homeomorphism.  The following 
is proved:
   
For each irrational number $\beta$, there exist uncountably many 
pairs $\{\,(U_\alpha,h_{\beta_\alpha})\,\}_\alpha$, where $U_\alpha$ is a 
simply connected, bounded domain in the plane, and $h_{\beta_\alpha}$ is a 
homeomorphism of the plane onto itself keeping $U_\alpha$ invariant such that 

\roster

\item  $h_{\beta_\alpha}\restriction U_\alpha $ is conjugate to the 
irrational, counter-clockwise rotation of angle $\beta$ on the interior of 
the unit disk, $D$, 

\item For $\alpha \neq \beta , Bd(U_{\alpha})$ is not homeomorphic to 
$Bd(U_{\beta})$, and 

\item The induced prime end map on the boundary of the unit disk is 
conjugate to a Denjoy homeomorphism. (This shows that $Cl(U_\alpha)$ may be taken 
to be a disk, although we also construct a Denjoy homeomorphism on a disk precisely.) 

See [B] to review the terminology of prime ends.
\endroster

Note that 
the boundaries of our examples are decomposable, and thus, by the results of Rogers,
none of these can be turned into an example of a Siegel disk with boundary which is 
not a simple closed curve.  

The author is indebted to Timo Erkama for bringing the Siegel Disk Problem to her 
attention, and for pointing out the references [He], [Moe], and [PR].  She also
wishes to express her gratitude to Jaime Zapata for the beautiful 
computer drawings which he prepared for this paper.

\heading 2.  Denjoy homeomorphisms of the circle \endheading

Let $R_\beta$ denote the counter-clockwise rotation of the unit circle, $S$, 
by angle $\beta$.  Let $Y = \bigcup \{\,y_n \,\}_{n\in Z}$, where $Y$ is a 
full orbit of the point $y_0$ under $R_{\beta}$, and $y_n = R_\beta (y_0)$.

Let $S^\prime$ and $S^{\prime\prime}$ be two copies of S.  On  $S^\prime$, let 
$I = \bigcup \{\,I_n \,\}_{n\in Z}$ be the union of the closures of the collection 
of complementary intervals of some Cantor set. 

Let $f:S^{\prime} \longrightarrow S^{\prime\prime}$ be an onto map whose 
nondegenerate inverses are precisely the closed intervals $I_n \subset I$, 
and let $f(I) = X = \bigcup \{\,x_n\,\}_{n\in Z}$.

Let $g:S^{\prime\prime} \longrightarrow S$ be a homeomorphism taking $X$ onto $Y$. 
Re-label the points of $X$ with the same subscripts as their images under $g$.  Then  
re-label the intervals of $I$ with the same subscripts as their inverses under $f$.

Let $H_{\beta} : S^{\prime} \longrightarrow S^{\prime}$ be the homeomorphism 
defined by $H_{\beta} = (gf)^{-1} R_{\beta} (gf)$ on $C(I)$, and extend by limits on 
the endpoints of $I$, and linearly on the interiors of the intervals of $I$.  
$H_{\beta}$ is called a {\bf Denjoy homeomorphism} of the circle. 

\vskip .05 in

Recall the following definition:  Let $h:S^1 \longrightarrow S^1$ be an orientation preserving 
homeomorphism of the unit circle onto itself.  Let $x \in S^1$, and let 
$H:R \longrightarrow R$ be a lift of $h$.  The {\bf rotation number} of $h$, 
denoted $\rho (h)$, is defined by 

$$\rho (H) = lim_{n\to \infty} \frac {|H^n(\tilde x)|}{n}$$  where $\tilde x$ is 
any lift of $x$, and 

$$\rho (h) = \rho (H) (mod 1)$$
It can be shown that $\rho (h)$ is independent of both $x$ and of the particular 
lift of $h$ that is chosen.  (See [vK] for some properties of the rotation number.) 

Then our 
homeomorphism $H_{\beta}$ has the same rotation number as the rotation $R_{\beta}$.  

Figure 1 illustrates the process described above.


\heading 3. Theorems  \endheading


\proclaim{3.1. Theorem} For each irrational number $\beta$, there exists a 
homeomorphism $h_{\beta}$ of the closed unit disk $D$ onto itself such that 

\roster
\item $h_{\beta}\restriction Int(D)$ is topologically conjugate to the 
counterclockwise irrational rotation of angle $\beta$ on $Int(D)$, and 

\item $h_{\beta}\restriction Bd(D)$ is a Denjoy homeomorphism.
\endroster

\endproclaim

\pr The proof becomes transparent, if one refers to Figure 2 while reading 
the following.

\subheading {Notation}

Let $D$ be the unit disk with boundary $S$.

Let $R_\beta$ be the counterclockwise, irrational rotation of angle $\beta$ on $S$, 
keeping the set $Y$ invariant as in Figure 1.

Let $\hat R_{\beta}$ be the homeomorphism that cones $R_\beta$ over the disk $D$, 
that is, the counterclockwise, irrational rotation of angle $\beta$ on $D$.

Let $D^\prime$ be a copy of the unit disk with boundary $S^\prime$.

Let $H_\beta$ be a Denjoy homeomorphism on $S^\prime$ \lq\lq rotating" the arcs 
$\{I_i\}_{i\in Z}$, as in Figure 1.  

Let $\hat H_{\beta}$ be the homeomorphism that cones $H_\beta$ over the disk 
$D^\prime$.


\subheading {Procedure}

\roster

\item Construct an arbitrary sequence of disks, $\{{B^\prime}_i\}_{i=1}^{\infty}$, 
in the \lq\lq thumb disk" over $I_0$, as indicated in Figure 2.  (We call this a 
{\it thumb disk} because it is shaped like the tip of one's thumb.)  However, the 
largest disk of the sequence should be in the closed sector of the unit disk, 
$D^{\prime}$, determined by the radii from the origin to the endpoints of $I_0$.

\item Copy $\{{B^\prime}_i\}_{i=n}^{\infty}$ on thumb disks over $I_n$ and 
$I_{-n}$, using the Denjoy disk homeomorphism $\hat H_\beta$.  For example, the 
thumb disk over $I_{-2}$ would consist of ${\hat H_\beta}^{-2} (B^{\prime}_i)$ for 
each $i \geq 2$. Note that the thumb disks are small enough so that the thumb disks 
over $I_i$ are disjoint from the thumb disks over $I_j$, for $i \neq j$. 

\item Construct an arbitrary \lq\lq Hawaiian earring" sequence of disks at $y_0$; 
that is, construct a sequence of disks, $\{B_i\}_{i=1}^{\infty}$, 
as indicated in Figure 2.  (We call $B_n$ an {\it Hawaiian disk} because  
$\bigcup_{i=n}^{\infty} Bd(B_i)$ looks like the well-known Hawaiian earring example.)
However, the disk $B_n$ must be small enough so that the $2n+1$ disks of the collection
$\{\hat R_{\beta}^i (B_n)\}_{i=-n}^n$ are disjoint.

\item Copy $\{B_i\}_{i=n}^{\infty}$ at the points $y_n$ and $y_{-n}$ using 
$\hat R_\beta$ in the same way that $\hat H_\beta$ is used in number (2) above.  

\item Map the thumb disk ${B^\prime}_1$ homeomorphically onto the Hawaiian 
disk $B_1$, preserving the diagram; that is, the boundary of ${B^\prime}_i$ maps to 
the boundary of $B_i$, for all $i \geq 1$.  Call this map $\phi _1$.

\item Copy $\phi _1$ to get 
$\phi _2 :\bigcup _{i=-\infty}^\infty$ \{thumb disks over $I_i$\} 
$\longrightarrow \bigcup _{i=-\infty}^\infty$ \{Hawaiian disks over $I_i$\}.  
For example, on the Hawaiian disk over $y_{-2}$, one would put 
${\hat H_\beta}^{-2} (B^{\prime}_i)$ for each $i \geq 2$.

\item Extend $\phi _2$ to get 
$\phi ^* : (S^\prime \cup$ \{thumb disks\}) $\longrightarrow (S \cup $ \{Hawaiian disks\}).

\item Observe that $D^\prime - (S^\prime \cup $ \{thumb disks\}) is an open disk bounded 
by a simple closed curve, say $C$.  Thus the map $\phi ^*$ can be extended to 
all of $D^\prime$ by extending $\phi ^* \restriction C$ to the disk which it 
bounds.  

Call this map $\phi$.  

\endroster

\subheading {Result}

Then $\phi : D^\prime \longrightarrow D$ is a 
map which is one-to-one on all of $D^\prime$, except for the intervals of $I$, 
which map to the points of $Y$.
We define $h_\beta : D^\prime \longrightarrow D^\prime$ by 

$$h_\beta (x) = \left\{
\aligned &\phi^{-1}R_\beta \phi (x) \quad \text{for} \quad x \in C(I)\\
  &H_{\beta} (x) \qquad\quad \text{for} \quad x \in I.  \endaligned
\right. 
$$

Then $h_\beta : D^\prime \longrightarrow D^\prime$ is the desired (onto) 
homeomorphism.  \qd



\proclaim{3.2 Theorem} For each $\beta$, there exist uncountably many pairs 
$\{\,(U_{\alpha},h_{\beta_{\alpha}})\,\}_{\alpha}$ such that 

\roster
\item $U_{\alpha}$ is a bounded, simply connected domain, with $Cl(U_\alpha) = M_\alpha$

\item $Bd(U_{\alpha})$ is {\it not} a simple closed curve, 

\item $h_{\beta_{\alpha}}\restriction U_{\alpha}$ is topologically conjugate 
to the counterclockwise rotation of angle $\beta$ on $Int(D)$.

\endroster

In fact, the elements of the collection $\{M{\alpha}\}$ may be taken to 
be topologically distinct.

\endproclaim

\pr In $D^{\prime}$ of Figure 2, replace $I_0$ by a chainable continuum different 
from an arc, say $P_0$, and having two 
endpoints, and \lq\lq copy" $P_0$, using the 
Denjoy homeomorphism $\hat H_{\beta}$ extended to the entire plane, to replace 
$I_i$ by $P_i$.  This produces a continuum, $M$, different from the disk, since its 
boundary is not a simple closed curve. 

In order to get uncountably many examples, we may proceed in either of two ways: 

\roster
\item In [L], Wayne Lewis describes uncountably many inequivalent embeddings 
$\{P_{\alpha}\}$ 
of the pseudo arc in the plane, each embedding with two endpoints.  For each $\alpha$, 
we get an example by replacing $I_0$ with $P_{\alpha}$, and then copying the continuum 
in the other $I_i$'s, using the 
Denjoy homeomorphism $\hat H_{\beta}$ extended to the entire plane. This produces 
uncountably many examples, $M_\alpha$.

\item In [A], J.J. Andrews describes uncountably many inequivalent chainable continua, each 
with two endpoints.  One could use these continua, as in (1), to obtain uncountably many 
inequivalent examples.

\endroster

See Figure 3 for a graphical representation of the result.
\qd

\proclaim {3.3 Theorem} In each of the examples of Theorem 3.2, the homeomorphism 
induced by $h_{\beta_\alpha}$ on the prime end circle associated with $Bd(U_{\alpha})$ 
($ = Bd(M_\alpha)$)
is a Denjoy homeomorphism conjugate to $H_{\beta}$.

\endproclaim 
\pr In Theorem 3.1, a $C$-map (or conformal map) from $D^\prime$ to the unit disk $D$ is 
approximated by a homeomorphism from the subdisk of $D^\prime$ bounded by $C$.  Closer 
approximations are obtained by enlarging this subdisk by sequentially moving to the 
next thumb disk over each $I_i$ at each stage.  In the case of Theorem 3.1, the 
identity homeomorphism is a $C$-map, so the Denjoy homeomorphism is induced.  

In Theorem 3.2, there is a subdisk of $M_\alpha$ corresponding to the subdisk of $D^\prime$ 
bounded by $C$.  Since $M_\alpha$ is not a disk, we can no longer use the identity 
homeomorphism into the unit disk.  But, clearly, there is a homeomorphism, 
${\psi}^{\prime\prime}$ into the 
unit disk which approximates the diagram.  We then extend this homeomorphism to $Int(M)$, 
by performing a $C$-map extension to the interior of $B^{\prime}_1$, and copying the 
resulting map, say ${\psi}^\prime$, using ${\hat R_\beta}^n {\psi}^\prime {\hat H_\beta}^{-n}$, 
to the interiors of the copies of $B^{\prime}_n$ over $P_n$ and $P_{-n}$.  From this, it 
is clear that the induced homemorphism, $\psi$, on $Bd(D)$ is a Denjoy homeomorphism.
\qd


\Refs

\ref \key A \by James J. Andrews \paper A chainable continuum no two of whose 
nondegenerate subcontinua are homeomorphic \jour Bull. Amer. Math. Soc. \vol 12 
\yr 1961 \pages 333-334
\endref

\ref \key BG \by Marcy Barge and Richard M. Gillette 
\paper Rotation and periodicity in plane separating continua 
\jour Ergodic Theory and Dynamical Systems \yr 1991 \vol 11 \pages 619-631 
\endref

\ref \key B \by Beverly Brechner \paper On stable homeomorphisms and imbeddings of 
the pseudo arc \jour Illinois J. Math. \vol 22 \yr 1978 \pages 630-661
\endref

\ref \key BGM \by B. L. Brechner, M. D. Guay, and J. C. Mayer  
\paper The rotational dynamics of cofrontiers \jour These proceedings 
\endref

\ref \key C \by M. Charpentier 
\paper Sur quelques propri\acuteaccent et\acuteaccent es des courbes de M. Birkhoff 
\jour Bull. Soc. Math. France \yr 1934 \vol 62 \pages 193-224
\endref

\ref \key Ha \by Michael Handel \paper A pathological area preserving $C^{\infty}$ 
diffeomorphism of the plane \jour Proc. Amer. Soc. \vol 86 \yr 1982 \pages 163-168
\endref

\ref \key He \by Michael Herman \paper Construction of some curious diffeomorphisms 
of the Riemann sphere \jour J. London Math. Soc. \vol 34 \yr 1986 \pages 375-384
\endref

\ref \key L \by Wayne Lewis \paper Embeddings of the pseudo arc in $E^2$ 
\jour Pac. J. Math \vol 93 \yr 1981 \pages 115-120
\endref

\ref \key MO \by John C. Mayer and Lex G. Oversteegen 
\paper Denjoy meets rotation on an indecomposable cofrontier
\yr 1992  \jour These proceedings  
\endref

\ref \key Moe \by Richard Moeckel \paper Rotations of the closures of some simply 
connected domains \jour Complex Variables \vol 4 \yr 1985 \pages 285-294
\endref

\ref \key PR \by Ch. Pommerenke and B. Rodin \paper Intrinsic rotations of 
simply connected regions, II \jour Complex Variables \vol 4 \yr 1985 \pages 223-232
\endref

\ref \key R1 \by James T. Rogers, Jr. \paper Is the boundary of a Siegel disk a 
Jordan curve?  \jour Bull. Amer. Math. Soc. \vol 27 \yr 1992  \pages 284-287 
\endref

\ref \key R2 \by James T. Rogers, Jr. \paper Singularities in the boundaries of 
local Siegel disks \jour preprint \yr 1992
\endref

\ref \key vK \by E.R. van Kampen \paper The topological transformations of a 
simple closed curve into itself \jour Amer. J. Math. \vol 57 \yr 1935 \pages 142-152
\endref

\endRefs

\enddocument
