\section{Introduction}  \label{S:Intro} 


By definition, a \emph{metric quasicircle} is the \qsc\ image of the unit circle $\mfS^1$.  (See \rf{S:Prelims} for definitions and basic terminology.)  We exhibit a catalog that contains a \bl\ copy of each metric quasicircle.  This is a metric space analog of recent work by Steffen Rohde \cite{Rohde-qcircles-mod-bl}, so we briefly describe his result.  He constructed a collection $\mcR$ of snowflake type planar curves with the intriguing property that each planar quasicircle (the image of $\mfS^1$ under a global \qc\ self-\homeo\ of the plane) is \bl\ equivalent to some curve in $\mcR$.

Rohde's catalog is $\mcR:=\bigcup \mcR_p$, where $p\in[1/4,1/2)$ is a \emph{snowflake parameter}.  Each curve in $\mcR_p$ is built in a manner reminiscent of the construction of the von Koch snowflake.  Thus, each $R\in\mcR_p$ is the limit of a sequence $(R^n)$ of polygons where $R^{n+1}$ is obtained from $R^n$ by using the replacement rule illustrated in \rf{f:Rohde_snow}:  for each of the $4^n$ edges $E$ of $R^n$ we have two choices, either we replace $E$ with the four line segments obtained by dividing $E$ into four arcs of equal diameter, or we replace $E$  by a similarity copy of the polygonal arc $A_p$ pictured at the top right of \rf{f:Rohde_snow}.  In both cases $E$ is replaced by four new segments, each of these with diameter $(1/4)\diam(E)$ in the first case or with diameter $p\diam(E)$ in the second case.  The second type of replacement is done so that the ``tip'' of the replacement arc points into the exterior of $R^n$.  This iterative process starts with $R^1$ being the unit square, and the snowflake parameter, thus the polygonal arc $A_p$, is fixed throughout the construction.  See the discussion at the beginning of \rf{s:C} for more details.

The sequence $(R^n)$ of polygons converges, in the Hausdorff metric, to a planar quasicircle $R$ that we call a \emph{Rohde snowflake} constructed with snowflake parameter $p$.  Then $\mcR_p$ is the collection of all Rohde snowflakes that can be constructed with snowflake parameter $p$.

Rohde \cite[Theorem~1.1]{Rohde-qcircles-mod-bl} proved the following.
\begin{noname}
  A planar Jordan curve is a quasicircle \ifff it is \\ the image of some Rohde snowflake under a \bl \\ self-homeomorphism of the plane.
\end{noname}


\begin{figure}[t] 

  \centering
  \begin{overpic}[width=12cm, tics=10]{Rohde_snow.eps}
  \put(9,13.5){An edge $E$ of $R^n$}
  \put(71.7,15){The arc $A_p$}
    \put(66.5,23){$\scriptstyle{p}$}
    \put(91,23){$\scriptstyle{p}$}
    \put(74.5,28){$\scriptstyle{p}$}
    \put(83.5,28){$\scriptstyle{p}$}
    \put(63,3){$\scriptstyle{1/4}$}
    \put(72.5,3){$\scriptstyle{1/4}$}
    \put(82.3,3){$\scriptstyle{1/4}$}
    \put(92.1,3){$\scriptstyle{1/4}$}
  \end{overpic}
  \caption{Construction of a Rohde-snowflake.}
  \label{f:Rohde_snow}
\end{figure}


Thanks to a celebrated theorem of Ahlfors \cite{Ahlfors-QCrflxns}, there is a \emph{simple geometric criterion} that characterizes planar quasicircles: a planar Jordan curve $\Gam$ is a quasicircle if and only if it satisfies the \emph{bounded turning condition}, which means that there is a constant $C\geq 1$ such that for each pair of points $x,y$ on $\Gam$, the smaller diameter subarc $\Gam[x,y]$ of $\Gamma$ that joins $x,y$ satisfies
\begin{equation}  \label{eq:bt}
  \diam(\Gam[x,y])\le C\,|x-y| \,.  \tag{BT}
\end{equation}
We say $\Gamma$ is \emph{$C$-bounded turning} to emphasize the constant $C$.

Tukia and \Va\ \cite{TV-qs} introduced the notion of a \emph{\qsy} between metric spaces.  In this same paper they established the following metric space analog of Ahlfors' result.
\begin{noname}
A metric Jordan curve is a metric quasicircle \ifff it is both \bt\ and doubling (that is, of finite Assouad dimension).
\end{noname}

\medskip

Our catalog $\mcS$ of metric snowflake curves is a collection of metric circles $(\mfS^1,d)$ where the metrics $d$ are given in a simple way by specifying the diameter of each dyadic subarc of $\mfS^1$.  See \eqref{E:dist} and the end of \rf{s:DDF} for precise details.

Our catalog is $\mcS:=\bigcup \mcS_\sig$, and we also employ an auxiliary \emph{snowflake parameter} $\sig\in[1/2,1]$.  Each $(\mfS^1,d_\sig)$ in $\mcS_\sig$ has a metric $d_\sig$ that is obtained by the assignment of diameters to each dyadic subarc of $\mfS^1$.  As in Rohde's construction, at each step there are two choices: the diameter (\wrt $d_\sig$) of a given dyadic subarc is either one-half, or $\sig$, times the diameter of its parent subarc.

Each $(\mfS^1,d_\sig)$ is a \bt\ circle.  Moreover, when $\sig<1$, $(\mfS^1,d_\sig)$ has Assouad dimension $\alpha \leq \log 2/ \log(1/\sigma)<\infty$ (so, $2^{-1/\alpha}\le \sig<1$), hence $(\mfS^1,d_\sig)$ is doubling and thus a metric quasicircle; see \rf{L:dist}(e).  In fact, each collection $\mcS_\sig$ (with $\sig<1$) contains a \bl\ copy of every metric quasicircle with Assouad dimension strictly less than $\log(2)/\log(1/\sig)$.  In addition, the sub-catalog $\mcS_1$ contains a \bl\ copy of every \bt\ circle.

Here is our main result.


\begin{thm*} 

  Let $\Gam$ be a metric Jordan curve.
  \begin{itemize}
    \item[(A)]  If $\,\Gam$ is \bt, then $\Gam$ is \bl\ equivalent to some curve in $\mcS_1$.
    \item[(B)]  If $\,\Gam$ is a metric quasicircle with Assouad dimension $\alf:=\dimA(\Gam)$ and $\sig\in(2^{-1/\alf},1)$, then $\Gam$ is \bl\ equivalent to a curve in $\mcS_\sig$.
    \item[(C)]  A metric quasicircle is \bl\ equivalent to a \emph{planar quasicircle} if and only if it has Assouad dimension strictly less than two.   


  \end{itemize}
\end{thm*}


\noindent
This result is quantitative in that the \bl\ constants depend only on the given data.  For example, if $\Gam$ is $C$-\bt, then the \bl\ constant in \rm(A) is
$$
  L=8\,C \max\{\diam(\Gam),\diam(\Gam)^{-1}\}.
$$
Minor modifications to our proofs reveal that the analogous results hold for \bt\ Jordan arcs and metric quasiarcs.

In addition, we explain how to recover Rohde's theorem from our result.  This provides an alternative proof of Rohde's result that avoids the technical construction of a ``uniform doubling measure'' appearing in \cite[Theorem~1.2]{Rohde-qcircles-mod-bl}.  In view of this, our argument somewhat simplifies the proof of Rohde's theorem.

We mention that Bonk, Heinonen, and Rohde have established a result that gives metric quasicircles as metric boundaries of certain metric disks; see \cite[Lemma~3.7]{BHR-cfml}.

\smallskip

The novel ideas in our approach include the following.  We make extensive use of the fact that every \bt\ metric space is \bl\ equivalent to its associated diameter distance space; see \rf{L:BT iff BL}.  In particular, this permits us to restrict attention to 1-\bt\ Jordan curves.  In this setting, the metrics are characterized, up to \bl\ equivalence, by knowledge of the diameters of certain subarcs, provided we have a sufficiently plentiful collection of subarcs; see \rf{L:dist}.  Finally, there is a straightforward way to build a \bl\ \homeo\ from one of our model curves onto such a metric Jordan curve; see \rf{P:subdiv_homeo} and \rf{L:d BL_to_d_D_variant2k}.


This document is organized as follows.  \rf{S:Prelims} contains preliminary information including background material on Assouad dimension (in \rf{s:assouad-dimension}) and on \qsc\ \homeo s (in \rf{s:QS}).   We prove a result about dividing an arc into subarcs of equal diameter (in \rf{s:dividing-arcs}) and (in \rf{s:shrinking}) give a useful tool for constructing \homeo s between Jordan curves.  We construct our dyadic models in \rf{s:DDF} and prove our Theorem in \rf{S:Proof}.
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