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Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.

If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a paper of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?

To state the question more precisely, by the space of Jordan curves here I mean one-to-one continuous maps from $\textbf{S}^1$ to $\textbf{R}^2$, modulo homeomorphisms of $\textbf{S}^1$. So two Jordan curves are close provided that they admit parameterizations which are point wise close.

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.

If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a paper of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?

By the space of Jordan curves here I mean one-to-one continuous maps from $\textbf{S}^1$ to $\textbf{R}^2$, modulo homeomorphisms of $\textbf{S}^1$. So two Jordan curves are close provided that they admit parameterizations which are pointwise close.

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.

If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a paper of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.

If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a paper of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?

To state the question more precisely, by the space of Jordan curves here I mean one-to-one continuous maps from $\textbf{S}^1$ to $\textbf{R}^2$, modulo homeomorphisms of $\textbf{S}^1$. So two Jordan curves are close provided that they admit parameterizations which are point wise close.