I. N. Galay, A. V. Tetenov, “Bi-Lipschitz isomorphisms of self-similar Jordan arcs”, Russian Math. Surveys, 77:4 (2022), 756

A system $\mathcal S=\{S_i,i\in J\}$ of contracting similarities of $\mathbb R^n$, where $J=\{1,\dots,m\}$, is called a self-similar zipper [1] with vertices $\{z_0,\dots,z_m\}$ and signature ${\boldsymbol\varepsilon} \in \{0,1\}^m$ if $S_i(z_0)=z_{i-1+\varepsilon_i}$ and $S_i(z_m)=z_{i-{\varepsilon_i}}$ for all $i\in J$. A non-empty compact set $K\subset \mathbb R^n$ is an attractor of the zipper $\mathcal S$ if $K=S_1(K)\cup\cdots\cup S_m(K)$. An attractor $K$ exists and is uniquely defined by the system $\mathcal S$ (see [2]). If $0=t_0 < t_1 <\cdots< t_m=1$ is a set of points in the interval $I=[0,1] \subset \mathbb R$, then a self-similar zipper ${\mathcal T}=\{T_1,\dots,T_m\}$ with vertices $\{t_0,\dots,t_m\}$ and signature ${\boldsymbol\varepsilon} \in \{0,1\}^m$ is called linear. The attractor of a linear zipper is the interval $I$. For any zipper ${\mathcal S}=\{S_1,\dots,S_m\}$ with vertices $\{z_0,\dots,z_m\}$ and any linear zipper with the same signature $\boldsymbol \varepsilon$ there exists a unique continuous mapping $g\colon I \to K$ such that $g(t_i)=z_i$ and $S_i\circ g=g\circ T_i$ for each $i\in J$. This mapping $g$ is Hölder continuous and $g(I)=K$. Such mappings $g$ are called structural parametrizations of the attractor of the zipper ${\mathcal S}$. A zipper ${\mathcal S}$ is a Jordan zipper if some (hence each) structural parametrization of its attractor $K$ maps the interval $I=[0,1]$ homeomorphically onto $K$ (in this case $K$ is a Jordan arc $\gamma$) [1]. Note that the inverse homeomorphism of a structural parametrization can fail to be Hölder continuous for any Hölder exponent [4].

An important property of subarcs of the attractor $\gamma$ of a Jordan zipper $\mathcal S$ is as follows: if $J^*=\{{\boldsymbol j}=(j_1,\dots,j_n), j_k\in J, n\in \mathbb N\}$ is the set of all multi-indices over $J$, and if $S_{\boldsymbol j}=S_{j_1}\circ\cdots\circ S_{j_n}$ and $\gamma_{\boldsymbol j}=S_{\boldsymbol j}(\gamma)$, then the systems of subarcs $\{\gamma_{\boldsymbol j},{\boldsymbol j}\in J^n,n\in \mathbb N\}$ form a refining sequence of partitions of the arc $\gamma$ such that $\gamma_{\boldsymbol i}\supseteq \gamma_{\boldsymbol j}$ if and only if ${\boldsymbol i}\sqsubseteq{\boldsymbol j}$. If ${\boldsymbol i}\not\sqsubseteq{\boldsymbol j}$ and ${\boldsymbol i}\not\sqsupseteq{\boldsymbol j}$, then $\gamma_{\boldsymbol i}\cap \gamma_{\boldsymbol j}$ is either empty or is the common endpoint of the subarcs $\gamma_{\boldsymbol i}$ and $\gamma_{\boldsymbol j}$.

We say that $\gamma\subset \mathbb R^n$ is an arc of bounded turning ([3], p. 100) if there exists $M>0$ such that for any $x,y\in \gamma$ the diameter $|\gamma_{xy}|$ of the subarc $\gamma_{xy}\subset\gamma$ with endpoints $x$ and $y$ is at most $M\|x-y\|$.

Let $\mathcal S=\{S_i,i\in J\}$ and $\mathcal S'=\{S'_i,i\in J\}$ be self-similar Jordan zippers, and let $\gamma$ and $\gamma'$ be their attractors, respectively. A homeomorphism $f\colon\gamma\to \gamma'$ is said to agree with the systems $\mathcal S$ and $\mathcal S'$ if $f\circ S_i=S'_i\circ f$ for any $i\in J$. In this case one says that $f$ defines an isomorphism of the zippers $\mathcal S$ and $\mathcal S'$. Note that $f(\gamma_{\boldsymbol j})=\gamma'_{\boldsymbol j}$ for any ${\boldsymbol j}\in J^*$.

The following result is a consequence of the uniqueness and invertibility of the structural parametrization of Jordan zippers.

Zippers $\mathcal S$ and $\mathcal S'$ are said to be bi-Hölder (bi-Lipschitz) isomorphic if the homeomorphism $f$ is bi-Hölder (bi-Lipschitz). The following result holds.

Proof. Let $p_{\min}=\min{\{p_1,\dots,p_m\}}$, and let $f\colon\gamma\to\gamma'$ be a homeomorphism which agrees with $\mathcal S$ and $\mathcal S'$. Let $x,y\in\gamma$, $x'=f(x)$, and $y'=f(y)$. Then there exists $M>0$ such that, for the arcs $\gamma_{xy}\subset\gamma$ and $f(\gamma_{xy})=\gamma'_{x'y'}$ we have

$$ \begin{equation} |\gamma_{xy}| \leqslant M\|x-y\|\quad\text{and}\quad |\gamma'_{x'y'}| \leqslant M\|x'-y'\|. \end{equation} \tag{1} $$

By the properties of the families $\{\gamma_{\boldsymbol j},{\boldsymbol j}\in J^*\}$ and $\{\gamma'_{\boldsymbol j},{\boldsymbol j}\in J^*\}$ there are only two possibilities for the subarcs $\gamma_{xy}$ and $\gamma'_{xy}$.

(a) There exist $\boldsymbol i=(i_1,\dots,i_k)\in J^*$ and $i_{k+1}\in I$ such that $\gamma_{{\boldsymbol i}i_{k+1}}\subset \gamma_{xy}\subset \gamma_{\boldsymbol i}$ and $\gamma'_{{\boldsymbol i}i_{k+1}}\subset \gamma'_{x'y'}\subset \gamma'_{\boldsymbol i}$. Hence $p_{{\boldsymbol i}i_{k+1}}|\gamma|\leqslant |\gamma_{xy}|\leqslant p_{\boldsymbol i}|\gamma|$ and $q_{{\boldsymbol i}i_{k+1}}|\gamma'|\leqslant|\gamma'_{x'y'}| \leqslant q_{\boldsymbol j}|\gamma'|$. By (1) we have $p_{{\boldsymbol i}}\leqslant |\gamma_{xy}|/(|\gamma|p_{\min})\leqslant M\|x-y\|/(|\gamma|p_{\min})$. Since $q_{\boldsymbol k}\leqslant p_{\boldsymbol k}^\alpha$ for any multi-index $\boldsymbol k$, it follows that

$$ \begin{equation*} \|x'-y'\|\leqslant|\gamma'_{x'y'}|\leqslant p_{\boldsymbol i}^\alpha |\gamma'|\leqslant M^\alpha|\gamma'|(p_{\min}|\gamma|)^{-\alpha}\|x-y\|^\alpha. \end{equation*} \notag $$

(b) There exist multi-indices $\boldsymbol i=(i_1,\dots,i_k)$ and $\boldsymbol j=(j_1,\dots,j_l)$ and indices $i_{k+1}$ and $j_{l+1}$ such that $\gamma_{{\boldsymbol i}i_{k+1}}\cup \gamma_{{\boldsymbol j}j_{l+1}}\subset\gamma_{xy}\subset \gamma_{\boldsymbol i}\cup \gamma_{\boldsymbol j}$ and $\gamma_{{\boldsymbol i}i_{k+1}}\cap \gamma_{{\boldsymbol j}j_{l+1}}=\gamma_{\boldsymbol i}\cap \gamma_{\boldsymbol j}$ is a singleton; $\gamma'_{x'y'}$ satisfies similar relations.

The inequalities $\max\{p_{{\boldsymbol i}i_{k+1}},p_{{\boldsymbol j}j_{l+1}}\} |\gamma|\leqslant|\gamma_{xy}|\leqslant(p_{\boldsymbol i}+ p_{\boldsymbol j})|\gamma|$ and $|\gamma'_{x'y'}|\leqslant (q_{\boldsymbol i}+q_{\boldsymbol j})|\gamma'|$ imply that $\|x'-y'\|\leqslant|\gamma'_{x'y'}|\leqslant 2[\max\{p_{{\boldsymbol i}},p_{\boldsymbol j}\}]^\alpha|\gamma'|$. Next, we have $\max\{p_{\boldsymbol i},p_{\boldsymbol j}\} \leqslant M\|x- y\|/(|\gamma|p_{\min})$, hence

$$ \begin{equation*} \|x'-y'\|\leqslant 2M^\alpha|\gamma'| (p_{\min}|\gamma|)^{-\alpha}\|x-y\|^\alpha. \end{equation*} \notag $$

If $i\in J$ is such that $\alpha=\ln q_i/\ln p_i$, then for $x=S_i^k(z_0)$ and $y=S_i^k(z_m)$ we have $\|x'-y'\|=M_0\|x-y\|^\alpha$, where $M_0=\|z'_m-z'_0\|/\|z_m-z_0\|^\alpha$. The same argument applies to the mapping $f^{-1}\!\colon\gamma'\to\gamma$. So $f$ and $f^{-1}$ are Hölder continuous with (smallest possible) exponent $\alpha$. $\Box$

The case when $\alpha=1$ implies the following condition for bi-Lipschitz equivalence of self-similar zippers.

Theorem 2. Let $\mathcal{S}=\{S_1,\dots,S_m\}$ and $\mathcal{S'}=\{S'_1,\dots,S'_m\}$ be self-similar Jordan zippers in $\mathbb{R}^n$ with attractors $\gamma$ and $\gamma'$, signatures $\boldsymbol\varepsilon$ and $\boldsymbol\varepsilon'$, and similarity coefficients $\operatorname{Lip}S_i$ and $\operatorname{Lip} S_i'$, $i\in J$, respectively.

1. If $\boldsymbol \varepsilon=\boldsymbol \varepsilon'$ and $\operatorname{Lip}S_i=\operatorname{Lip}S_i'$ for any $i\in J$, and if $\gamma$ and $\gamma'$ are arcs of bounded turning, then $\mathcal{S}$ and $\mathcal{S'}$ are bi-Lipschitz isomorphic.

2. If $\mathcal{S}$ and $\mathcal{S'}$ are bi-Lipschitz isomorphic, and $\gamma$ is an arc of bounded turning, then $\boldsymbol\varepsilon=\boldsymbol \varepsilon'$ and $\operatorname{Lip} S_i=\operatorname{Lip} S_i'$ for any $i\in J$, and $\gamma'$ is an arc of bounded turning.

Citation in format AMSBIB

\Bibitem{GalTet22}

\by I.~N.~Galay, A.~V.~Tetenov

\paper Bi-Lipschitz isomorphisms of self-similar Jordan arcs

\jour Russian Math. Surveys

\yr 2022

\vol 77

\issue 4

\pages 756--758

\mathnet{http://mi.mathnet.ru//eng/rm10062}

\crossref{https://doi.org/10.4213/rm10062e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4461388}

\zmath{https://zbmath.org/?q=an:1521.28009}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2022RuMaS..77..756G}

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