Krzysztof Leśniak
Krzysztof Leśniak: Enriching IFS fractals with symmetry
23/05/22 08:00
Thursday, May 26, 2022 17:15
Location: room P.01 C-11
Speaker: Krzysztof Leśniak (Nicolaus Copernicus University in Toruń)
Title: Enriching IFS fractals with symmetry
Abstract: Let \(\mathcal{F}=(X; f_i:i\in I)\) be an iterated function system (IFS) consisting of a finite number of Banach contractions \(f_i\) acting on a complete metric space \(X\). According to the seminal result of Hutchinson (1981), \(F\) admits an attractor, denoted by \(A_{\mathcal{F}}\). Let \(g:X\to X\) be a \(p\)-periodic isometry, \(p>1\), which admits a (not necessarily unique) fixed point.
Proposition: The IFS \(\widetilde{\mathcal{F}} =F \cup \{g\}\) admits a semiattractor \(A^{\flat}\) (in the Lasota—Myjak sense) which is compact and \(g\)-symmetric.
The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.
Theorem (L & Snigireva): \(A^{\flat}\) is an attractor of any of the following contractive IFSs
\begin{eqnarray*}
\mathcal{G}
= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),
%\label{eq:IFS-Gconj}
\\
GF
= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),
%\label{eq:IFS-GF}
\\
\widehat{\mathcal{G}} =
(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).
%\label{eq:IFS-Gdoubletilde}
\end{eqnarray*}
Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).
Remark: \(\widehat{\mathcal{G}}\) appears in Symmetry in Chaos by Field & Golubitsky, cf. http://larryriddle.agnesscott.org/ifskit/IFShelp/howtoCreateSymmetricFractal.html by L.R. Riddle
The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).
Location: room P.01 C-11
Speaker: Krzysztof Leśniak (Nicolaus Copernicus University in Toruń)
Title: Enriching IFS fractals with symmetry
Abstract: Let \(\mathcal{F}=(X; f_i:i\in I)\) be an iterated function system (IFS) consisting of a finite number of Banach contractions \(f_i\) acting on a complete metric space \(X\). According to the seminal result of Hutchinson (1981), \(F\) admits an attractor, denoted by \(A_{\mathcal{F}}\). Let \(g:X\to X\) be a \(p\)-periodic isometry, \(p>1\), which admits a (not necessarily unique) fixed point.
Proposition: The IFS \(\widetilde{\mathcal{F}} =F \cup \{g\}\) admits a semiattractor \(A^{\flat}\) (in the Lasota—Myjak sense) which is compact and \(g\)-symmetric.
The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.
Theorem (L & Snigireva): \(A^{\flat}\) is an attractor of any of the following contractive IFSs
\begin{eqnarray*}
\mathcal{G}
= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),
%\label{eq:IFS-Gconj}
\\
GF
= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),
%\label{eq:IFS-GF}
\\
\widehat{\mathcal{G}} =
(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).
%\label{eq:IFS-Gdoubletilde}
\end{eqnarray*}
Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).
Remark: \(\widehat{\mathcal{G}}\) appears in Symmetry in Chaos by Field & Golubitsky, cf. http://larryriddle.agnesscott.org/ifskit/IFShelp/howtoCreateSymmetricFractal.html by L.R. Riddle
The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).