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Problem. Is each Peano continuum a topological fractal?

A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that for any sequence $(g_i)_{i\in\omega}\subset\{f_1,\dots,f_n\}^{\omega}$ the intersection $\bigcap_{m\in\omega} g_0\circ\dots\circ g_m(X)$ is a singleton.

(The problem was posed on 16.11.2014 by Taras Banakh on page 3 of Volume 0 of the Lviv Scottish Book).

The prize for solution: A lunch in "Szkocka".

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    $\begingroup$ Karasova and Vejnar proved in (arxiv.org/abs/2209.15394) that a Peano continuum is a topological fractal if it has uncountably many local cut points. $\endgroup$
    – Taras Banakh
    Commented Oct 6, 2022 at 8:43

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