Seminarium 24.10.2024
Michał Popławski
A finite family $\mathcal{F} = \{f_1, f_2, . . . , f_n\}$ of continuous selfmaps on a given metric space $(X, d)$ is called an iterated function system (IFS). In [1] authors asked whether for any IFS $\mathcal{F}$ there is a remetrization $(X, \rho)$ of $(X, d)$ (i.e. $d$ and $\rho$ are equivalent) making $\mathcal{F}$ Lipschitz, i.e. each $f_k$ is Lipschitz, for $k = 1, . . . , n$. In [1] there is a positive answer for $\mathcal{F}$ consisting of one function $f : [0, 1] \to [0, 1]$ with some extra assumptions on $f$. We give a general positive answer. Moreover, our technique works for some infinite families of continuous selfmaps.
References
[1] K. Leśniak, N. Snigreva, F. Strobin, A. Vince, Highly non-contractive iterated function systems on euclidean space can have an attractor, J. Dyn. Differ. Equ: 1-18, 2024.
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