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Mapping Theorems for Rigid Continua and Their Inverse Limits

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Abstract

We give mapping theorems for certain families of rigid continua; i.e., we prove a mapping theorem for stars, paths and cycles of Cook continua. We also introduce the degree of rigidity of a continuum, the notion of \(\frac{1}{n}\)-rigid continua and prove some existence theorems for \(\frac{1}{n}\)-rigid continua. We also construct a non-trivial infinite family of pairwise non-homeomorphic continua X with the property that for any sequence \((f_n)\) of continuous surjections \(f_n:X\rightarrow X\), the inverse limit \(\varprojlim \{X,f_i\}_{i=1}^{\infty }\) is homeomorphic to X. Explicitly, we show that for each positive integer n, every \(\frac{1}{n}\)-rigid continuum has this property.

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Acknowledgements

The authors thank the anonymous referee for careful reading and constructive remarks that helped us improve the paper.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000, Maribor, Slovenia

    Iztok Banič

  2. Physics and Mechanics, Institute of Mathematics, Jadranska 19, SI-1000, Ljubljana, Slovenia

    Iztok Banič

  3. Andrej Marušič Institute, University of Primorska, Muzejski trg 2, SI-6000, Koper, Slovenia

    Iztok Banič

  4. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000, Maribor, Slovenia

    Teja Kac

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  1. Iztok Banič
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  2. Teja Kac
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Correspondence to Teja Kac.

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Banič, I., Kac, T. Mapping Theorems for Rigid Continua and Their Inverse Limits. Qual. Theory Dyn. Syst. 21, 117 (2022). https://doi.org/10.1007/s12346-022-00647-1

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