Mapping Theorems for Rigid Continua and Their Inverse Limits

• Iztok Baniˇc ^[1] ; Teja Kac ^[1]
  1. [1] University of Maribor

     University of Maribor

     Eslovenia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • 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 1 n -rigid continua and prove some existence theorems for 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 ( fn) of continuous surjections fn : X → X, the inverse limit lim←−{X, fi}∞ i=1 is homeomorphic to X. Explicitly, we show that for each positive integer n, every 1 n -rigid continuum has this property.

• Referencias bibliográficas
  • 1. Ancel, F.D., Singh, S.: Rigid finite dimensional compacta whose squares are manifolds. Proc. Amer. Math. Soc. 887, 342–346 (1983)
  • 2. Bing, R.H.: A homogeneous plane continuum. Duke. Math. J. 15, 729–742 (1948)
  • 3. Cook, H.: Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund. Math. 60, 241–249 (1967)
  • 4. Hernandez-Gutierrez, R., Martinez-de-la-Vega, V.: Rigidity of symmetric products. Topol. Appl. 160, 1577–1587 (2013)
  • 5. Hernandez-Gutierez, R., Illanes, A., Martinez-de-la-Vega, V.: Rigidity of Hiperspaces. Rocky Mt. J. Math. 45, 213–236 (2015)
  • 6. Ingram, W.T.: An Introduction to Inverse Limits with Set-valued Functions. Springer, New York (2012)
  • 7. Ma´ckowiak, T.: Singular arc-like continua. Diss. Math. 257, 1–40 (1986)
  • 8. Kennedy, J.: Infinite product of cook continua. Topol. Proc. 14, 89–111 (1989)
  • 9. Kennedy, J.: Positive entropy homeomorphisms on the pseudoarc. Mich. Math. J. 36, 181–191 (1989)
  • 10. Mardeši´c, S.: Chainable Continua and Inverse Limits. Glasnik Mat. Fizicki Astronomski 14, 219–232 (1959)
  • 11. van Mill, J.: A rigid space X for which X × X is homogeneous; an application of infinite-dimensional topology. Proc. Amer. Math. Soc....
  • 12. Nadler, S.B.: Continuum Theory. An Introduction. Marcel Dekker Inc, New York (1992)
  • 13. Reed, J.H.: Inverse limits and indecomposable continua. Pac. J. Math. 23, 597–600 (1967)
  • 14. Terasawa, J.: Rigid continua with many embeddings. Canad. Math. Bull. 35, 557–559 (1992)