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PreprintCC BY-NC-ND 4.0HAL

INTERVAL MAPS WITH DENSE PERIODICITY

  • Jozef Bobok
  • Jernej Činč
  • Piotr Oprocha
  • Serge Troubetzkoy
Open at sourcePDFCite

Abstract

We consider the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence. Besides studying basic topological properties and density results in the spaces CP and Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent. Furthermore, we prove that a strong topological expansion property called topological exactness (or leo property) is attained on the open dense set of maps in CP and on a residual set in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is rich in a sense that it contains uncountably many conjugacy classes. An analogous conclusion also holds in the setting of interval maps preserving any fixed non-atomic probability measure with full support. Finally, we give a detailed description of the structure of periodic points of generic maps in CP and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing property.

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Cite

BibTeXRIS
BibTeX
@misc{bobok2026interval,
  title = {{INTERVAL MAPS WITH DENSE PERIODICITY}},
  author = {Jozef Bobok and Jernej Činč and Piotr Oprocha and Serge Troubetzkoy},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2402.05638},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-04444509v3},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - INTERVAL MAPS WITH DENSE PERIODICITY
AU  - Jozef Bobok
AU  - Jernej Činč
AU  - Piotr Oprocha
AU  - Serge Troubetzkoy
PY  - 2026
DA  - 2026/09/10
PB  - HAL
UR  - https://hal.science/hal-04444509v3
AB  - We consider the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence. Besides studying basic topological properties and density results in the spaces CP and Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent. Furthermore, we prove that a strong topological expansion property called topological exactness (or leo property) is attained on the open dense set of maps in CP and on a residual set in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is rich in a sense that it contains uncountably many conjugacy classes. An analogous conclusion also holds in the setting of interval maps preserving any fixed non-atomic probability measure with full support. Finally, we give a detailed description of the structure of periodic points of generic maps in CP and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing property.
ER  -