On the distality and expansivity of certain maps on spheres

Abstract

Any affine map on the (n + 1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity.

Keywords:

• Topological dynamics
• homeomorphisms
• spheres
• distality
• expansivity

MSC 2020:

• Primary:37B05
• 37C25
• Secondary:37E10

Acknowledgments

This work started at the International Centre for Theoretical Sciences (ICTS) when the authors went there for the programme Probabilistic Methods in Negative Curvature ICTS/pmnc2019 and the authors are grateful for the hospitality of the ICTS and the nice environment they provide during any programme. The authors are also thankful to ICTS for organizing the second programme with the same theme Probabilistic Methods in Negative Curvature ICTS/pmnc2021/3 which was also helpful to the authors. G. Faraco wishes to thank Ursula Hamenstädt for invaluable support during the last six months of his period at Bonn. One of the authors A. K. Yadav would like to thank R. Shah for introducing this problem and having rigorous discussions during his Ph.D. at JNU, New Delhi and later, and also help in proving Proposition 2.3. A. K. Yadav would also thank National Board for Higher Mathematics, India for Post-doctoral fellowship, and T. Das for local hospitality in the Department of Mathematics, University of Delhi during this fellowship. The authors are very much thankful to the referee for many insightful suggestions which led to a significant improvement in the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).