Sharkovsky Ordering

Alexander M. Blokh is a Full Professor at the Department of Mathematics at the University of Alabama at Birmingham, AL, USA and the Director of the Math Department Outreach Program. He mostly does research in dynamical systems, but he is also interested in (mainly planar) topology, and in beautiful results and surprising discoveries in general. Oleksandr M. Sharkovsky, DSc. is the Head of the Department of Dynamical Systems Theory of the Institute of Mathematics at the National Academy of Sciences of Ukraine in Kiev. His research fields include dynamical systems, differential and difference equations, mathematical physics and topology. He is best known for the discovery of the Sharkovsky ordering.

• Preface.- 1 Coexistence of Cycles for Continuous Interval Maps.- 2 Combinatorial Dynamics on the Interval.- 3 Coexistence of Cycles for One-dimensional Spaces.- 4 Multidimensional Dynamical Systems.- 5 Historical Remarks.- 6 Appendix.

“This short book describes the role of Sharkovsky ordering in the theory of dynamical systems, and in particular the way Sharkovsky’s work opened the field of combinatorial dynamics. ... A highlight of the book is a short chapter at the end written by Sharkovsky that describes how his results developed, how they contradicted common opinions of the time that one-dimensional mappings were simple, could be easily understood … .” (Bill Satzer, MAA Reviews, March 30, 2024)“It is brief but develops the theory carefully and with motivation. It also includes an account of the history of the results and some of the context and later developments by the second author. It will be useful as an introduction to low-dimensional dynamics and combinatorial dynamics, and would be suitable for students.” (Thomas B. Ward, zbMATH 1505.37001, 2023)