This book provides a comprehensive survey of the Sharkovsky ordering, its different aspects and its role in dynamical systems theory and applications. It addresses the coexistence of cycles for continuous interval maps and one-dimensional spaces, combinatorial dynamics on the interval and multidimensional dynamical systems. Also featured is a short chapter of personal remarks by O.M. Sharkovsky on the history of the Sharkovsky ordering, the discovery of which almost 60 years ago led to the inception of combinatorial dynamics. Now one of cornerstones of dynamics, bifurcation theory and chaos theory, the Sharkovsky ordering is an important tool for the investigation of dynamical processes in nature. Assuming only a basic mathematical background, the book will appeal to students, researchers and anyone who is interested in the subject.

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.