This book presents recent research results on Lyapunov stability and attraction for semigroup actions in a pedagogical format, providing the reader with numerous modern ideas and mathematical formulations for dynamical concepts in the transformation group theory.

In recent decades, many fundamental concepts of dynamical systems have been extended to the general framework of transformation semigroups. Limit sets, attractors, isolated invariant sets, prolongational limit sets, and stable sets now have semigroup theoretical analogues. This monograph consolidates recent advancements in this field in a way that makes it accessible to graduate students. An effort was made to relate the presented results to important recurrence notions, for contextual clarity.

A rudimentary understanding of group theory and topology, including the concepts of semigroup action, orbit, fiber bundle, compactness, and connectedness, is a prerequisite for reading this text. As a valuable resource for research projects and academic dissertations on topological dynamics, geometry, and mathematical analysis, this work can potentially open new avenues for further research.

Victor H. L. Rocha is a Professor at the São Paulo State University, Brazil. He completed his PhD studies at the State University of Maringá, Brazil. His research interests include geometry and topology, more specifically topological dynamics, dynamical systems, geometric control theory, and Lie theory.

Josiney A. Souza is a Professor at the State University of Maringá, Brazil. He holds a PhD in Mathematics from the State University of Campinas, Brazil. Dr. Souza’s research interests lie in control system, topological dynamics, and Lie theory. He has published two books and more than fifty articles on attractors, Morse decomposition, chain transitivity, stability, controllability, recursiveness, dispersiveness, and uniformities.