Causality: The p-adic Theory

This book delves into the mathematical theory of causal functions over discrete time, offering a fresh perspective on causality beyond its philosophical roots. By exploring the intricate world of p-adic 1-Lipschitz functions, this volume bridges the gap between abstract mathematical concepts and their practical applications in fields such as automata theory, combinatorics, and applied computer science.

Readers will uncover a wealth of insights as the book investigates key topics including the nature of causal functions, the role of discrete time in causality, and the application of non-Archimedean metrics. With contributions from eminent scholars, this work invites readers to ponder critical questions: How do we define causality in mathematical terms? What are the implications of using p-adic analysis in understanding complex systems especially quantum ones? The author's unique approach makes this book an essential read for anyone interested in the intersection of mathematics and real-world applications.

Ideal for researchers and practitioners with a background in mathematics, computer science, or physics, this book is a valuable resource for those seeking to deepen their understanding of causal functions. Whether you're a scholar exploring theoretical perspectives or a professional looking to apply these concepts practically, this volume offers a comprehensive guide to navigating the complexities of causality. Part of an ongoing series on advanced mathematical theories, it is an indispensable addition to any academic library.

Foreword. - Preface. - Background. - Part I. Basics of p-Adic Analysis. - Rings and Fields of p-Adic Numbers. - p-Adic Calculus. - p-Adic Series. - 1-Lipschitz functions. - Special Classes of 1-Lipschitz Functions. - Part II. The p-Adic Ergodic Theory. - Ergodic Theory: Preliminaries for the p-Adic Case. - The Main Ergodic Theorem for p-Adic 1-Lipschitz Maps. - 1-Lipschitz Ergodicity on Zp. - Measure-Preservation and Ergodicity of Uniformly Differentiable Functions. - 1-Lipschitz Ergodicity on Subspaces. - Plots of 1-Lipschitz Functions in Euclidean Space. - Part III. Applications. - Applications to Automata Theory. - Applications to Computer Science. - Application to Combinatorics. - Applications to Foundations of Quantum Theory. - References. - Index.