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This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras

This volume of the Encyclopedia of Complexity and Systems Science, Second Edition, is a unique collection of concise overviews of state-of-art, theoretical and experimental findings, prepared by the world leaders in unconventional computing. Topics covered include bacterial computing, artificial chemistry, amorphous computing, computing with Solitons, evolution in materio, immune computing, mechanical computing, molecular automata, membrane computing, bio-inspired metaheuristics, reversible computing, sound and music computing, enzyme-based computing, structural machines, reservoir computing, infinity computing, biomolecular data structures, slime mold computing, nanocomputers, analog computers, DNA computing, novel hardware, thermodynamics of computation, and quantum and optical computing. Topics added to the second edition include: social algorithms, unconventional computational problems, enzyme-based computing, inductive Turing machines, reservoir computing, Grossone Infinity computing, slime mould computing, biomolecular data structures, parallelization of bio-inspired unconventional computing, and photonic computing.Unconventional computing is a cross-breed of computer science, physics, mathematics, chemistry, electronic engineering, biology, materials science and nanotechnology. The aims are to uncover and exploit principles and mechanisms of information processing in, and functional properties of, physical, chemical and living systems, with the goal to develop efficient algorithms, design optimal architectures and manufacture working prototypes of future and emergent computing devices.

The first edition of the Encyclopedia of Complexity and Systems Science (ECSS, 2009) presented a comprehensive overview of granular computing (GrC) broadly divided into several categories: Granular computing from rough set theory, Granular Computing in Database Theory, Granular Computing in Social Networks, Granular Computing and Fuzzy Set Theory, Grid/Cloud Computing, as well as general issues in granular computing. In 2011, the formal theory of GrC was established, providing an adequate infrastructure to support revolutionary new approaches to computer/data science, including the challenges presented by so-called big data. For this volume of ECSS, Second Edition, many entries have been updated to capture these new developments, together with new chapters on such topics as data clustering, outliers in data mining, qualitative fuzzy sets, and information flow analysis for security applications. Granulations can be seen as a natural and ancient methodology deeply rooted in the human mind. Many daily "things" are routinely granulated into sub "things": The topography of earth is granulated into hills, plateaus, etc., space and time are granulated into infinitesimal granules, and a circle is granulated into polygons of infinitesimal sides. Such granules led to the invention of calculus, topology and non-standard analysis. Formalization of general granulation was difficult but, as shown in this volume, great progress has been made in combing discrete and continuous mathematics under one roof for a broad range of applications in data science.