TIME OPERATOR: INNOVATION AND COMPLEXITY
CONTENTS 1 PREDICTABILITY AND INNOVATION 1 1 1 DYNAMICAL SYSTEMS 2 1 2 DYNAMICAL SYSTEMS ASSOCIATED WITH MAPS 4 1 3 THE ERGODIC HIERARCHY 6 1 4 EVOLUTION OPERATORS 9 1 5 ERGODIC PROPERTIES OF DYNAMICAL SYSTEMS OPERATOR APPROACH 12 1 6 INNOVATION AND TIME OPERATOR 14 2 TIME OPERATOR OF KOLMOGOROV SYSTEMS 21 3 TIME OPERATOR OF THE BAKER MAP 29 4 TIME OPERATOR OF RELATIVISTIC SYSTEMS 33 5 TIME OPERATOR OF EXACT SYSTEMS 35 V VI CONTENTS 5 1 EXACT SYSTEMS 35 5 2 TIME OPERATOR OF UNILATERAL SHIFT 38 5 3 TIME OPERATOR OF BILATERAL SHIFT 44 6 TIME OPERATOR OF THE RENYI MAP AND THE HAAR WAVELETS 47 6 1 NON UNIFORM TIME OPERATOR OF THE RENYI MAP 48 6 2 THE DOMAIN OF THE TIME OPERATOR 52 6 3 THE HAAR WAVELETS ON THE INTERVAL 54 6 4 RELATIONS BETWEEN THE TIME OPERATORS OF THE RENYI AND BAKER MAPS 56 6 5 THE UNIFORM TIME OPERATOR FOR THE RENYI MAP 58 7 TIME OPERATOR OF THE CUSP MAP 61 8 TIME OPERATOR OF STATIONARY STOCHASTIC PROCESSES 65 8 1 TIME OPERATOR OF THE STOCHASTIC PROCESSES STATIONARY IN WIDE SENSE 68 8 2 TIME OPERATORS OF STRICTLY STATIONARY PROCESSES FOCK SPACE 73 9 TIME OPERATOR OF DIFFUSION PROCESSES 77 9 1 TIME OPERATORS FOR SEMIGROUPS AND INTERTWINING 78 9 2 INTERTWINING OF THE DIFFUSION EQUATION WITH THE UNILATERAL SHIFT 79 9 3 THE TIME OPERATOR OF THE DIFFUSION SEMIGROUP 82 9 4 THE SPECTRAL RESOLUTION OF THE TIME OPERATOR 85 9 5 AGE EIGENFUNCTIONS AND SHIFT REPRESENTATION OF THE SOLUTION OF THE DIFFUSION EQUATION 86 9 6 TIME OPERATOR OF THE TELEGRAPHIST EQUATION 87 9 7 NONLOCAL ENTROPY OPERATOR FOR THE DIFFUSION EQUATION 88 10 TIME OPERATOR OF SELF SIMILAR PROCESSES 91 11 TIME OPERATOR OF MARKOV PROCESSES 97 11 1 MARKOV PROCESSES AND MARKOV SEMIGROUPS 99 11 2 CANONICAL PROCESS 101 11 3 TIME OPERATORS ASSOCIATED WITH MARKOV PROCESSES 102 12 TIME OPERATOR AND APPROXIMATION 107