Strong Uniformity and Large Dynamical Systems

It is the first book about a new aspect of Uniform distribution, called Strong Uniformity. Besides developing the theory of Strong Uniformity, the book also includes novel applications in the underdeveloped field of Large Dynamical Systems.

Contents:

• From Uniform Distribution to the Time-Evolution of Large Off-Equilibrium Systems:

  • Traditional Uniform Distribution and Weyl's Criterion
  • Strong Uniformity
  • High-Dimensional Configuration Space of Large Systems and Unrealistic Time Scale
  • Dimension-Free Strong Uniformity on a Realistic Time Scale
  • Rapid Approach and Long-Term Stability of Square-Root Equilibrium
  • Non-ergodic Time-flow: Closed Orbit Spherical Systems
  • Closed Orbit Polar Systems
  • Snapshot Randomness (I): Poisson
  • Proofs of Theorems 4.2 and 4.3

• General Models:

  • General Model: Unique Ergodicity via Typical Rotations
  • Asymptotic Time-Lapse Randomness
  • Short-Term Time-Lapse Randomness: Multiple Mixedupness (I)
  • Extensions of Theorem 4.2 beyond the Gaussian Case
  • Extensions of Theorem 4.2 to Nonlinear Curves on the Plane

• More Applications of Theorem 4.2:

  • Snapshot Randomness (II): Central Limit Theorem
  • Snapshot Randomness (III) Case of Closed Orbits
  • Time-Lapse Randomness vs Snapshot Randomness (I): A Fundamental Difference
  • Time-Lapse Randomness vs Snapshot Randomness (II): A Fundamental Difference
  • CLT Time-Lapse Randomness: Upper Bound

• More Results about Randomness and Stability in Equilibrium:

  • Simultaneous Square-Root Equilibrium Relative to Nice Sets (I)
  • Simultaneous Square-Root Equilibrium Relative to Nice Sets (II)
  • Simultaneous Square-Root Equilibrium Relative to Nice Sets (III)
  • On the Square-Root Logarithmic Threshold in the Gaussian Case
  • Beyond the Applications of Theorem 4.2
  • The Case of Singular Underlying Measure

• More Proofs:

  • Proof of Theorem 4.1
  • Starting the Proofs of Theorems 13.1–13.4
  • Completing the Proof of Lemma 27.2
  • Finishing the Proofs of Theorems 13.1–13.4
  • Starting the Proof of Theorem 14.1
  • Finishing the Proof of Theorem 14.1
  • Proof of Theorem 14.2
  • Multiple Mixedupness (II): Proof of Lemma 12.2
  • Multiple Mixedupness (III): Proof of Theorem 12.2

Readership: Researchers in dynamical systems and ergodic theory and statistical physics.
Key Features:

• It is the first book about the subject, so it is unique
• The book is basically my contribution to the field