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Key research themes
1. How do nonsingular transformations generalize classical ergodic properties and what invariant classifications arise?
This research theme focuses on extending classical ergodic theory, originally developed for measure-preserving transformations, to the broader context of nonsingular transformations that preserve measure classes but not necessarily the measure itself. Such transformations model non-equilibrium and infinite measure systems and exhibit richer dynamical behavior requiring new classifications (e.g., type II1, II∞, III) and invariants. Understanding basic ergodic properties, conservativeness, ergodic decomposition, and invariants in nonsingular settings illuminates how classical theorems generalize and what new phenomena emerge.
Key finding: Introduces a comprehensive framework for invertible nonsingular ergodic systems on σ-finite measure spaces, defining canonical types II1, II∞, and III based on invariant measures or their absence. The paper provides direct... Read more
Key finding: Constructs examples of infinite measure-preserving rank-one transformations that are weakly doubly ergodic and rigid but whose two-fold Cartesian products are not ergodic, providing counterexamples to the naive extension of... Read more
Key finding: Surveys advanced topics in ergodic theory including spectral theory, periodic approximation, orbit equivalence, and structural properties of measure-preserving transformations, emphasizing novel constructions and spectral... Read more
2. What ergodic theorems hold under generalized measures and non-Markovian dynamics, and how do they impact statistical mechanics and stochastic differential equations?
This theme addresses generalizations of classical ergodic theorems and asymptotic laws to settings involving non-additive measures (e.g., lower probabilities), systems with extrinsic memory or non-Markovian noise (e.g., fractional Brownian motion-driven SDEs), and their applications in statistical mechanics and random dynamical systems. The work investigates convergence properties, uniqueness of invariant or stationary measures, and the extension of results like the strong law of large numbers and mean/pointwise ergodic theorems under these complex conditions, expanding the scope of ergodic theory to modern probabilistic models.
Key finding: Traces the historical development and mathematical proofs of von Neumann's mean ergodic theorem and Birkhoff's pointwise ergodic theorem, establishing their foundational significance in statistical mechanics by providing... Read more
Key finding: Develops a theory extending classical ergodicity results from Markovian stochastic differential equations to non-Markovian cases driven by fractional Brownian motion with Hurst parameter H > 1/2, introducing generalized... Read more
Key finding: Extends classical ergodic theorems to the setting of lower probabilities, which are non-additive monotone set functions arising in economics and statistics, by defining multiple appropriate notions of invariance for such... Read more
3. How does ergodic behavior manifest in infinite measure and deterministic rotation systems, and what are the limiting distributional properties of ergodic sums?
This theme explores ergodic properties, limit theorems, and diffusion-like asymptotic behavior in infinite measure preserving transformations and deterministic systems such as circle rotations, which lack classical ergodicity properties due to arithmetic constraints or infinite measure. Studies include existence and properties of ergodic indices, rigidity phenomena, and almost sure invariance principles (ASIP) for ergodic sums along carefully chosen subsequences, providing new insights into limit distributions, variances growth, and strong approximations relevant for both infinite ergodic theory and low-dimensional dynamical systems.
Key finding: Constructs rank-one infinite measure-preserving transformations that are weakly doubly ergodic and rigid with conservative Cartesian squares that nevertheless fail to have ergodic Cartesian squares, revealing subtle... Read more
Key finding: Analyzes variance growth and establishes an Almost Sure Invariance Principle (ASIP) for ergodic sums of bounded variation functions over irrational circle rotations with unbounded partial quotients. Provides explicit... Read more
Key finding: Derives necessary and sufficient conditions for various types of ergodicity—geometric, strong, and polynomial—for the GI/G/1-type Markov chains by exploiting block-structured matrix methods. Demonstrates explicit connections... Read more