This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundam...

Abstract. We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-spacevalued additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides in par...

The Local Ergodic Theorem (also known as the ‘Fundamental Theorem’) gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However, the proof of that theorem relies upon a delicate a...

Local Ergodic Theorem (also known as ‘Fundamental Theorem’) gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption...

This essay investigates Furstenberg’s proof of Szemerédi’s Theorem. The necessary concepts and results from ergodic theory are introduced, including the Poincaré and Mean Ergodic Theorems which are proved in full. The Ergodic Decomposition Theorem is also discussed. Furstenberg’s Multiple Recurrence Theorem is then stated and shown to imply Szemerédi’s Theorem. The remainder of the essay concen...

in this paper, a local approach to the concept of hudetz $g$-entropy is presented. the introduced concept is stated in terms of hudetz $g$-entropy. this representation is based on the concept of $g$-ergodic decomposition which is a result of the choquet's representation theorem for compact convex metrizable subsets of locally convex spaces.

Szemerédi’s Theorem asserts that any positive-density subset of the integers must contain arbitrarily long arithmetic progressions. It is one of the central results of additive combinatorics. After Szemeredi’s original combinatorial proof, Furstenberg noticed the equivalence of this result to a new phenomenon in ergodic theory that he called ‘multiple recurrence’. Furstenberg then developed som...

We characterize the points that satisfy Birkhoff’s ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Löf random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the...