Ergodicity in Riesz Spaces

Homann, Jonathan; Kuo, Wen-Chi; Watson, Bruce A.

doi:10.1007/978-3-030-70974-7_9

Cited by 5 publications

(3 citation statements)

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“…In this setting analogues of the Hopf-Garsia, Birkhoff and Wiener-Kakutani-Yoshida ergodic theorems were given. More recently mixing processes were considered, see [14] and [8], which revisited the concept of ergodicity in Riesz space, see [9]. The current work builds on this foundation to consider, in the Riesz space setting, a Poincaré Recurrence Theorem and Kac formula for the (conditional) mean of the recurrence time of a conditionally ergodic process.…”

Section: Introductionmentioning

confidence: 99%

“…where the above limit is the order limit and f ∈ E S where E S is the set of f ∈ E for which the above order limit exists. Thus L S : E S → E. The set of S-invariant f ∈ E will be denoted [9,Theorem 2.4]. These ideas are further expanded in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…However, as S is a Riesz homomorphism, the S invariant elements form a Riesz subspace generated by the S invariant components of e. Applying Freudenthal's Theorem and using that e is a weak order unit, we have that I S ⊂ R (T ) if and only of each S invariant component of e is in R(T ). Thus we have Definition 1.3 of conditional ergodicity in a Riesz space, which is equivalent to that given in [9], since L S f = f if and only if f ∈ I S . Definition 1.3 (Ergodicity).…”

Section: Introductionmentioning

confidence: 99%

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The Kac formula and Poincaré recurrence theorem in Riesz spaces

Azouzi

Amor

Homann

et al. 2023

Proc. Amer. Math. Soc. Ser. B

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Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in L 1 ( Ω , A , μ ) L^1(\Omega ,{\mathcal A},\mu ) , where μ \mu is a probability measure, new conditional versions of the above theorems are obtained.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Kac formula and Poincaré recurrence theorem in Riesz spaces

Azouzi

Amor

Homann

et al. 2023

Proc. Amer. Math. Soc. Ser. B

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On a Problem of Conrad on Riesz Space Structures

Lenzi

2024

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This paper is concerned with Riesz space structures on a lattice ordered abelian group, continuing a line of research conducted by the author and the collaborators Antonio Di Nola and Gaetano Vitale. First we prove a statement in a paper of Paul Conrad (given without proof) that every non-archimedean totally ordered abelian group has at least two Riesz space structures, if any. Then, as a main result, we prove that there is a non-archimedean lattice ordered abelian group with strong unit having only one Riesz space structure. This gives a solution to a problem posed in a paper of Conrad dating back to 1975. Then we combine these results and the categorial equivalence between lattice ordered abelian groups with strong unit and MV-algebras (due to Daniele Mundici) and the one between Riesz spaces with strong unit and Riesz MV-algebras (due to Di Nola and Ioana Leustean). By combining these tools, we prove that every non-semisimple totally ordered MV-algebra has at least two Riesz MV-algebra structures, if any, and that there is a non-semisimple MV-algebra with only one Riesz MV-algebra structure.

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