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Reply to the Comments on: Tian Zhao et al. The Principle of Least Action for Reversible Thermodynamic Processes and Cycles. Entropy 2018, 20, 542
Open AccessArticle

Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Entropy 2018, 20(12), 987; https://doi.org/10.3390/e20120987
Received: 20 November 2018 / Revised: 9 December 2018 / Accepted: 11 December 2018 / Published: 19 December 2018
Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n k = 0 n 1 λ n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov. View Full-Text
Keywords: ergodic theorems; C*-dynamical systems ergodic theorems; C*-dynamical systems
MDPI and ACS Style

Fidaleo, F. Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems. Entropy 2018, 20, 987.

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