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Journal articles on the topic 'Ergodic theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 July 2024

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1

Gray, R. M. "Ergodic theory." Proceedings of the IEEE 74, no. 2 (1986): 380. http://dx.doi.org/10.1109/proc.1986.13473.

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2

Kida, Yoshikata. "Ergodic group theory." Sugaku Expositions 35, no. 1 (April 7, 2022): 103–26. http://dx.doi.org/10.1090/suga/470.

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3

Moore, Calvin C. "Ergodic theorem, ergodic theory, and statistical mechanics." Proceedings of the National Academy of Sciences 112, no. 7 (February 17, 2015): 1907–11. http://dx.doi.org/10.1073/pnas.1421798112.

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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 fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.

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4

JONES, ROGER L., ROBERT KAUFMAN, JOSEPH M. ROSENBLATT, and MÁTÉ WIERDL. "Oscillation in ergodic theory." Ergodic Theory and Dynamical Systems 18, no. 4 (August 1998): 889–935. http://dx.doi.org/10.1017/s0143385798108349.

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In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

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5

Boyd, David W., Karma Dajani, and Cor Kraaikamp. "Ergodic Theory of Numbers." American Mathematical Monthly 111, no. 7 (August 2004): 633. http://dx.doi.org/10.2307/4145181.

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6

Walters, Peter. "TOPICS IN ERGODIC THEORY." Bulletin of the London Mathematical Society 28, no. 2 (March 1996): 221–23. http://dx.doi.org/10.1112/blms/28.2.221.

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7

Sinai, Ya G., and Barry Simon. "Topics in Ergodic Theory." Physics Today 47, no. 10 (October 1994): 74–75. http://dx.doi.org/10.1063/1.2808677.

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8

Barnes, Julie, Lorelei Koss, and Rachel Rossetti. "The Ergodic Theory Café." Math Horizons 26, no. 3 (December 28, 2018): 5–9. http://dx.doi.org/10.1080/10724117.2018.1518099.

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9

Kozlov, V. V. "Coarsening in ergodic theory." Russian Journal of Mathematical Physics 22, no. 2 (April 2015): 184–87. http://dx.doi.org/10.1134/s1061920815020053.

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10

Jones, Roger L., Joseph M. Rosenblatt, and Máté Wierdl. "Counting in Ergodic Theory." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 996–1019. http://dx.doi.org/10.4153/cjm-1999-044-2.

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AbstractMany aspects of the behavior of averages in ergodic theory are a matter of counting the number of times a particular event occurs. This theme is pursued in this article where we consider large deviations, square functions, jump inequalities and related topics.

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11

Barnes, Julia, and Lorelei Koss. "The Ergodic Theory Carnival." Mathematics Magazine 83, no. 3 (June 2010): 180–90. http://dx.doi.org/10.4169/002557010x494823.

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12

Qi, Weiwei, Zhongwei Shen, Shirou Wang, and Yingfei Yi. "Towards mesoscopic ergodic theory." Science China Mathematics 63, no. 9 (August 6, 2020): 1853–76. http://dx.doi.org/10.1007/s11425-019-1642-5.

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13

Fivaz, Roland. "Ergodic theory of communication." Systems Research 13, no. 2 (June 1996): 127–44. http://dx.doi.org/10.1002/(sici)1099-1735(199606)13:2<127::aid-sres67>3.0.co;2-q.

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14

Biswas, HR, and MS Islam. "Ergodic theory of one dimensional Map." Bangladesh Journal of Scientific and Industrial Research 47, no. 3 (December 21, 2012): 321–26. http://dx.doi.org/10.3329/bjsir.v47i3.13067.

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In this paper we study one dimensional linear and non-linear maps and its dynamical behavior. We study measure theoretical dynamical behavior of the maps. We study ergodic measure and Birkhoff ergodic theorem. Also, we study some problems using Birkhoff's ergodic theorem. DOI: http://dx.doi.org/10.3329/bjsir.v47i3.13067 Bangladesh J. Sci. Ind. Res. 47(3), 321-326 2012

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15

ROY, EMMANUEL. "Poisson suspensions and infinite ergodic theory." Ergodic Theory and Dynamical Systems 29, no. 2 (April 2009): 667–83. http://dx.doi.org/10.1017/s0143385708080279.

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AbstractWe investigate the ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure-preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite-measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.

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16

Ellis, Robert. "Topological dynamics and ergodic theory." Ergodic Theory and Dynamical Systems 7, no. 1 (March 1987): 25–47. http://dx.doi.org/10.1017/s0143385700003795.

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AbstractIt is shown that when viewed properly some concepts in topological dynamics and ergodic theory are not merely analogous but equivalent. Also the Mackey-Halmos-von Neumann theorem on ergodic processes with discrete spectrum is generalized and an account of the Mackey-Zimmer theory of minimal cocycles is given in a more general setting.

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17

Mitkowski, Paweł, and Wojciech Mitkowski. "Ergodic theory approach to chaos: Remarks and computational aspects." International Journal of Applied Mathematics and Computer Science 22, no. 2 (June 1, 2012): 259–67. http://dx.doi.org/10.2478/v10006-012-0019-4.

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Ergodic theory approach to chaos: Remarks and computational aspectsWe discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.

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18

Yuzvinsky, Sergey. "Rokhlin's School in ergodic theory." Ergodic Theory and Dynamical Systems 9, no. 4 (December 1989): 609–18. http://dx.doi.org/10.1017/s0143385700005241.

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Ergodic theory is one of several fields of mathematics where the name Vladimir Abramovich Rokhlin (spelled also ‘Rochlin’ and ‘Rohlin’) is well known to the specialists. That name is attached to some very often used theorems, but the goal of this paper is not just to remind the reader of these theorems. I put them in the context of the general development of ergodic theory during the thirty years 1940–70. Most of all, I want to emphasize that Rokhlin was not only a researcher producing powerful results but also a founder of schools at the two best Universities in the USSR. For at least 10 years (1958–68) these schools dominated ergodic theory. This paper is not biographical. Rokhlin's life certainly deserves a better biographer. However, I mention certain circumstances of a non-mathematical nature where it seems to be appropriate.

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19

Bergelson, Vitaly, Nikos Frantzikinakis, Terence Tao, and Tamar Ziegler. "Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory." Oberwolfach Reports 9, no. 4 (2012): 2985–3059. http://dx.doi.org/10.4171/owr/2012/50.

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20

Ammari, Habib, Liliana Borcea, Thorsten Hohage, and Barbara Kaltenbacher. "Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory." Oberwolfach Reports 9, no. 4 (2012): 3061–127. http://dx.doi.org/10.4171/owr/2012/51.

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21

Svitek, Miroslav. "Wave probabilistic functions, entanglement and quasi-non-ergodic models." ECTI Transactions on Electrical Engineering, Electronics, and Communications 7, no. 2 (December 26, 2008): 55–62. http://dx.doi.org/10.37936/ecti-eec.200972.171892.

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The paper presents the theory of wave probabilistic models together with their important features like inclusion-exclusion rule, product rule and entanglement. These features are athematically described and the illustrative example is shown to demonstrate the possible applications of the theory. The presented theory can be also applied for modeling of quasi-non-ergodic probabilistic systems. First of all we show the new methodology on binary non-ergodic time series. The theory is extended into M-dimensional non-ergodic n-valued systems with linear ergodicity evolution that are called quasi-on-ergodic probabilistic systems.

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22

Johansson, Anders, Anders Öberg, and Mark Pollicott. "Ergodic theory of Kusuoka measures." Journal of Fractal Geometry 4, no. 2 (2017): 185–214. http://dx.doi.org/10.4171/jfg/49.

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23

O'Malley. "Ergodic Theory and First Returns." Real Analysis Exchange 17, no. 1 (1991): 63. http://dx.doi.org/10.2307/44152185.

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24

ORNSTEIN, D. S. "Ergodic Theory, Randomness, and "Chaos"." Science 243, no. 4888 (January 13, 1989): 182–87. http://dx.doi.org/10.1126/science.243.4888.182.

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25

Weiss, Benjamin. "ALGEBRAIC IDEAS IN ERGODIC THEORY." Bulletin of the London Mathematical Society 24, no. 5 (September 1992): 505. http://dx.doi.org/10.1112/blms/24.5.505.

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26

Kitchens, Bruce. "Book Review: Computational ergodic theory." Bulletin of the American Mathematical Society 44, no. 01 (October 2, 2006): 147–56. http://dx.doi.org/10.1090/s0273-0979-06-01120-7.

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27

Feldman, Jack. "Book Review: Basic ergodic theory." Bulletin of the American Mathematical Society 33, no. 03 (July 1, 1996): 345–47. http://dx.doi.org/10.1090/s0273-0979-96-00659-3.

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28

Ruelle, D. "Ergodic Aspects of Turbulence Theory." Physica Scripta T9 (January 1, 1985): 147–50. http://dx.doi.org/10.1088/0031-8949/1985/t9/024.

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29

Runde, Volker, and Ami Viselter. "Ergodic theory for quantum semigroups." Journal of the London Mathematical Society 89, no. 3 (April 25, 2014): 941–59. http://dx.doi.org/10.1112/jlms/jdu015.

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30

Anosov, D. V. "Spectral multiplicity in ergodic theory." Proceedings of the Steklov Institute of Mathematics 290, S1 (September 15, 2015): 1–44. http://dx.doi.org/10.1134/s0081543815070019.

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31

Kilochytska, Т. V. "Evolution of the Ergodic Theory." Nauka ta naukoznavstvo 4 (2019): 102–15. http://dx.doi.org/10.15407/sofs2019.04.102.

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32

Woś, Janusz. "Approximate Convergence in Ergodic Theory." Proceedings of the London Mathematical Society s3-53, no. 1 (July 1986): 65–84. http://dx.doi.org/10.1112/plms/s3-53.1.65.

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33

BOYARSKY, ABRAHAM, and PAWEŁ GÓRA. "AN ERGODIC THEORY OF CONSCIOUSNESS." International Journal of Bifurcation and Chaos 19, no. 04 (April 2009): 1397–400. http://dx.doi.org/10.1142/s0218127409023731.

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It has been suggested that the properties of "integration" and "differentiation" are necessary for the emergence of consciousness. We present a dynamical system model that is based on these two conditions. The collection of neurons are partitioned into clusters on which we define a map that reflects the communication between clusters. Such a map displays the forward and backward circuitry between clusters in a probabilistic manner. The presence of "re-entry" guarantees that the map is sufficiently complex, that is, nonlinear and chaotic, to possess numerous invariant sets of clusters, which are referred to as agglomerations. We suggest that an agglomeration that is mixing characterizes a conscious state. The model establishes a theoretical framework that may structure and encourage experimental work.

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34

Sinai, Ya G. "Kolmogorov's Work on Ergodic Theory." Annals of Probability 17, no. 3 (July 1989): 833–39. http://dx.doi.org/10.1214/aop/1176991249.

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35

Jones, Roger L., Iosif V. Ostrovskii, and Joseph M. Rosenblatt. "Square functions in ergodic theory." Ergodic Theory and Dynamical Systems 16, no. 2 (April 1996): 267–305. http://dx.doi.org/10.1017/s0143385700008816.

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AbstractGiven the usual averages in ergodic theory, let n1 ≤ n2 ≤ … and . There is a strong inequality ‖Sf‖2 ≤ 25‖f‖2 and there is a weak inequality m{Sf > λ} ≤ (7000/λ)‖f‖1. Related results and questions for other variants of this square function are also discussed.

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36

Rota, Gian-Carlo. "Classification problems in ergodic theory." Advances in Mathematics 56, no. 3 (June 1985): 319. http://dx.doi.org/10.1016/0001-8708(85)90038-6.

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37

Rota, Gian-Carlo. "Ergodic theory and dynamical systems." Advances in Mathematics 56, no. 3 (June 1985): 319. http://dx.doi.org/10.1016/0001-8708(85)90040-4.

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38

Rota, Gian-Carlo. "Ergodic theory and related topics." Advances in Mathematics 59, no. 1 (January 1986): 95. http://dx.doi.org/10.1016/0001-8708(86)90042-3.

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39

Rota, Gian-Carlo. "An introduction to ergodic theory." Advances in Mathematics 59, no. 2 (February 1986): 184. http://dx.doi.org/10.1016/0001-8708(86)90052-6.

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40

Rota, Gian-Carlo. "Ergodic theory and differentiable dynamics." Advances in Mathematics 80, no. 2 (April 1990): 270. http://dx.doi.org/10.1016/0001-8708(90)90033-j.

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41

Lin, Dongdai, Tao Shi, and Zifeng Yang. "Ergodic theory over F2〚T〛." Finite Fields and Their Applications 18, no. 3 (May 2012): 473–91. http://dx.doi.org/10.1016/j.ffa.2011.11.001.

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42

Saitô, Nobuhiko. "New concepts in ergodic theory." Vistas in Astronomy 37 (January 1993): 137–39. http://dx.doi.org/10.1016/0083-6656(93)90023-d.

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43

Urbański, Mariusz, and Christian Wolf. "Ergodic Theory of Parabolic Horseshoes." Communications in Mathematical Physics 281, no. 3 (May 15, 2008): 711–51. http://dx.doi.org/10.1007/s00220-008-0498-1.

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44

Thomine, Damien. "Keplerian shear in ergodic theory." Annales Henri Lebesgue 3 (July 30, 2020): 649–76. http://dx.doi.org/10.5802/ahl.42.

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45

Kieffer, John C., and Y. Kifer. "Ergodic Theory of Random Transformations." Journal of the American Statistical Association 83, no. 402 (June 1988): 563. http://dx.doi.org/10.2307/2288883.

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46

Demir, Sakin. "Logarithmic Summability in Ergodic Theory." Journal of Computer and Mathematical Sciences 9, no. 5 (June 5, 2018): 330–33. http://dx.doi.org/10.29055/jcms/779.

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47

Goldman, William M. "Ergodic Theory on Moduli Spaces." Annals of Mathematics 146, no. 3 (November 1997): 475. http://dx.doi.org/10.2307/2952454.

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48

Parczyk, Krystyna. "Ergodic theory and differentiable dynamics." Reports on Mathematical Physics 28, no. 3 (December 1989): 439–41. http://dx.doi.org/10.1016/0034-4877(89)90074-8.

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49

Avigad, Jeremy. "The metamathematics of ergodic theory." Annals of Pure and Applied Logic 157, no. 2-3 (February 2009): 64–76. http://dx.doi.org/10.1016/j.apal.2008.09.001.

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50

Chen, Mufa. "Eigenvalues, inequalities and ergodic theory." Chinese Science Bulletin 45, no. 9 (May 2000): 769–74. http://dx.doi.org/10.1007/bf02887400.

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