PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 3 | 1 |
Tytuł artykułu

On the origin and development of some notions of entropy

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Discrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper we are dealing mainly with the original ideas of entropy in Thermodinamics and their evolution until the appearing in the twenty century of the notions of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn such notions have to evolve to other recent situations where it is necessary to give some extended versions of them adapted to the new problems.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-06-02
zaakceptowano
2015-08-13
online
2015-11-19
Twórcy
  • Departamento de Matemáticas, Universidad de Murcia, 30100-Murcia, Spain
Bibliografia
  • [1] Adler R.L., Konheim A.G. and McAndrew M.H., Topological entropy, Trans.Am.Math.Soc. 114-309 (1965).[WoS]
  • [2] Alsedá L. Llibre J. and Misiurewicz M.Combinatorial dynamics and entropy in dimension one, Advances Series in NonlinearDynamics, World Scientific, Singapore (1993).
  • [3] Amigó J.M. and Giménez A. A simplified algorithm for the topological entropy of multimodal maps, Entropy, 16(2), (2014),627-644.
  • [4] Balibrea F. and Oprocha P., Weak mixing and chaos in nonautonomous discrete systems, Applied Mathematical Letters, 25(2012), 1135-1141.
  • [5] Block L., Guckenheimer J., Misiurewicz M. and Young L.S. Periodic orbits and topological entropy of one-dimensional mapsGlobal Theory of Dynamical Systems, Lecture Notes in Math.,vol 819, Springer-Verlag, New York, (1980), 18-34.
  • [6] Block L., Keesling J., Li S. and Peterson K. An Improved Algorithm for Computing Topological Entropy, Journal of StatisticalPhysics, 55(5/6), (1989), 929-939.
  • [7] Blokh A., On sensitive mappings of the interval, Russian Math.Surveys, 37:2 (1982), 189-190.[Crossref]
  • [8] Balmforth N.J. and Spiegel E.A. Topological entropy of ene-dimensional maps: approximations and bounds, Physical ReviewLetters. vol 72, number 1, (1994), 80-83.
  • [9] Boltzmann L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wiener Berichte 66, (1872), 275-370
  • [10] Boltzmann L. Bermerkungen über einige Probleme der mechanische Wärmetheorie, Wiener Berichte, 75: 62-100; in WA II,paper 39 (1877).
  • [11] Boltzmann L. Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnungrespektive den Sätzen über das Wärmegleichgewicht, Wiener Berichte, 76: 373-435; in WA II, paper 42(1877).
  • [12] Bowen R., Entropy for group of endomorphisms and homogeneous spaces, Trans.Am.Math.Soc. 153, 401-414 (1971); Errata:181(1973), 509-510.
  • [13] Brandaõ P. On the structure of Lorenz maps, arXiv: 1402.2862vi [math.DS], 12/02/2014.
  • [14] Carnot S., Reflections on the Motive Power of Fire and on Machines Fitted to develop that power, Paris: Bachelier. Frenchtitle: Réflections sur la puissance motrice du feu et sur les machines propres à développer cette puissance.
  • [15] Clausius R., Über die bewegende Kraft der Wärme, Part I, Part II, Annalen der Physik, 79, (1850) 368-397, 500-524. Englishtranslation: On the moving Force of Heat and the Laws regarding the Nature of Heat itself which are deducible therefrom,Phil.Mag. (1851),2, 1-21, 102-119.
  • [16] Collet P., Crutchfield P. and Eckmann J.P.Computing the topological entropy of maps, Communications in MathematicalPhysics, 88(2), (1983), 257-262.
  • [17] Downarowicz T. Entropy in Dynamical Systems, New Mathematical Monographs, number 18,Cambridge University Press(2011)
  • [18] Gantmacher W.F. Theory of Matrices, Vol 1, Chelsea Publishing Co. (1959).
  • [19] Glendining P. and Hall T., Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity 9 (1996),999-1014.
  • [20] Gomez P., Franco N, or Silva L., Syllabe Permutations and Hyperbolic Lorenz Knots, to appear in Applied Mathematics andInformation Sciences.
  • [21] Gressman P. and Strain M., Global existence of classical solutions and rapid time decay Proccedings of the National Academyof Sciences of the U.S.A., Vol 107, no. 13, (2010), 5744-5749.
  • [22] Gressman P. and Strain M.,Global classical solutions of the Boltzmann equation with angular cut-off, J.Amer.Math.Soc. 24(2011), 771-847.[WoS][Crossref]
  • [23] Góra P. and Boyarsky A., Computing the topological entropy of general one-dimensional maps, Trans.Am.Math.Soc., 323,1,(1991), 39-49.
  • [24] Hasselblatt B. and Katok A., Principal structures in Handbook of Dynamical Systems, North-Holland, Amsterdam IA, (2002),1-208.
  • [25] Hsu C.S. and Kim M.C., On topological entropy, Pys. Rev. A, 31, (1985), 3253-3260.
  • [26] Katok A., Fifty years on entropy in dynamics: 1958-2007, Journal of Modern Dynamics, Volume 1, No. 4, (2007), 545-596.
  • [27] Katok A. and Hasselblatt B., Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge,(1995).
  • [28] Kawan C. Metric entropy of non-autonomous dynamical systems (arXiv:1304.5682v2 [math.DS] (2013).
  • [29] Kolyada S. and Snoha L., Topological entropy of non-autonomous dynamical systems, Random and Computational Dynamics,4, (1996), 205-233.
  • [30] Kolmogorov A.N., On dynamical systems with an integral invariant on the torus, Doklady Akademii Nauk. SSSR (N.S.), 93(1953), 763-766.
  • [31] Lind D. and Marcus B. An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, reprinted in (1999).
  • [32] Misiurewicz M. and Szlenk W. Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
  • [33] Milnor J. and Thurston W. Dynamical Systems, Lecture Notes in Mathematics 1342, Edited by A.Dold and B.Eckmann,Springer-Verlag (1988).
  • [34] Oliveira H., Symbolic dynamics of odd discontinuous bimodal maps, to appear in Applied Mathematics and InformationSciences.
  • [35] Peris. A. Transitivity, dense orbits and discontinuous functions, Bull. Belg. Math. Soc. 6(1999), 391-394.
  • [36] Shannon C. A Mathematical theory of communication, Bell System Tech., (1948), 379-423, 623-656.
  • [37] Walters P. An Introduction to Ergodic Theory, Springer Graduate Texts in Math., 79, New-York, (1982).
  • [38] Smorodinsky M. Information, entropy and Bernouilli systems, Development of mathematics 1950-2000, Birkhäuser, Basel(2000).
  • [39] Tsallis C., Introduction to Non-extensive Statistical Mechanics- Approaching a Complex World, Springer-Verlag, New-York(2009).
  • [40] Tsallis C., The Nonadditivite Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks, Entropy, Vol 13, (2011),1765-1804.[WoS]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_taa-2015-0006
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.