Abstract
In this paper we study analytic properties of orbits given by real rational functions. We introduce some comparison methods which allow us to compare the real rational dynamics with automata given by (max, +) functions, passing through a kind of scale transform in tropical geometry. Such a scale transform gives a one-to-one correspondence of presentations between automata and real rational functions. We study invariant properties of the real rational dynamics under change of presentations of automata.
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R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
Hirota R., Yahagi H.: Recurrence equations, an integrable system. Journal of Phys. Soc. Japan 71, 2867–2872 (2002)
Kato T.: Operator dynamics in molecular biology, in “Proceedings of the First International Conference on Natural Computation”. Springer LN in Computer Science 3611, 974–989 (2005)
Kato T.: Interacting maps, symbolic dynamics and automorphisms in microscopic scale. Int. Journal of Pure and Appl. Mathematics 25(3), 311–374 (2005)
T. Kato, Pattern formation from projectively dynamical systems and iterations by families of maps, MPI preprint (2006).
G. Litvinov, V. Maslov, The correspondence principle for idempotent calculus and some computer applications, Idempotency, (J. Gunawardena, ed.), Cambridge Univ. Press (1998), 420–443.
W. de Melo, S. van Strien, One Dimensional Dynamics, Springer (1993).
G. Mikhalkin, Amoebas and tropical geoemtry, in “Different Faces of Geometry” (S. Donaldson, Y. Eliashberg, M. Gromov, eds), Kluwer/Plenum Publ. (2004), 257–300.
C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, Studies in Adv. Math., CRC Press (1999).
D. Takahashi, M. Iwao, Geometrical dynamics of an integrable piecewise-linear mapping, in “Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete” (NATO Science Series II: Mathematics, Physics and Chemistry, (L.D. Faddeev, P.V. Moerbeke, F. Lambert, eds) 201 (2005), 291–300.
Viro O.: Dequantization of real algebraic geometry on logarithmic paper, European Congress of Mathematics, Vol. I (Barcelona, 2000). Birkhäuser Progr. Math. 201, 135–146 (2001)
S. Wolfram, Cellular Automata and Complexity, Addison Wesley (1994).
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Kato, T. Deformations of Real Rational Dynamics in Tropical Geometry. Geom. Funct. Anal. 19, 883–901 (2009). https://doi.org/10.1007/s00039-009-0023-5
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DOI: https://doi.org/10.1007/s00039-009-0023-5