On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

Albeverio, S., Karwowski, W.: A random walk on p-adics, the generator and its spectrum. Stoch. Process. Their Appl. 53, 1–22 (1994)

Albeverio, S., Zhao, X.: On the relation between different constructions of random walks on p-adics. Markov Processes Relat. Fields 6, 239–256 (2000)

Albeverio, S., Zhao, X.: Measure-valued branching processes associated with random walks on p-adics. Ann. Probab. 28, 1680–1710 (2000)

Albeverio, S., Khrennikov, A., Kloeden, P.E.: Memory retrieval as a p-adic dynamical system. Biosystems 49, 105–115 (1999)

Anashin, V., Khrennikov, A.: Applied Algebraic Dynamics. Walter de Gruyter, Berlin, New York (2009)

Areféva, I.Ya., Dragovic, B., Volovich, I.V.: p −adic summability of the anharmonic ocillator. Phys. Lett. B 200, 512–514 (1988)

Areféva, I.Ya., Dragovic, B., Frampton, P.H., Volovich, I.V.: The wave function of the Universe and p −adic gravity. Int. J. Modern Phys. A 6, 4341–4358 (1991)

Arrowsmith, D.K., Vivaldi, F.: Some p −adic representations of the Smale horseshoe. Phys. Lett. A 176, 292–294 (1993)

Arrowsmith, D.K., Vivaldi, F.: Geometry of p-adic Siegel discs. Physica D 71, 222–236 (1994)

Avetisov, V.A., Bikulov, A.H., Kozyrev, S.V.: Application of p-adic analysis to models of spontaneous breaking of the replica symmetry. J. Phys. A: Math. Gen. 32, 8785–8791 (1999)

Beltrametti, E., Cassinelli, G.: Quantum mechanics and p −adic numbers. Found. Phys. 2, 1–7 (1972)

Benedetto, R.: Hyperbolic maps in p-adic dynamics. Ergod. Theory Dyn. Syst. 21, 1–11 (2001)

Besser, A., Deninger, C.: p-adic Mahler measures. J. Reine Angew. Math. 517, 19–50 (1999)

Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79, 473–482 (1995)

Del Muto, M., Fig $\grave{a}$ -Talamanca, A.: Diffusion on locally compact ultrametric spaces. Expo. Math. 22, 197–211 (2004)

Dobrushin, R.L.: The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct. Anal. Appl. 2, 302–312 (1968)

Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theor. Probab. Appl. 15, 458–486 (1970)

Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge Univ. Press, Cambridge (1997)

Fan, A.H., Li, M.T., Yao, J.Y., Zhou, D.: Strict ergodicity of affine p-adic dynamical systems on Z p . Adv. Math. 214, 666–700 (2007)

Fan, A.H., Liao, L.M., Wang, Y.F., Zhou, D.: p-adic repellers in Q p are subshifts of finite type. C. R. Math. Acad. Sci. Paris 344, 219–224 (2007)

Freund, P.G.O., Olson, M.: Non-Archimedian strings. Phys. Lett., B 199, 186–190 (1987)

Ganikhodjaev, N.N.: On pure phases of the three-state ferromagnetic Potts model on the Bethe lattice order two. Theor. Math. Phys. 85, 163–175 (1990)

Ganikhodjaev, N.N., Rozikov, U.A.: Description of periodic extreme Gibbs measures of some lattice model on the Cayley tree. Theor. Math. Phys. 111, 480–486 (1997)

Ganikhodjaev, N.N., Mukhamedov, F.M., Rozikov, U.A.: Phase transitions of the Ising model on ${{\mathbb Z}}$ in the p-adic number field. Uzbek. Math. Jour. 4, 23–29 (Russian) (1998)

Ganikhodjaev, N.N., Mukhamedov, F.M., Rozikov, U.A.: Existence of a phase transition for the Potts p-adic model on the set Z. Theor. Math. Phys. 130, 425–431 (2002)

Georgii, H.O.: Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin (1988)

Herman, M., Yoccoz, J.-C.: Generalizations of some theorems of small divisors to non-Archimedean fields. In: Geometric Dynamics (Rio de Janeiro, 1981). Lecture Notes in Math, vol. 1007, pp. 408–447. Springer, Berlin (1983)

Kaneko, H., Kochubei, A.N.: Weak solutions of stochastic differential equations over the field of p-adic numbers. Tohoku Math. J. 59, 547–564 (2007)

Khamraev, M., Mukhamedov, F.M.: On p-adic λ-model on the Cayley tree. J. Math. Phys. 45, 4025–4034 (2004)

Khamraev, M., Mukhamedov, F.M.: On a class of rational p-adic dynamical systems. J. Math. Anal. Appl. 315, 76–89 (2006)

Katsaras, A.K.: Extensions of p-adic vector measures. Indag. Math. N. S. 19, 579–600 (2008)

Katsaras, A.K.: On spaces of p-adic vector measures. P-Adic Numbers, Ultrametric Analysis, Appl. 1, 190–203 (2009)

Katsaras, A.K.: On p-adic vector measures. J. Math. Anal. Appl. 365, 342–357 (2010)

Khrennikov, A.Yu.: p-adic valued probability measures. Indag. Math. N. S. 7, 311–330 (1996)

Khrennikov, A.Yu.: p-adic description of chaos. In: Alfinito, E., Boti, M. (eds.) Nonlinear Physics: Theory and Experiment, pp. 177–184. WSP, Singapore (1996)

Khrennikov, A.Yu.: p-adic Valued Distributions in Mathematical Physics. Kluwer Academic Publisher, Dordrecht (1994)

Khrennikov, A.Yu.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publisher, Dordrecht (1997)

Khrennikov, A.Yu.: The Bernoulli theorem for probabilities that take p-adic values. Dokl. Math. 55(N. 3), 402–405 (1998)

Khrennikov, A.Yu.: p-adic behaviour of Bernoulli probabilities. Statist. Probab. Lett. 37, 375–380 (1998)

Khrennikov, A.Yu.: Generalized probabilities taking values in non-Archimedean fields and in topological groups. Russian J. Math. Phys. 14, 142–159 (2007)

Khrennikov, A.Yu., Kozyrev, S.V.: Wavelets on ultrametric spaces. Appl. Comput. Harmonic Anal. 19, 61–76 (2005)

Khrennikov, A.Yu., Kozyrev, S.V.: Ultrametric random field. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 199–213 (2006)

Khrennikov, A.Yu., Kozyrev, S.V.: Replica symmetry breaking related to a general ultrametric space I, II, III. Physica, A 359, 222–240, 241–266 (2006); 378, 283–298 (2007)

Khrennikov, A.Yu., Ludkovsky, S.: Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields. Markov Processes Relat. Fields 9, 131–162 (2003)

Khrennikov, A., Mukhamedov, F., Mendes, J.F.F.: On p-adic Gibbs measures of countable state Potts model on the Cayley tree. Nonlinearity 20, 2923–2937 (2007)

Khrennikov, A.Yu., Nilsson, M.: p-adic Deterministic and Random Dynamical Systems. Kluwer, Dordreht (2004)

Khrennikov, A.Yu., Yamada, S., van Rooij, A.: Measure-theoretical approach to p-adic probability theory. Annals Math. Blaise Pascal 6, 21–32 (1999)

Koblitz, N.: p-adic Numbers, p-adic Analysis and Zeta-Function. Berlin, Springer (1977)

Kochubei, A.N.: Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields. Mongr. Textbooks Pure Appl. Math. 244 Marcel Dekker, New York (2001)

Kozyrev, S.V.: Wavelets and spectral analysis of ultrametric pseudodifferential operators. Sbornik Math. 198, 97–116 (2007)

Lubin, J.: Nonarchimedean dynamical systems. Composito Math. 94, 321–346 (1994)

Ludkovsky, S.V.: Non-Archimedean valued quasi-invariant descending at infinity measures. Int. J. Math. Math. Sci. 2005(N. 23), 3799–3817 (2005)

Marinary, E., Parisi, G.: On the p-adic five point function. Phys. Lett., B 203, 52–56 (1988)

Mahler, K.: p-adic numbers and their functions. Cambridge Tracts in Mathematics, vol. 76. Cambridge Univ. Press, Cambridge, New York (1981)

Monna, A., Springer, T.: Integration non-Archimedienne 1, 2. Indag. Math. 25, 634–653 (1963)

Mukhamedov, F.M.: On the existence of generalized Gibbs measures for the one-dimensional p-adic countable state Potts model. Proc. Steklov Inst. Math. 265, 165–176 (2009)

Mukhamedov, F.: On p-adic quasi Gibbs measures for q + 1-state Potts model on the Cayley tree. P-adic Numbers, Ultametric Anal. Appl. 2, 241–251 (2010)

Mukhamedov, F.M., Rozikov, U.A.: On Gibbs measures of p-adic Potts model on the Cayley tree. Indag. Math. N. S. 15, 85–100 (2004)

Mukhamedov, F.M., Rozikov, U.A.: On inhomogeneous p-adic Potts model on a Cayley tree. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8, 277–290 (2005)

Peitgen, H.-O., Jungers, H., Saupe, D.: Chaos Fractals. Springer, Heidelberg-New York (1992)

Peruggi, F., di Liberto, F., Monroy, G.: Phase diagrams of the q-state Potts model on Bethe lattices. Phys. A 141, 151–186 (1987)

Peruggi, F., di Liberto, F., Monroy, G.: The Potts model on Bethe lattices. I. General results. J. Phys. A 16, 811–827 (1983)

Rivera-Letelier, J.: Dynamics of rational functions over local fields. Astérisque 287, 147–230 (2003)

Robert, A.M.: A Course of p-adic Analysis. Springer, New York (2000)

van Rooij, A.: Non-Archimedean Functional Analysis. Marcel Dekker, New York (1978)

Schikhof, W.H.: Ultrametric Calculus. Cambridge University Press, Cambridge (1984)

Silverman, J.H.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics 241. New York, Springer (2007)

Shiryaev, A.N.: Probability. Nauka, Moscow (1980)

Thiran, E., Verstegen, D., Weters, J.: p-adic dynamics. J. Stat. Phys. 54, 893–913 (1989)

Vivaldi, F.: www.maths.qmw.ac.uk/~fv/database/algdyn.bib . Accessed 15 September 2012

Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)

Volovich, I.V.: Number theory as the ultimate physical theory. P-adic Numb. Ultr. Anal. Appl. 2, 77–87 (2010)

Volovich, I.V.: p −adic string. Class. Quan. Grav. 4, L83–L87 (1987)

Yasuda, K.: Extension of measures to infinite-dimensional spaces over p-adic field. Osaka J. Math. 37, 967–985 (2000)

Woodcock, C.F., Smart, N.P.: p-adic chaos and random number generation. Experiment Math. 7, 333–342 (1998)

Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)

Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11, 894–903 (1983)