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ON pADIC GIBBS MEASURES OF COUNTABLE STATE POTTS MODEL ON THE CAYLEY TREE
, 705
"... Abstract. In the present paper we consider countable state padic Potts model on the Cayley tree. A construction of padic Gibbs measures which depends on weights λ is given, and an investigation of such measures is reduced to examination of an infinitedimensional recursion equation. Studying of th ..."
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Abstract. In the present paper we consider countable state padic Potts model on the Cayley tree. A construction of padic Gibbs measures which depends on weights λ is given, and an investigation of such measures is reduced to examination of an infinitedimensional recursion equation. Studying of the derived equation under some condition on weights, we prove absence of the phase transition. Note that the condition does not depend on values of the prime p, and an analogues fact is not true when the number of spins is finite. For homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one padic Gibbs measure µλ. The boundedness of the measure is also established. Moreover, continuous dependence the measure µλ on λ is proved. At the end we formulate one limit theorem for µλ.
ON PADIC QUASI GIBBS MEASURES FOR Q+ 1STATE POTTS MODEL ON THE CAYLEY TREE
, 2010
"... In the present paper we introduce a new class of padic measures, associated with q +1state Potts model, called padic quasi Gibbs measure, which is totally different from the padic Gibbs measure. We establish the existence padic quasi Gibbs measures for the model on a Cayley tree. If q is divisi ..."
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In the present paper we introduce a new class of padic measures, associated with q +1state Potts model, called padic quasi Gibbs measure, which is totally different from the padic Gibbs measure. We establish the existence padic quasi Gibbs measures for the model on a Cayley tree. If q is divisible by p, then we prove the occurrence of a strong phase transition. If q and p are relatively prime, then there is a quasi phase transition. These results are totally different from the results of [F.M.Mukhamedov, U.A. Rozikov, Indag. Math. N.S. 15(2005) 85–100], since q is divisible by p, which means that q + 1 is not divided by p, so according to a main result of the mentioned paper, there is a unique and bounded padic Gibbs measure (different from padic quasi Gibbs measure). MIRAMARE – TRIESTE
ON DYNAMICAL SYSTEMS AND PHASE TRANSITIONS FOR Q+ 1STATE PADIC POTTS MODEL ON THE CAYLEY TREE
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PHASE TRANSITIONS FOR PADIC POTTS MODEL ON THE CAYLEY TREE OF ORDER THREE
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BOOK REVIEW
"... Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on ..."
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Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on a manifold, and the local properties of such a map are largely governed by the behaviour of the derivative. Thus one quickly arrives at the simplest model system to study: iterates of a linear map on R n. Specializations and other model systems abound; those most relevant here are rational maps on the Riemann sphere (complex dynamics), attractors for expansive maps, automorphisms of compact groups (algebraic dynamics) and iteration of polynomials in an arithmetic setting. Each of these contributes motivation, ideas, and suggestions for analogies, to the topics in this book. 1. padic dynamics in vivo Before turning to the willful pursuit of padic dynamics, it is important to recognize that padic phenomena cannot be avoided – they arise in the familiar setting of smooth maps on manifolds. The map
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on ..."
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Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on a manifold, and the local properties of such a map are largely governed by the behaviour of the derivative. Thus one quickly arrives at the simplest model system to study: iterates of a linear map on R n. Specializations and other model systems abound; those most relevant here are rational maps on the Riemann sphere (complex dynamics), attractors for expansive maps, automorphisms of compact groups (algebraic dynamics) and iteration of polynomials in an arithmetic setting. Each of these contributes motivation, ideas, and suggestions for analogies to the topics in this book. 1. padic dynamics in vivo Before turning to the willful pursuit of padic dynamics, it is important to recognize that padic phenomena cannot be avoided – they arise in the familiar setting of smooth maps on manifolds. The map