Browsing by Author "Mohd Ali Khameini bin Ahmad"

In this thesis, we did a research on cubic equations over p-adic field Qp where p is a prime number. In the first part, we studied the general cubic equations over Qp where p >3. We described the location of roots of the general cubic equation without knowing their exact values. Furthermore, we gave the solvability criterion and the number of roots of the general cubic equation over Zp; Zp \ Zp; Qp nZp and Qp. In the second part, we discussed the depressed cubic equations over p-adic field. We provided the exact location of roots of the depressed cubic equation over p-adic field for p > 3. For p = 3, we provided the solvability criterion and the number of roots of the depressed cubic equation over Z3; Z3 \ Z3; Q3 \ Z3 and Q3. While, for p D 2, we provided the solvability criterion of the depressed cubic equation over Z2; Z2 \ Z2; Q2 \ Z2 and Q2. In the last part, we presented the application of the cubic equations in the p-adic Potts model. We described all translation invariant p-adic Gibbs measures for the Potts model on Cayley tree of order three by giving all possible forms of boundary functions. We showed that the boundary functions can be described in terms of roots of some general cubic equations over some domains of p-adic field.

This thesis is devoted to the study of the q-state Potts model over Q_p on Cayley trees. Specifically, we investigate the p-adic Gibbs measures of the Potts model on the Cayley trees of orders 3 and 4 and their related p-adic dynamical systems. In the first part, we describe the existence of the translation-invariant p-adic Gibbs measures of the Potts model on the Cayley tree of order 4. The existence of translation-invariant p-adic Gibbs measures is equivalent to the existence of fixed points of a rational map called Potts–Bethe mapping. The Potts–Bethe mapping is derived from the recurrent equation of a Q_p-valued function in the construction of the p-adic Gibbs measures of the Potts model on Cayley trees. In order to describe the existence of these translation-invariant p-adic Gibbs measures, we find the solutions of some quartic equation in some domains Ep c Qp. In general, we also provide some solvability conditions for the depressed quartic equation over Q_p. In the second part, we study the dynamics of the Potts–Bethe mapping of degrees 3 and 4. First, we describe the Potts–Bethe mapping having good reduction. For a Potts–Bethe mapping with good reduction, the projective P1(Q_p) can be decomposed into minimal components and their attracting basins. However, the Potts–Bethe mapping associated to the Potts model on the Cayley trees of orders 3 and 4 have bad reduction. For many prime numbers p, such Potts–Bethe mappings are chaotic. In fact, for these primesp, we prove that restricted to their Julia sets, the Potts–Bethe mappings are topologically conjugate to the full shift dynamics. For other primes p, restricted to their Julia sets, the Potts–Bethe mappings are not topologically conjugate to the full shift dynamics. The chaotic property of the Potts-Bethe mapping implies the vastness of the set of the p-adic Gibbs measures, and hence implies the phase transition. As application, for many prime numbers p, the Potts models over Q_p on the Cayley trees of orders 3 and 4 have phase transition. We also remark the statement that phase transition implies chaos is not true.