On dynamical systems and phase transitions for Q+1Q+1-state PP-adic Potts model on the Cayley tree

In the present paper, we introduce a new kind of $p$-adic measures for $q+1$-state Potts model, called {\it $p$-adic quasi Gibbs measure}. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define one dimensional fractional $p$-adic dynamical system. In ferromagnetic case, we establish that if $q$ is divisible by $p$, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If $q$ is not divisible by $p$, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.Comment: 29 pages, 1 figur

Renormalization method in pp-adic λ\lambda-model on the Cayley tree

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic \l-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter \r\in\bq_p, for the \l-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.Comment: 18 page

On multiparameter Weighted ergodic theorem for Noncommutative L_{p}-spaces

In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von Neumann algebra \M with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight \{a(\kb)\}_{\kb\in\bn^d} and every x\in L_{p}(\M), ($p>1$) the averages A_{\Nb}(x)=\frac{1}{|\Nb|}\sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x). converge bilaterally almost uniformly in L_{p}(\M).Comment: 8 pages. submitte

On infinite dimensional Volterra type operators

In this paper we study Volterra type operators on infinite dimensional simplex. It is provided a sufficient condition for Volterra type operators to be bijective. Furthermore it is shoved that the condition is not necessary.Comment: 10 page