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Farrukh Mukhamedov

Publications and source records attributed to Farrukh Mukhamedov.

137 records · Page 8Linked to original sources
arXiv2006

On Rational $P$-Adic Dyanamical Systems

In the paper we investigate the behavior of trajectory of rational $p$-adic dynamical system in complex $p$-adic filed $\C_p$. It is studied Siegel disks and attractors of such dynamical systems. We show that Siegel disks may either coincide or disjoin for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. It is proved that such kind of dynamical system is not ergodic on a unit sphere with respect to the Haar measure.

math.DS↗
arXiv2006

On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $ł$-Model on the Cayley Tree

We consider a nearest-neighbor $p$-adic $ł$-model with spin values $\pm 1$ on a Cayley tree of order $k\geq 1$. We prove for the model there is no phase transition and as well as the unique $p$-adic Gibbs measure is bounded if and only if $p\geq 3$. If $p=2$ then we find a condition which guarantees nonexistence of a phase transition. Besides, the results are applied to the $p$-adic Ising model and we show that for the model there is a unique $p$-adic Gibbs measure.

math-ph↗
arXiv2006

On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree

In the paper we considere three state $p$-adic Potts model with competing interactions on a Cayley tree of order two. We reduce a problem of describing of the $p$-adic Gibbs measures to the solution of certain recursive equation, and using it we will prove that a phase transition occurs if and only if $p=3$ for any value (non zero) of interactions. As well, we completely solve the uniqueness problem for the considered model in a $p$-adic context. Namely, if $p\neq 3$ then there is only a unique Gibbs measure the model.

math-ph↗
arXiv2005

On Mixing and Completely Mixing Properties of Positive $L^1$-Contractions of Finite Von Neumann Algebras

Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be a positive contraction. Assume that for $z\in L^1(X,{\cf},\m)$ the sequence $(T^nz)$ converges weakly in $L^1(X,{\cf},\m)$, then either $\lim\limits_{n\to\infty}\|T^nz\|=0$ or there exists a positive function $h\in L^1(X,{\cf},\m)$, $h\neq 0$ such that $Th=h$. In the paper we prove an extension of this result in finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative $L^1$-space has no non zero positive invariant element, then its mixing property implies completely mixing property one.

math.OA↗
arXiv2005

A few remarks on mixing properties of $C^*$-dynamical systems

We consider strictly ergodic and strictly weak mixing $C^*$-dynamical systems. We prove that the system is strictly weak mixing if and only if its tensor product is strictly ergodic, moreover strictly weak mixing too. We also investigate some other mixing properties of the system.

math.OA↗
arXiv2005

N Infinite Dimensional Quadratic Volterra Operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.

math.FA↗
arXiv2005

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We completely describe the set of all periodic Gibbs measures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III$_λ$ then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too. \\[10mm] {\bf Keywords:} Cayley tree, Ising model, competing interactions, Gibbs measure, GNS-construction, Hamiltonian, von Neumann algebra.

math-ph↗