# On chaotic behavior of the $P$-adic generalized Ising mapping and its application

## TL;DR

This paper investigates the chaotic dynamics of the $p$-adic generalized Ising mapping related to the $p$-adic Ising-Vannemenus model on a Cayley tree, revealing fixed points, attractors, and chaos, and establishing periodic $p$-adic Gibbs measures.

## Contribution

It demonstrates the chaotic behavior of the $p$-adic Ising mapping and links it to the existence of periodic $p$-adic Gibbs measures, providing new insights into $p$-adic dynamical systems.

## Key findings

- Existence of fixed points and attractors for the $p$-adic Ising mapping
- Mapping is topologically conjugate to symbolic shift, indicating chaos
- Existence of periodic $p$-adic Gibbs measures

## Abstract

In the present paper, by conducting research on the dynamics of the $p$-adic generalized Ising mapping corresponding to renormalization group associated with the $p$-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic $p$-adic Gibbs measures for the $p$-adic Ising-Vannemenus model.