On the Chaotic Properties of Quadratic Maps Over Non-Archimedean Fields

Rational map ax+1/x on the projective line over $\mathbb{Q}_2$

“…On the other hand, the polynomial x p −x p on Z p is proved to be topologically conjugate to the full shift on the symbolic space with p symbols and thus exhibits chaos ( [23]). This chaotic property has also been studied for quadratic polynomials in Q p ( [22,6]) and for some general expanding polynomials in Q p ( [8]).…”

Rational map $ax+1/x$ on the projective line over $\mathbb{Q}\_{p}$

“…A general method was also proposed in [15] to find subshifts of finite type subsystems in a p-adic polynomial dynamical system. We remark that Thiran, Verstegen and Weyers [28] and Dremov, Shabat and Vytnova [9] studied the chaotic behavior of p-adic quadratic polynomial dynamical systems. Woodcock and Smart [29] proved that the so-called p-adic logistic map x p −x p is topologically conjugate to the full shift on the symbolic system with p symbols.…”

On the chaotic behavior of a generalized logistic p-adic dynamical system

“…Even for a quadratic function f (x) = x 2 + c, c ∈ Q p its chaotic behavior is complicated (see [47,4,46]). In [46,16] the Fatou and Julia sets of such a p-adic dynamical system were found. Certain ergodic and mixing properties of monomial and perturbated dynamical systems have been considered in [1], [18].…”