Dynamical Systems of the p-Adic (2, 2)-Rational Functions with Two Fixed Points

Rozikov, U. A.; Sattarov, I. A.

doi:10.1007/s00025-020-01227-y

Cited by 8 publications

(4 citation statements)

References 29 publications

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“…where B = x 2 2 + cx 2 + d and D = 2x 2 + c. In [13] the case x 2 = 0 is studied. Thus in this paper we consider the case x 2 = 0 in (2.2).…”

Section: A Function Conjugate To (11)mentioning

confidence: 99%

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Ergodicity and periodic orbits of $p$-adic $(1,2)$-rational dynamical systems with two fixed points

Sattarov¹,

Aliev²

2023

Preprint

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We consider (1, 2)-rational functions given on the field of p-adic numbers Qp.In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then (1, 2)-rational function is conjugate to a two-parametric (1, 2)-rational function. Depending on these two parameters we determine type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study ergodicity properties of the function on each invariant set. We describe 2-and 3-periodic orbits of the p-adic dynamical systems generated by the two-parametric (1, 2)-rational functions.

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“…where B = x 2 2 + cx 2 + d and D = 2x 2 + c. In [13] the case x 2 = 0 is studied. Thus in this paper we consider the case x 2 = 0 in (2.2).…”

Section: A Function Conjugate To (11)mentioning

confidence: 99%

“…In this paper we study dynamical systems generated by a (1.2-)rational function. Our investigations based on methods of [1], [3], [13]- [17]. For motivations of the study see [2], [4]- [6], [10]- [12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity and periodic orbits of $p$-adic $(1,2)$-rational dynamical systems with two fixed points

Sattarov¹,

Aliev²

2023

Preprint

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“…The proof of this lemma follows easily from the fact that f is an isometry. In [17], [26], [28]- [30] the dynamics of various rational functions were considered. In these papers, we have shown that dynamics is an isometry on invariant spheres.…”

Section: Dynamical System Of Isometry On the Invariant Spherementioning

confidence: 99%

Group structure of the $p$-adic ball and dynamical system of isometry on a sphere

Sattarov¹

2022

Preprint

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In this paper the group structure of the p-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations ⊕ and ⊙ on a ball and sphere respectively, and prove that this sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in Zp is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere, and show that the trajectory of any initial point that is not a fixed point isn't convergent. We study ergodicity of this p-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For p ≥ 3 we prove that the dynamical systems are not ergodic. But for p = 2 under some conditions the dynamical system may be ergodic.

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“…In this paper we continue our study of p-adic dynamical systems generated by rational functions (see [1]- [10] and references therein for motivations and history of p-adic dynamical systems).…”

Section: Introductionmentioning

confidence: 99%