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2011•arXiv (Cornell University)Open access

A multi-dimensional-time dynamical system

U. A. Rozikov

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Abstract

In this paper we give a concept of multi-dimensional-time dynamical system (MDTDS). Such dynamical system is generated by a finite family of functions $\{f_i\}$. The multi-dimensional-time space is taken as a free group. Using the subgroups of the free group we define periodic orbits of MDTDS and construct such orbits. We study fixed points of MDTDS and show that the set of the fixed points is the intersection of the set of fixed points of each $f_i$. The $ω$-limit set of the MDTDS is defined and some properties of the set is studied. Moreover we construct a MDTDS for income of a deposit from several banks and construct a MDTDS on circle. For these MDTDSs we describe the set of fixed and periodic points. We discuss several open problems.

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In this paper we give a concept of multi-dimensional-time dynamical system (MDTDS). Such dynamical system is generated by a finite family of functions $\{f_i\}$. The multi-dimensional-time space is taken as a free group. Using the subgroups of the free group we define periodic orbits of MDTDS and construct such orbits. We study fixed points of MDTDS and show that the set of the fixed points is the intersection of the set of fixed points of each $f_i$. The $ω$-limit set of the MDTDS is defined and some properties of the set is studied. Moreover we construct a MDTDS for income of a deposit from several banks and construct a MDTDS on circle. For these MDTDSs we describe the set of fixed and periodic points. We discuss several open problems.

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Available abstract

In this paper we give a concept of multi-dimensional-time dynamical system (MDTDS). Such dynamical system is generated by a finite family of functions $\{f_i\}$. The multi-dimensional-time space is taken as a free group. Using the subgroups of the free group we define periodic orbits of MDTDS and construct such orbits. We study fixed points of MDTDS and show that the set of the fixed points is the intersection of the set of fixed points of each $f_i$. The $ω$-limit set of the MDTDS is defined and some properties of the set is studied. Moreover we construct a MDTDS for income of a deposit from several banks and construct a MDTDS on circle. For these MDTDSs we describe the set of fixed and periodic points. We discuss several open problems.

Key concepts: Intersection (aeronautics), Fixed point, Construct (python library), Limit set, Dynamical systems theory, Set (abstract data type), Mathematics, Periodic orbits

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