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November  2003, 9(6): 1411-1422. doi: 10.3934/dcds.2003.9.1411

Noncommutative dynamical systems with two generators and their applications in analysis

Boris Paneah 1,

1. 

Department of Mathematics, Technion, Haifa, 32000

Received  May 2002 Revised  February 2003 Published  September 2003

In this paper, some new dynamical systems which are determined by a semigroup $\Phi$ of maps in a closed interval $I$ are studied.The main peculiarity of these systems is that $\Phi$ is generated by two noncommuting maps. Introducing certain closed subsets $\mathcal T_1$ and $\mathcal T_2$ in $I$ makes it possible to determine some specific orbits corresponding to $\Phi$ and some specific attractors in $I$. These orbits play a crucial role in solving a wide variety problems in such diverse fields of analysis as functional and functional-integral equations, integral geometry, boundary problems for hyperbolic partial differential equations of higher $(>2)$ order. In the first part of this work we describe some conditions which ensure the existence of attractors in question of a special structure. In the second part several new problems in the above-mentioned fields of analysis are formulated, and we trace how the above dynamic approach works in solving this problems.
Keywords: attractor, integral geometry., functional equation, Dynamical system, orbit.
Mathematics Subject Classification: 37N99, 39B22, 35L3.
Citation: Boris Paneah. Noncommutative dynamical systems with two generators and their applications in analysis. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1411-1422. doi: 10.3934/dcds.2003.9.1411
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