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Symbolic dynamics for iterated smooth functions. (English) Zbl 1221.37023

Summary: We consider the discrete dynamical system \(({\mathcal A},T)\), where \({\mathcal A}\) is a class of smooth real functions defined on some interval and \(T:{\mathcal A}\to{\mathcal A}\) is an operator \(T\varphi:=f\circ\varphi\), where \(f\) is a function on the real line. Iteration of smooth maps appears naturally in the study of continuous difference equations and boundary value problems. In our work we develop some techniques of symbolic dynamics for the discrete dynamical system \(({\mathcal A},T)\). We analyze in detail the case in which \(T\) arises form unimodal interval maps.

MSC:

37B10 Symbolic dynamics
39B12 Iteration theory, iterative and composite equations
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
26A18 Iteration of real functions in one variable
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