Abstract
Before we study parametrized families of maps, we want to analyze individual maps. We are interested in the possible behavior of the successive images of an initial point x0 on the interval [-1,1] for a fixed map f. For this we first outline a graphical method for determining the iterates \({\rm x}_{{\rm n}} = {\rm f}^{{\rm n}}({\rm x}_0)\). Here, we define \({\rm f}_{{\rm n}}({\rm x}_0) = {\rm f}({\rm f}^{{\rm n-1}}({\rm x}_0))\). The following figure 1.5 shows how this is done through the rule: Go from x0 to the graph of the function, from the graph to the diagaonal, from the diagonal to the graph,….
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© 2009 Birkhäuser Boston
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Collet, P., Eckmann, JP. (2009). Typical Behavior for One Map. In: Iterated Maps on the Interval as Dynamical Systems. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4927-2_2
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DOI: https://doi.org/10.1007/978-0-8176-4927-2_2
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Publisher Name: Birkhäuser Boston
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Online ISBN: 978-0-8176-4927-2
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