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A precise calculation of the Feigenbaum constants. (English) Zbl 0744.58053

Let \(f_{\mu,z}(x)=1-\mu| x|^ z\). If for a particular parameter value there exists a stable \(p\)-cycle, then as \(\mu\) is increased the period will be observed to double, so that a stable \(2p\)- cycle appears. Denote the critical \(\mu\) value at which the \(2^ j\) cycle first appears by \(\mu_ j\) and let \(d_ j\) be the value of the nearest cycle element to 0 in \(2^ j\) cycle. The author calculates to high precision the Feigenbaum constants \[ \delta_ z:=\lim_{j\to \infty}(\mu_ j-\mu_{j-1})/(\mu_{j+1}-\mu_ j)\hbox{ and } \lim_{j\to \infty}d_ j/d_{j+1} \] associated with period doubling bifurcations for maps \(f_{\mu,z}\) for integers \(2\leq z\leq 12\). Multiple-precision floating-point techniques are used to find a solution of Feigenbaum’s functional equation, and hence the constants.
Reviewer: M.C.Zdun (Kraków)

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37B99 Topological dynamics
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
11Y60 Evaluation of number-theoretic constants
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