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p-adic dynamics. (English) Zbl 0672.58019

The quadratic map \(x\to f(x)=x^ 2+a\) is studied over p-adic numbers. It is shown that p-adic dynamics exhibits many similarities with real or complex dynamics. In particular, the notion of attractive or indifferent cycles, quasiperiodicity and chaos are still applicable. On the other hand in the case of p-adic dynamics chaos does not emerge through a cascade of period-doubling bifurcations. The procedure for finding all attractive cycles is given.
The quadratic map is then studied in some detail near an indifferent fixed point by showing that, under certain conditions, it is topologically conjugated to a linear map. The analysis here is much simpler than in the complex case. It is proven that for \(| a|_ p>1\) most points of \(Q_ p\) eventually end up at infinity under iterations of f; points with a bounded orbit exist only if \(a=-\gamma^ 2\). They belong to the Cantor set \(\Lambda\) on which f takes a form of shift map. Due to the simplicity of this map it can be completely understood. In this way it is proven that the p-adic quadratic map with \(| a|_ p>1\) is chaotic on the Cantor set \(\Lambda\).
Reviewer: P.Maslanka

MSC:

37B99 Topological dynamics
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
43A99 Abstract harmonic analysis
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