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An example of a genuinely discontinuous generically chaotic transformation of the interval. (English) Zbl 0848.54028

A (not necessarily continuous) transformation \(f\) from the unit interval \(I\) into itself is called generically chaotic if the set of all \((x,y) \in I^2\) for which \(\liminf_{n \to \infty} |f^n (x) - f^n (y) |= 0\) and \(\limsup_{n \to \infty} |f^n (x) - f^n (y) |> 0\) is residual in \(I^2\), i.e., if its complement is of the first category. In the paper a class of such transformations is found. Then it is shown that the Gauss map defined by \(f(0) = 0\) and \(f(x) = 1/x \pmod 1\), \(0 < x \leq 1\), belongs to the class and so is generically chaotic.

MSC:

54H20 Topological dynamics (MSC2010)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A18 Iteration of real functions in one variable
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