I know there are many chaotic flows with fractal basin of attraction. Also 2D maps with fractal basin of attraction. Is there any piecewise continuous 1-D map with fractal basin of attraction? For example I know the famous Logistic map doesn’t have such thing.
I question your claim that the logistic map does not have a fractal basin of attraction, unless you mean to constrain the parameter to A = 4. For example, with A = 4.5, every iteration of the map causes the middle third of the x values to be "thrown out" of the interval 0 to 1, so the effect is that only the points within a (fractal) Cantor set will stay in this interval; all others will escape. I assume a piecewise continuous map could be constructed with similar properties. Such a fractal basin, of course, is a set of measure zero, as is the resulting attractor.
dear Sajad, the answer is YES, and it is wrong to state that in the logistic map this does not occur. For example, consider the parameter corresponding to an attracting cycle of period three, then consider the third iterate of the map for which there are three attracting fixed points, then they have fractal basins of attraction, and the same occurs for any attracting cycle after the Feigenbaum point.
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