Cantor-like sets

Cantor strange repeller

You see in Fig.1 that for c < -2 interval (BC) is mapped outside invariant interval I_c and all points go eventually to infinity. Two intervals [AB] and [BC] are mapped onto I_c. So similar to the tent map Cantor strange repeller with zero measure appears in quadratic maps.

Cantor strange repeller in regular dynamics window

We meet complicated Cantor-like structures for c = -1.7542 corresponding to period-3 window of regular dynamics. For almost all x in interval I_c points are attracted to period-3 orbits (these points lie in black circles). All the rest points (after cutting these circles) make Cantor strange repeller with zero measure. It includes unstable periodic orbits and chaotic continuum.

The basic dichotomy for real quadratic maps

For almost every c in [-2, 1/4], the quadratic map f_c: x → x² + c is either regular or stochastic [1]
For quadratic maps it is proven that the set of c values for which attractor is chaotic has positive Lebesque measure and attracting periodic orbits are dense in the set. I.e. between any two chaotic parameter values there is always a periodic interval.

"Fat" Cantor sets

We will get a general Cantor set if in the "1/3 cutting" process we cut the central 1/3 piece, then i.g. 1/9, then 1/27, etc. Resulting set is topologically equivalent to the standard Cantor set, but as since holes decrease in size very fast therefore the "fat" Cantor set has positive Lebesque measure and fractal dimension 1 .
In the real interval -2 < c <1/4 , regions with chaotic dynamics have nonzero Lebesgue measure and make a "fat" Cantor set. You can see below that regular dynamics regions (black M-midgets) are dense on real axis.

[1] Mikhail Lyubich The Quadratic Family as a Qualitatively Solvable Model of Chaos Notices of the AMS, 47, 1042-1052 (2000)

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updated 8 Nov 2006