# 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 → x*^{2} + 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