On the space of Cantor subsets of R3 Gartside, Paul; Kovan-Bakan, Merve
Topology and its applications,
06/2013, Letnik:
160, Številka:
10
Journal Article
Recenzirano
Odprti dostop
The space of Cantor subsets of R3, denoted C(R3), is a Polish space. We prove this space is path connected and locally path connected. The group of autohomeomorphisms of R3, denoted Aut(R3), acts on ...C(R3) naturally. This action gives us natural invariant classes of Cantor sets and we show that these classes are in the lower levels of the Borel hierarchy, in fact they are open, closed, Fσ or Gδ in C(R3). Moreover, we prove that the classification problem of Cantor sets arising from this action is at least as complicated as the classification of countable linear orders.
In 1994, J. Cobb described a Cantor set in R3 each of whose projections into 2-planes is one-dimensional. A series of works by other authors developing this field followed. We present another very ...simple series of Cantor sets in R3 all of whose projections are connected and one-dimensional. These are self-similar Cantor sets which go back to the work of Louis Antoine, and we celebrate their centenary birthday in 2020–2021.
IFS attractors and Cantor sets Crovisier, Sylvain; Rams, Michał
Topology and its applications,
05/2006, Letnik:
153, Številka:
11
Journal Article
Recenzirano
Odprti dostop
We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set
K in
R
3
such that every ...homeomorphism
f of
R
3
which preserves
K coincides with the identity on
K.
It is shown that, in any dimension
d
≥ 3, there exist diffeomorphisms of compact
d
-manifolds with one-dimensional expanding attractors which are conjugate on these attractors but not conjugate on ...their neighborhoods.
PDF
