Limit cycles for successive projections onto hyperplanes in Rn

Random iteration and projection method

“…(Note that for N = 1, a single map, this is an elementary exercise). Related results together with a panorama of examples can be found in [3].…”

“…We finish this Section by giving motivating examples where the projection method invites a nonexpansive IFSs viewpoint (cf. [3]). Specifically Example 4 suggests the way we should interpret the result of the projection algorithm in general.…”

Contractive Iterated Function Systems Enriched with Nonexpansive Maps

“…Such IFSs were considered in the literature, for example in [8,9,12,17] or [20]. In fact, item (3) of the above Remark for nonexpansive maps was noted in [12] (see also [1]).…”

“…For example, for k ∈ N, take f k (x) = 1 2 x+k, x ∈ R, and consider F := {f k , k ∈ N}. Then each element of F is a Matkowski contraction (as Lip(f k ) = 1 2 ), but F is not bounded. Observe that here each F-invariant set is necessarily unbounded, so F does not have any Hutchinson attractor.…”

“…The existence of the semiattractors of N , M 1 , M 2 and H is guaranteed by item 2. of Theorem 2.19 as the IFS F is contained in each of those IFSs. We first prove item (1). In view of Theorem 2.19, we only have to prove that…”