We consider [0,1]^N, the unit cube of R^N, N geq 1. Let S = {S_1,...,S_M} be a finite set of contraction maps from X to itself. A non-empty subset E of X is an attractor (or an invariant set) for the iterated function system (IFS) S if E = cup_{i=1}^M S_i(E). We construct, for each s ∈ ]0, N ], a nowhere dense perfect set E contained in [0, 1]^N , with Hausdorff dimension s, which is not an attractor for any iterated function system composed of weak contractions from [0, 1]^N to itself.
Non self-similar sets in [0,1]^N of arbitrary dimension
D'ANIELLO, Emma
2017
Abstract
We consider [0,1]^N, the unit cube of R^N, N geq 1. Let S = {S_1,...,S_M} be a finite set of contraction maps from X to itself. A non-empty subset E of X is an attractor (or an invariant set) for the iterated function system (IFS) S if E = cup_{i=1}^M S_i(E). We construct, for each s ∈ ]0, N ], a nowhere dense perfect set E contained in [0, 1]^N , with Hausdorff dimension s, which is not an attractor for any iterated function system composed of weak contractions from [0, 1]^N to itself.File in questo prodotto:
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