TY - JOUR

T1 - On the 'Mandelbrot set' for pairs of linear maps

T2 - Asymptotic self-similarity

AU - Solomyak, Boris

PY - 2005/9/1

Y1 - 2005/9/1

N2 - We continue the investigation of iterated function systems (IFS) {λz, λz + 1} in the complex plane, depending on a parameter λ in the open unit disc. Let Aλ be the attractor, and let denote the connectedness locus, that is, the set of λ for which Aλ is connected. We show that the set is locally asymptotically self-similar and asymptotically similar to the attractor of the IFS {λz - 1, λz, λz + 1}, for certain 'landmark' points on the boundary of . We also study the parameters for which the attractor A λ is tree-like, and prove that there are uncountable many such λ, answering a question of Bandt.

AB - We continue the investigation of iterated function systems (IFS) {λz, λz + 1} in the complex plane, depending on a parameter λ in the open unit disc. Let Aλ be the attractor, and let denote the connectedness locus, that is, the set of λ for which Aλ is connected. We show that the set is locally asymptotically self-similar and asymptotically similar to the attractor of the IFS {λz - 1, λz, λz + 1}, for certain 'landmark' points on the boundary of . We also study the parameters for which the attractor A λ is tree-like, and prove that there are uncountable many such λ, answering a question of Bandt.

UR - http://www.scopus.com/inward/record.url?scp=23444434931&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/18/5/003

DO - 10.1088/0951-7715/18/5/003

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AN - SCOPUS:23444434931

SN - 0951-7715

VL - 18

SP - 1927

EP - 1943

JO - Nonlinearity

JF - Nonlinearity

IS - 5

ER -