Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS a...

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Bibliographic Details
Published in:Nonlinearity 2022-01, Vol.35 (1), p.245-277
Main Authors: Dániel Prokaj, R, Simon, Károly
Format: Article
Language:English
Subjects:
graph directed iterated function system
Hausdorff dimension
piecewise linear dynamics
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Summary:In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ac355e