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Geometry of Measures in Random Systems with Complete Connections

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Abstract

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and place-dependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps \(\mathcal S\), and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system \(\mathcal S\).

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Authors and Affiliations

  1. Institute of Mathematics of the Romanian Academy, Calea Grivitei 21, P.O Box 1-764, 014700, Bucharest, Romania

    Eugen Mihailescu

  2. Department of Mathematics, University of North Texas, Denton, TX, 76203-1430, USA

    Mariusz Urbański

Authors
  1. Eugen Mihailescu
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  2. Mariusz Urbański
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Correspondence to Eugen Mihailescu.

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Research of the first author supported by Project PN-III-P4-ID-PCE-2020-2693 “Dimensions and invariance in dynamical systems” from Ministry of Research and Innovation, CNCS/CCCDI-UEFISCDI Romania. Research of the second author supported by the NSF Grant DMS 0400481.

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Mihailescu, E., Urbański, M. Geometry of Measures in Random Systems with Complete Connections. J Geom Anal 32, 162 (2022). https://doi.org/10.1007/s12220-022-00899-y

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