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Dimension theory of iterated function systems
Author(s) -
Feng DeJun,
Hu Huyi
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20276
Subject(s) - mathematics , iterated function system , ergodic theory , hausdorff dimension , conformal map , attractor , lyapunov exponent , minkowski–bouligand dimension , pure mathematics , hausdorff space , entropy (arrow of time) , invariant (physics) , mathematical analysis , discrete mathematics , fractal , fractal dimension , mathematical physics , physics , nonlinear system , quantum mechanics
Let { S i } i = 1 be an iterated function system (IFS) on ℝ d with attractor K . Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, }. We define the projection entropy function h π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K . We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.