Dimension theory of iterated function systems

By De-Jun Feng, Huyi Hu. 2009

202 citations recorded by the source corpus.

Let {Si} i = 1𝓁 be an iterated function system (IFS) on ℝd with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, 𝓁}. We define the projection entropy function hπ 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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/507319b2eed581b9a6dc6e4b41c7f0148f3034db

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In Mathematical Physics

Citations in the corpus

On self-similar sets with overlaps and inverse theorems for entropy (2012, background): “We remark that lim inf 1 n H ( θ, D n ) ≥ dim θ for any probability measure θ , and the limit exists and is equal to dim θ when θ is exact dimensional, which is the case for self-similar measures [9].”
Dimension of invariant measures for affine iterated function systems (2019, methodology): “In [28], the partitions were made on the one-sided shift space due to the simple structure of Oseledets splitting subspaces.”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “It is known (see [16]) that for any λ, the limit”
Recent progress on Bernoulli convolutions (2016, background): “If μ is exact dimensional (which is the case for Bernoulli convolutions [21]), then the limit (7) exists and is equal to dimμ.”
On the dimension of Bernoulli convolutions for all transcendental parameters (2018, background): “Feng and Hu [8] proved that Bernoulli convolutions are exact dimensional, that is to say, there is number, which we denote by dim νλ, such that”
Absolute continuity of self-similar measures, their projections and convolutions (2014, background): “We start by recalling that Hausdorff and entropy dimensions coincide for self-similar measures (see [8]) (in fact, the proofs of all results we have stated for the Hausdorff dimension of self-similar measures actually apply to entropy dimension).”