Journal Article10.1088/0951-7715/14/6/310
Rigidity of multi-dimensional conformal iterated function systems
TL;DR: For a regular iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space d with d ≥ 3, satisfying the open set condition, the Radon-Nikodym derivative dµ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and µ is the unique probability invariant measure equivalent with m.
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Abstract: This paper starts with an appropriate version of the bounded distortion theorem. We show that for a regular iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space d with d≥3, satisfying the `open set condition', the Radon-Nikodym derivative dµ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and µ is the unique probability invariant measure equivalent with m. Next, within this context we explore the concept of the essential affinity of iterated function systems providing the several necessary and sufficient conditions. We prove the following rigidity result. If d≥3 and h, a topological conjugacy between two not essentially affine systems F and G sends the conformal measure mF to a measure equivalent with the conformal measure mG, then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, in exactly the same way as in Mauldin et al (2001 Compos. Math. to appear) we extend our rigidity result to the case of parabolic systems.
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Citations
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References
Lectures on hyperbolic geometry
Riccardo Benedetti,Carlo Petronio +1 more
- 01 Jan 1992
TL;DR: In this article, the authors provide an exposition of some fundamental results of hyperbolic manifolds, while being as self-contained, complete, detailed and unified as possible, and much space is devoted to the 3D case, based on the representation of three manifolds as glued ideal tetrahedra.
1K
Dimensions and Measures in Infinite Iterated Function Systems
TL;DR: In this article, the Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated.
Rigidity of Conformal Iterated Function Systems
TL;DR: In this article, the authors extend the rigidity of the mixing expanding repellers theorem of D. Sullivan to the case of iterated function systems of countably many holomorphic contractions and show that the Radon-Nikodym derivative dμ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and μ is the unique probability invariant measure equivalent with m.
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On the Uniqueness of the Density for the Invariant Measure in an Infinite Hyperbolic Iterated Function System
TL;DR: In this article, the authors consider a regular infinite hyperbolic iterated function satisfying a property which guarantees that the associated Frobenius-Perron operator ℒ is almost periodic.
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