Exponential separation of self-conformal systems
Abstract
On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is Tf=∇log|Df|. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are 𝒞2-open. On the line they are also dense, so both separation conditions are 𝒞2-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are 𝒞2-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for 𝒞2-generic planar systems with injective generators.
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BibTeX
@misc{kaenmaki2026exponential,
title = {{Exponential separation of self-conformal systems}},
author = {Antti Käenmäki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23743},
eprint = {2609.23743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23743},
}
RIS
TY - UNPB
TI - Exponential separation of self-conformal systems
AU - Antti Käenmäki
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23743
UR - https://arxiv.org/abs/2609.23743
AB - On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is T_f=∇log|Df|. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are 𝒞^2-open. On the line they are also dense, so both separation conditions are 𝒞^2-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are 𝒞^2-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for 𝒞^2-generic planar systems with injective generators.
ER -