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**When the weak separation condition implies the generalized finite type condition.**
*(English)*
Zbl 1466.28010

In this well-written paper, the authors prove that an iterated function system (IFS) of similarities on \(\mathbb R\) that satisfies the weak separation condition and has an interval \([0,1]\) as its self-similar set is of generalized finite type. But it is unknown if the assumption that the self-similar set is an interval is necessary. In addition, the IFS satisfies the convex generalized finite type condition if and only if it satisfies the finite neighbour condition.

Reviewer: Wen-hui Ai (Changsha)

### MSC:

28A80 | Fractals |

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\textit{K. E. Hare} et al., Proc. Am. Math. Soc. 149, No. 4, 1555--1568 (2021; Zbl 1466.28010)

### References:

[1] | Deng, Qi-Rong; Lau, Ka-Sing; Ngai, Sze-Man, Separation conditions for iterated function systems with overlaps. Fractal geometry and dynamical systems in pure and applied mathematics. I. Fractals in pure mathematics, Contemp. Math. 600, 1-20 (2013), Amer. Math. Soc., Providence, RI · Zbl 1321.28014 |

[2] | Falconer, Kenneth, Techniques in fractal geometry, xviii+256 pp. (1997), John Wiley & Sons, Ltd., Chichester · Zbl 0869.28003 |

[3] | Feng, De-Jun, Smoothness of the \(L^q\)-spectrum of self-similar measures with overlaps, J. London Math. Soc. (2), 68, 1, 102-118 (2003) · Zbl 1041.28004 |

[4] | Feng, De-Jun, The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers, Adv. Math., 195, 1, 24-101 (2005) · Zbl 1078.11062 |

[5] | Feng, De-Jun, Lyapunov exponents for products of matrices and multifractal analysis. II. General matrices, Israel J. Math., 170, 355-394 (2009) · Zbl 1181.37073 |

[6] | Feng, De-Jun, On the topology of polynomials with bounded integer coefficients, J. Eur. Math. Soc. (JEMS), 18, 1, 181-193 (2016) · Zbl 1332.11066 |

[7] | Feng, De-Jun; Lau, Ka-Sing, Multifractal formalism for self-similar measures with weak separation condition, J. Math. Pures Appl. (9), 92, 4, 407-428 (2009) · Zbl 1184.28009 |

[8] | Fraser, J. M.; Henderson, A. M.; Olson, E. J.; Robinson, J. C., On the Assouad dimension of self-similar sets with overlaps, Adv. Math., 273, 188-214 (2015) · Zbl 1317.28014 |

[9] | Hare, Kathryn E.; Hare, Kevin G.; Simms, Grant, Local dimensions of measures of finite type III-measures that are not equicontractive, J. Math. Anal. Appl., 458, 2, 1653-1677 (2018) · Zbl 1376.28008 |

[10] | HR Kathryn E. Hare and Alex Rutar, Local dimensions of self-similar measures satisfying the finite neighbour condition, preprint. 2101.07400. |

[11] | Hu, Tian-You; Lau, Ka-Sing, Multifractal structure of convolution of the Cantor measure, Adv. in Appl. Math., 27, 1, 1-16 (2001) · Zbl 0991.28008 |

[12] | Lau, Ka-Sing; Ngai, Sze-Man, Multifractal measures and a weak separation condition, Adv. Math., 141, 1, 45-96 (1999) · Zbl 0929.28007 |

[13] | Lau, Ka-Sing; Ngai, Sze-Man, A generalized finite type condition for iterated function systems, Adv. Math., 208, 2, 647-671 (2007) · Zbl 1113.28006 |

[14] | Lau, Ka-Sing; Ngai, Sze-Man; Rao, Hui, Iterated function systems with overlaps and self-similar measures, J. London Math. Soc. (2), 63, 1, 99-116 (2001) · Zbl 1019.28005 |

[15] | Ngai, Sze-Man; Wang, Yang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63, 3, 655-672 (2001) · Zbl 1013.28008 |

[16] | Nhu Nguyen, Iterated function systems of finite type and the weak separation property, Proc. Amer. Math. Soc., 130, 2, 483-487 (2002) · Zbl 0986.28010 |

[17] | Ru Alex Rutar, Geometric and combinatorial properties of self-similar multifractal measures, preprint.2008.00197. |

[18] | Shmerkin, Pablo, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal., 24, 3, 946-958 (2014) · Zbl 1305.28012 |

[19] | Va P\'eter P. Varj\'u, Recent progress on Bernoulli convolutions, Apollo-University of Cambridge Repository (Jan. 2018), https://www.repository.cam.ac.uk/handle/1810/270472. 1810.08905. |

[20] | Zerner, Martin P. W., Weak separation properties for self-similar sets, Proc. Amer. Math. Soc., 124, 11, 3529-3539 (1996) · Zbl 0874.54025 |

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