## Abstract

We compute the L^{q}-spectrum of self-similar measures defined by an iterated function system of the form S_{i} (x) = (x + i)/2, i = 0, 1, …, m, m ≥ 2. For an iterated function system of the form S_{i} (x) = (x + (N − 1)i)/N, i = 0, 1, …, N, N ≥ 3, the L^{q}-spectrum of a corresponding self-similar measure was computed by Lau and Ngai [Indiana Univ. Math. J. 49 (2000), 925–972] for q ≥ 0 and by Feng, Lau and Wang [Asian J. Math. 9 (2005), 473–488] for the case N = 3 and q < 0. The method used by Lau and Ngai fails if the contraction ratio is 1/2. We use some techniques by Feng, Lau and Wang to express the L^{q}-spectrum of µ as a limit of matrix products. By defining a sub-multiplicative sequence and using its properties, we obtain a formula for the L^{q}-spectrum, q ∈ R, under the assumption that some limit function r(q) exists when q < 0. We study systems for which the limit defining r(q) exists.

Original language | English |
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Pages (from-to) | 867-892 |

Number of pages | 26 |

Journal | Asian Journal of Mathematics |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - 2023 |

## Keywords

- L-spectrum
- iterated function systems with overlaps
- multifractal formalism
- self-similar measure
- sub-multiplicative sequence

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