## Abstract

We study L q -spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we disprove a theorem of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the L q -spectrum. As a further application we provide examples of self-affine measures whose L q -spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the L q -spectra, which in certain cases yield sharp results.

## Highlights

Introduction and summary of resultsThe Lq-spectrum is an important concept in multifractal analysis and quantifies global fluctuations in a given measure

We study Lq-spectra of planar self-affine measures generated by diagonal matrices

As a further application we provide examples of self-affine measures whose Lq-spectra exhibit new types of phase transitions

## Summary

The Lq-spectrum is an important concept in multifractal analysis and quantifies global fluctuations in a given measure. The paper [3] was mainly concerned with dimensions of self-affine sets, but towards the end it states a closed form expression for the generalised q-dimensions (these are a normalised version of the Lq-spectra) in a natural generic setting [3, theorem 4.1]. The proof of this result was just sketched and when the result appeared later in Miao’s thesis [8, theorem 3.11] the full proof was only given for 0 < q < 1 and the formula only conjectured to hold for q > 1. A key technical tool is the following growth result for split binomial sums: if one considers the binomial expansion of (1 + x)k, where x > 1 is fixed, and splits the sum in half, the ratio of the two halves grows exponentially in k, see theorem 2.1

## Full Text

### Topics from this Paper

- New Types Of Phase Transitions
- Self-affine Measures
- Spectra Of Measures
- Closed Form Expression
- Diagonal Matrices + Show 5 more

Create a personalized feed of these topics

Get Started#### Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call### Similar Papers

- Physical Review D
- Dec 26, 2012

- Geophysical Research Letters
- Dec 21, 2012

- Physics Letters A
- Mar 21, 1977

- Journal of High Energy Physics, Gravitation and Cosmology
- Jan 1, 2019

- Transactions of the American Mathematical Society
- Jun 24, 2015

- Biochimica et Biophysica Acta (BBA) - Biomembranes
- May 1, 2004

- Biochimica et Biophysica Acta (BBA) - Biomembranes
- May 1, 2004

- Phase Transitions
- Apr 1, 1991

- Journal of Physics C: Solid State Physics
- Oct 10, 1980

- Journal of the American Chemical Society
- Sep 29, 2007

- Journal of Functional Analysis
- Feb 1, 2011

- Solid State Communications
- Mar 15, 1975

- Physical Review Letters
- Feb 19, 1979

- Nature
- Aug 31, 2022

### Nonlinearity

- Nonlinearity
- Nov 17, 2023

- Nonlinearity
- Nov 17, 2023

- Nonlinearity
- Nov 16, 2023

- Nonlinearity
- Nov 15, 2023

- Nonlinearity
- Nov 13, 2023

- Nonlinearity
- Nov 13, 2023

- Nonlinearity
- Nov 10, 2023

- Nonlinearity
- Nov 10, 2023

- Nonlinearity
- Nov 8, 2023

- Nonlinearity
- Nov 6, 2023