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February  2014, 34(2): 709-732. doi: 10.3934/dcds.2014.34.709

On the derivative of the $\alpha$-Farey-Minkowski function

Sara Munday 1,

1. 

Department of Mathematics, University Walk, Clifton, Bristol BS8 1TW, United Kingdom

Received  February 2013 Revised  April 2013 Published  August 2013

In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0 : = \{x \in [0,1] : \theta_\alpha'(x)=0\}, \Theta_{\infty} : = \{ x \in [0,1] : \theta_\alpha'(x)=\infty \} $ and $\Theta_\sim : = \{ x \in [0,1] : \theta_\alpha'(x)\ does\ not\ exist \} $. The main result is that \[ \dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1, \] where $\sigma_\alpha(\log2)$ denotes the Hausdorff dimension of the level set $\{x\in [0,1]:\Lambda(F_\alpha, x)=\log2\}$ and $\Lambda(F_\alpha, x)$ is the Lyapunov exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\alpha$-Farey systems.
Keywords: multifractal analysis., Singular functions, dimension theory.
Mathematics Subject Classification: Primary: 26A30; Secondary: 37C4.
Citation: Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
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show all references

References:
[1]

Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[2]

Acta Arith., 74 (1996), 311-327.  Google Scholar

[3]

John Wiley & Sons, Ltd., Chichester, 1997.  Google Scholar

[4]

Trans. Amer. Math. Soc., 363 (2011), 313-330. doi: 10.1090/S0002-9947-2010-05326-7.  Google Scholar

[5]

Ergodic Theory Dynam. Systems, 32 (2012), 989-1017. doi: 10.1017/S0143385711000186.  Google Scholar

[6]

J. Reine Angew. Math., 605 (2007), 133-163. doi: 10.1515/CRELLE.2007.029.  Google Scholar

[7]

J. Number Theory, 128 (2008), 2663-2686. doi: 10.1016/j.jnt.2007.12.010.  Google Scholar

[8]

Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52. Google Scholar

[9]

Ph.D Thesis, University of St Andrews, 2011. Google Scholar

[10]

J. Math. Anal. Appl., 253 (2001), 107-125. doi: 10.1006/jmaa.2000.7064.  Google Scholar

[11]

J. Math. Anal. Appl., 329 (2007), 592-602. doi: 10.1016/j.jmaa.2006.06.082.  Google Scholar

[12]

Third edition, Prentice-Hall, New Jersey, 1988. Google Scholar

[13]

Trans. Amer. Math. Soc., 53 (1943), 427-439. doi: 10.1090/S0002-9947-1943-0007929-6.  Google Scholar

[14]

Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001.  Google Scholar

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