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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
References:
[1]

L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[2]

J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327.  Google Scholar

[3]

K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997.  Google Scholar

[4]

J. Jaerisch and M. Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc., 363 (2011), 313-330. doi: 10.1090/S0002-9947-2010-05326-7.  Google Scholar

[5]

M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017. doi: 10.1017/S0143385711000186.  Google Scholar

[6]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163. doi: 10.1515/CRELLE.2007.029.  Google Scholar

[7]

M. Kesseböhmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski's question mark function, J. Number Theory, 128 (2008), 2663-2686. doi: 10.1016/j.jnt.2007.12.010.  Google Scholar

[8]

H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52. Google Scholar

[9]

S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011. Google Scholar

[10]

J. Paradís and P. Viader, The derivative of Minkowski's $?(x)$ function, J. Math. Anal. Appl., 253 (2001), 107-125. doi: 10.1006/jmaa.2000.7064.  Google Scholar

[11]

J. Paradís, P. Viader and L. Bibiloni, Riesz-Nágy singular functions revisited, J. Math. Anal. Appl., 329 (2007), 592-602. doi: 10.1016/j.jmaa.2006.06.082.  Google Scholar

[12]

H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988. Google Scholar

[13]

R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439. doi: 10.1090/S0002-9947-1943-0007929-6.  Google Scholar

[14]

J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001.  Google Scholar

show all references

References:
[1]

L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[2]

J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327.  Google Scholar

[3]

K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997.  Google Scholar

[4]

J. Jaerisch and M. Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc., 363 (2011), 313-330. doi: 10.1090/S0002-9947-2010-05326-7.  Google Scholar

[5]

M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017. doi: 10.1017/S0143385711000186.  Google Scholar

[6]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163. doi: 10.1515/CRELLE.2007.029.  Google Scholar

[7]

M. Kesseböhmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski's question mark function, J. Number Theory, 128 (2008), 2663-2686. doi: 10.1016/j.jnt.2007.12.010.  Google Scholar

[8]

H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52. Google Scholar

[9]

S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011. Google Scholar

[10]

J. Paradís and P. Viader, The derivative of Minkowski's $?(x)$ function, J. Math. Anal. Appl., 253 (2001), 107-125. doi: 10.1006/jmaa.2000.7064.  Google Scholar

[11]

J. Paradís, P. Viader and L. Bibiloni, Riesz-Nágy singular functions revisited, J. Math. Anal. Appl., 329 (2007), 592-602. doi: 10.1016/j.jmaa.2006.06.082.  Google Scholar

[12]

H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988. Google Scholar

[13]

R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439. doi: 10.1090/S0002-9947-1943-0007929-6.  Google Scholar

[14]

J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001.  Google Scholar

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