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

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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

Abstract
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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 and 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.
[2]	J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327.
[3]	K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997.
[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.
[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.
[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.
[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.
[8]	H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52.
[9]	S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011.
[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.
[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.
[12]	H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988.
[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.
[14]	J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001.

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.
[2]	J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327.
[3]	K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997.
[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.
[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.
[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.
[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.
[8]	H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52.
[9]	S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011.
[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.
[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.
[12]	H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988.
[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.
[14]	J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001.

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