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

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

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.
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.

[1]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315

[2]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857

[3]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977

[4]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

[5]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[6]

Luis Barreira. Dimension theory of flows: A survey. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

[7]

Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025

[8]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[9]

Mihaela Roxana Nicolai, Dan Tiba. Implicit functions and parametrizations in dimension three: Generalized solutions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2701-2710. doi: 10.3934/dcds.2015.35.2701

[10]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135

[11]

Genghua Li, Shengjie Li, Manxue You. Asymptotic analysis of scalarization functions and applications. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022046

[12]

Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011

[13]

Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325

[14]

Yunping Wang, Ercai Chen, Xiaoyao Zhou. Mean dimension theory in symbolic dynamics for finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022050

[15]

Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024

[16]

Bernard Dacorogna. Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 257-263. doi: 10.3934/dcdsb.2001.1.257

[17]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006

[18]

Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859

[19]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks and Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[20]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]