# American Institute of Mathematical Sciences

April  2017, 10(2): 353-365. doi: 10.3934/dcdss.2017017

## Derivatives of slippery Devil's staircases

 1 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 2 Department of Mathematics, University of Bologna, Bologna, Italy

Received  December 2015 Revised  November 2016 Published  January 2017

In this paper we first give a survey of known results on the derivative of slippery Devil's staircase functions, that is, functions that are singular with respect to the Lebesgue measure and strictly increasing. The best known example of such a function is the Minkowski question-mark function, which was proved to be singular by Salem, in a paper which introduced some other constructions of singular functions. We describe all of these examples. Also we consider various generalisations of the Minkowski question-mark function, such as $α$-Farey-Minkowski functions. These examples all arise from one-dimensional dynamics. A few open questions and suggestions for filling minor gaps in the literature are proposed. Finally, we go back to ordinary Devil's staircases (i.e. non-decreasing singular functions) and discuss work done in that setting with the more general Hölder derivatives, and consider the outlook to extend those results to the strictly increasing situation.

Citation: Jun-Jie Miao, Sara Munday. Derivatives of slippery Devil's staircases. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 353-365. doi: 10.3934/dcdss.2017017
##### References:
 [1] A. Arroyo, Generalised Lüroth expansions and a family of Minkowski question-mark functions, Comptes Rendus Math., 353 (2015), 943-946.  doi: 10.1016/j.crma.2015.08.008. [2] J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arithm., 74 (1996), 311-327. [3] G. Cantor, De la puissance des ansembles parfaits de points, Acta Math., 4 (1884), 381-392.  doi: 10.1007/BF02418423. [4] R. Darst, The Hausdorff dimension of the non-differentiability set of the Cantor function is [log2/log3]2, Proc. Amer. Math. Soc., 119 (1993), 105-108.  doi: 10.2307/2159830. [5] A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17 (1938), 105-151. [6] A. A. Dushistova and N. G. Moshevitin, On the derivative of the Minkowski question mark function?(x), J. Math. Sci., 182 (2012), 463-471.  doi: 10.1007/s10958-012-0750-2. [7] A. A. Dushistova, I. D. Kan and N. G. Moshevitin, Differentiability of Minkowski's question mark function, J. Math. Anal. Appl., 401 (2013), 774-794.  doi: 10.1016/j.jmaa.2012.12.058. [8] K. J. Falconer, One-sided multifractal analysis and points of non-differentiability of devil's staircases, Math. Proc. Cambridge Philos. Soc., 136 (2004), 167-174.  doi: 10.1017/S0305004103006960. [9] K. Falconer, Fractal Geometry, John Wiley, New York, 1990. [10] T. Jordan, M. Kesseböhmer, M. Pollicott and B. O. Stratmann, Sets of non-differentiability for conjugacies between expanding interval maps, Fund. Math., 206 (2009), 161-183.  doi: 10.4064/fm206-0-10. [11] T. Jordan, S. Munday and T. Sahlsten, Pointwise perturbations of countable Markov maps, Preprint, available on Arxiv: http://arxiv.org/abs/1601.06591 [12] M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for α-Farey and α-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017.  doi: 10.1017/S0143385711000186. [13] 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. [14] 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. [15] M. Kesseböhmer and B. O. Stratmann, Hölder-differentiability of Gibbs distribution functions, Math. Proc. Cambridge Philos. Soc., 147 (2009), 489-503.  doi: 10.1017/S0305004109002473. [16] A. Ya Khinchin, Continued Fractions The University of Chicago Press, 1964. [17] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets Cambridge University Press, 2003. doi: 10.1017/CBO9780511543050. [18] H. Minkowski, Geometrie der Zahlen (German) Bibliotheca Mathematica Teubneriana, Band 40 Johnson Reprint Corp. , New York-London, 1968. [19] S. Munday, On the derivative of the α-Farey-Minkowski function, Discrete Contin. Dynam. Systems, 34 (2014), 709-732.  doi: 10.3934/dcds.2014.34.709. [20] 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. [21] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [22] 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. [23] J. F. Sánchez, P. Viader, J. Paradís and M. D. Carrillo, A singular function with a non-zero finite derivative, Nonlinear Anal., 75 (2012), 5010-5014.  doi: 10.1016/j.na.2012.04.015. [24] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820. [25] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758.  doi: 10.1090/S0002-9939-03-06927-2. [26] S. Troscheit, Hölder differentiability of self-conformal devil's staircases, Math. Proc. Cambridge Philos. Soc., 156 (2014), 295-311.  doi: 10.1017/S0305004113000698.

show all references

##### References:
 [1] A. Arroyo, Generalised Lüroth expansions and a family of Minkowski question-mark functions, Comptes Rendus Math., 353 (2015), 943-946.  doi: 10.1016/j.crma.2015.08.008. [2] J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arithm., 74 (1996), 311-327. [3] G. Cantor, De la puissance des ansembles parfaits de points, Acta Math., 4 (1884), 381-392.  doi: 10.1007/BF02418423. [4] R. Darst, The Hausdorff dimension of the non-differentiability set of the Cantor function is [log2/log3]2, Proc. Amer. Math. Soc., 119 (1993), 105-108.  doi: 10.2307/2159830. [5] A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17 (1938), 105-151. [6] A. A. Dushistova and N. G. Moshevitin, On the derivative of the Minkowski question mark function?(x), J. Math. Sci., 182 (2012), 463-471.  doi: 10.1007/s10958-012-0750-2. [7] A. A. Dushistova, I. D. Kan and N. G. Moshevitin, Differentiability of Minkowski's question mark function, J. Math. Anal. Appl., 401 (2013), 774-794.  doi: 10.1016/j.jmaa.2012.12.058. [8] K. J. Falconer, One-sided multifractal analysis and points of non-differentiability of devil's staircases, Math. Proc. Cambridge Philos. Soc., 136 (2004), 167-174.  doi: 10.1017/S0305004103006960. [9] K. Falconer, Fractal Geometry, John Wiley, New York, 1990. [10] T. Jordan, M. Kesseböhmer, M. Pollicott and B. O. Stratmann, Sets of non-differentiability for conjugacies between expanding interval maps, Fund. Math., 206 (2009), 161-183.  doi: 10.4064/fm206-0-10. [11] T. Jordan, S. Munday and T. Sahlsten, Pointwise perturbations of countable Markov maps, Preprint, available on Arxiv: http://arxiv.org/abs/1601.06591 [12] M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for α-Farey and α-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017.  doi: 10.1017/S0143385711000186. [13] 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. [14] 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. [15] M. Kesseböhmer and B. O. Stratmann, Hölder-differentiability of Gibbs distribution functions, Math. Proc. Cambridge Philos. Soc., 147 (2009), 489-503.  doi: 10.1017/S0305004109002473. [16] A. Ya Khinchin, Continued Fractions The University of Chicago Press, 1964. [17] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets Cambridge University Press, 2003. doi: 10.1017/CBO9780511543050. [18] H. Minkowski, Geometrie der Zahlen (German) Bibliotheca Mathematica Teubneriana, Band 40 Johnson Reprint Corp. , New York-London, 1968. [19] S. Munday, On the derivative of the α-Farey-Minkowski function, Discrete Contin. Dynam. Systems, 34 (2014), 709-732.  doi: 10.3934/dcds.2014.34.709. [20] 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. [21] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [22] 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. [23] J. F. Sánchez, P. Viader, J. Paradís and M. D. Carrillo, A singular function with a non-zero finite derivative, Nonlinear Anal., 75 (2012), 5010-5014.  doi: 10.1016/j.na.2012.04.015. [24] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820. [25] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758.  doi: 10.1090/S0002-9939-03-06927-2. [26] S. Troscheit, Hölder differentiability of self-conformal devil's staircases, Math. Proc. Cambridge Philos. Soc., 156 (2014), 295-311.  doi: 10.1017/S0305004113000698.
On the left, the functions $Q_1$, $Q_2$ and $Q_3$, and on the right, the Minkowski question-mark function, $Q:[0, 1]\to[0, 1]$
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