# American Institute of Mathematical Sciences

June  2020, 7(1): 83-121. doi: 10.3934/jcd.2020004

## Geometrical properties of the mean-median map

 School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

Received  January 2019 Published  December 2019

We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday [2] by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the set $[0,x,1]$ is greater than $1$.

Citation: Jonathan Hoseana, Franco Vivaldi. Geometrical properties of the mean-median map. Journal of Computational Dynamics, 2020, 7 (1) : 83-121. doi: 10.3934/jcd.2020004
##### References:
 [1] P. C. Allaart and K. Kawamura, The Takagi function: a survey, Real Analysis Exchange, 37 (2011), 1-54.   Google Scholar [2] F. Cellarosi and S. Munday, On two conjectures for M & m sequences, Journal of Difference Equations and Applications, 22 (2016), 428-440.  doi: 10.1080/10236198.2015.1102232.  Google Scholar [3] M. Chamberland and M. Martelli, The mean-median map, Journal of Difference Equations and Applications, 13 (2007), 577-583.  doi: 10.1080/10236190701264719.  Google Scholar [4] H. S. M. Coxeter, Projective Geometry, Springer, New York, 1987. Google Scholar [5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{th}$ edition, Oxford University Press, Oxford, 2008.   Google Scholar [6] J. Hoseana, The Mean-Median Map, MSc dissertation, Queen Mary University of London, 2015. Google Scholar [7] J. C. Lagarias, The Takagi function and its properties, in Functions in Number Theory and their Probabilistic Aspects (eds. K. Matsumoto, Editor in Chief, S. Akiyama, H. Nakada, H. Sugita, A. Tamagawa), RIMS Kôkyûroku Bessatsu B34, (2012), 153–189.  Google Scholar [8] S. Roman, Advanced Linear Algebra, 3$^{th}$ edition, Springer, California, 2007.  Google Scholar [9] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4$^{th}$ edition, Prentice Hall, Boston, 2010. Google Scholar [10] H. Schwerdtfeger, Geometry of Complex Numbers, Dover, New York, 1979.  Google Scholar [11] H. Schultz and R. Shiflett, M & m sequences, College Mathematics Journal, 36 (2005), 191-198.  doi: 10.2307/30044851.  Google Scholar

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##### References:
 [1] P. C. Allaart and K. Kawamura, The Takagi function: a survey, Real Analysis Exchange, 37 (2011), 1-54.   Google Scholar [2] F. Cellarosi and S. Munday, On two conjectures for M & m sequences, Journal of Difference Equations and Applications, 22 (2016), 428-440.  doi: 10.1080/10236198.2015.1102232.  Google Scholar [3] M. Chamberland and M. Martelli, The mean-median map, Journal of Difference Equations and Applications, 13 (2007), 577-583.  doi: 10.1080/10236190701264719.  Google Scholar [4] H. S. M. Coxeter, Projective Geometry, Springer, New York, 1987. Google Scholar [5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{th}$ edition, Oxford University Press, Oxford, 2008.   Google Scholar [6] J. Hoseana, The Mean-Median Map, MSc dissertation, Queen Mary University of London, 2015. Google Scholar [7] J. C. Lagarias, The Takagi function and its properties, in Functions in Number Theory and their Probabilistic Aspects (eds. K. Matsumoto, Editor in Chief, S. Akiyama, H. Nakada, H. Sugita, A. Tamagawa), RIMS Kôkyûroku Bessatsu B34, (2012), 153–189.  Google Scholar [8] S. Roman, Advanced Linear Algebra, 3$^{th}$ edition, Springer, California, 2007.  Google Scholar [9] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4$^{th}$ edition, Prentice Hall, Boston, 2010. Google Scholar [10] H. Schwerdtfeger, Geometry of Complex Numbers, Dover, New York, 1979.  Google Scholar [11] H. Schultz and R. Shiflett, M & m sequences, College Mathematics Journal, 36 (2005), 191-198.  doi: 10.2307/30044851.  Google Scholar
A computer-generated image of $m(x)$ for $x\in\left[\frac{1}{2},\frac{2}{3}\right]$. The limit function $m$ may be reconstructed entirely from its values in this interval, due to symmetries. The local minima occur at a prominent set of rational numbers
Early evolution of the bundle $[0,x,1]$. In each picture, the red function is the current median
Origin of a singularity in a regular bundle $\Xi_n$, if $n$ is odd ($n = 3$, left) and if $n$ is even ($n = 4$, right). The blue and red functions are the mean and median of the bundle, respectively. The latter is singular, due to the presence of the X-point $p$. In either case, the image function $Y_{n+1}$ will be singular at $p$
The decay of the proportion $P_n$ of fractions with denominator at most $n$ in the interval $\left[\frac{1}{2},\frac{2}{3}\right]$ which are X-points
The bundle $[0,x,1,1]$ in which the origin is an X-point of rank $2$ (left) and the bundle $\left[0,x,1,Y_4(x)\right]$, where $Y_4(x)$ is equal to $1$ for $x\leqslant0$ and to $\frac{x}{2}+1$ for $x>0$, in which it is an X-point of left-rank $2$ and right-rank $1$
Construction of the homology $\lambda$ determined by the triad $\left[U,L;Y\right]$. Here $U = I\lor I'$ and $L = J\lor J'$
Construction of the projective collineation $\lambda$ near a non-monotonic X-point $p$. The two branches $K$ and $K'$ of the auxiliary function of the pseudotriad do not meet on $P$
The flow chart for computing $p_{n+1}$, $q_{n+1}$ from $p_n$, $q_n$
Even-to-odd iteration (left) and odd-to-even iteration (right)
The transit time of $\hat{\Xi}$ for $x>0$
The medians $\mathcal{M}_{\tau\left(p_n\right)-2}$ (red), $\mathcal{M}_{\tau\left(p_n\right)-1}$ (blue), and $\mathcal{M}_{\tau\left(p_n\right)}$ (green), showing that $\tau\left(p_{n+r}\right) = \tau\left(p_n\right)+2$
The situation in a right-neighbourhood of an X-point $p$ of rank $1$ if the functional orbit stabilises. Theorem 5.6 first describes, in part i), the limit function in $\left[p,\ddot{p}\right)$. If the extra condition in part ii) holds, then we can extend the description to $\left[p,p'\right)$. The red, yellow, green, blue, and brown functions are the medians $\mathcal{M}_{t-2}$, $\mathcal{M}_{t-1}$, $\mathcal{M}_{t}$, $\mathcal{M}_{t+1}$, and $\mathcal{M}_{t+2}$, respectively
The situation in the left-tractability domain of the X-point $\frac{2}{3}$ of rank $1$ in the system $[0,x,1]$
The first five functions in the system $[0,x,-10x+1,-10x+1]$ and their median (red)
A violation of T2 on the right-hand side of the X-point $\frac{999}{1798}$ in the system $[0,x,1]$. The red function is the median $\mathcal{M}_{25}$
A violation of T3 on the right-hand side of the X-point $0$ in the system $\left[-5,-4,-3,Y_4(x),x,3,3\right]$, where $Y_4$ is defined in (52). The red function is the median $\mathcal{M}_{9}$
Least-square regression plot associated to (66)
The normal form orbit of order $55$ (red) and the corresponding median sequence (blue). The orbit behaves regularly up to $n = N_{55} = 122$, by lemma 6.1, which gives the explicit expressions for all terms up to this index. The horizontal dashed line represents the lower bound for the limit given by theorem 6.2, which is $740\frac{1}{2}$
Plots of $m_t$ (left) and $\tau_t$ (right) for $t\in\{5,7, \cdots ,95\}$ with the respective lower bounds given by theorem 6.2
Log-log plot of the variation of the limit function with respect to the Farey partition versus the size of the partition. The slope of the line is approximately $0.86$
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