Article Contents
Article Contents

# Geometrical properties of the mean-median map

• 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 $1$.

Mathematics Subject Classification: Primary: 37P99; Secondary: 11B75, 11J99, 26A27.

 Citation:

• Figure 1.  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

Figure 2.  Early evolution of the bundle $[0,x,1]$. In each picture, the red function is the current median

Figure 4.  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 $p$

Figure 3.  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

Figure 5.  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 $1$

Figure 6.  Construction of the homology $\lambda$ determined by the triad $\left[U,L;Y\right]$. Here $U = I\lor I'$ and $L = J\lor J'$

Figure 7.  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 $P$

Figure 9.  The flow chart for computing $p_{n+1}$, $q_{n+1}$ from $p_n$, $q_n$

Figure 8.  Even-to-odd iteration (left) and odd-to-even iteration (right)

Figure 10.  The transit time of $\hat{\Xi}$ for $x>0$

Figure 11.  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 $\tau\left(p_{n+r}\right) = \tau\left(p_n\right)+2$

Figure 12.  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

Figure 13.  The situation in the left-tractability domain of the X-point $\frac{2}{3}$ of rank $1$ in the system $[0,x,1]$

Figure 14.  The first five functions in the system $[0,x,-10x+1,-10x+1]$ and their median (red)

Figure 15.  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 $\mathcal{M}_{25}$

Figure 16.  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 $\mathcal{M}_{9}$

Figure 19.  Least-square regression plot associated to (66)

Figure 17.  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 $740\frac{1}{2}$

Figure 18.  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

Figure 20.  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 $0.86$

•  [1] P. C. Allaart and K. Kawamura, The Takagi function: a survey, Real Analysis Exchange, 37 (2011), 1-54. [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. [3] M. Chamberland and M. Martelli, The mean-median map, Journal of Difference Equations and Applications, 13 (2007), 577-583.  doi: 10.1080/10236190701264719. [4] H. S. M. Coxeter, Projective Geometry, Springer, New York, 1987. [5] G. H. Hardy and  E. M. Wright,  An Introduction to the Theory of Numbers, 6$^{th}$ edition, Oxford University Press, Oxford, 2008. [6] J. Hoseana, The Mean-Median Map, MSc dissertation, Queen Mary University of London, 2015. [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. [8] S. Roman, Advanced Linear Algebra, 3$^{th}$ edition, Springer, California, 2007. [9] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4$^{th}$ edition, Prentice Hall, Boston, 2010. [10] H. Schwerdtfeger, Geometry of Complex Numbers, Dover, New York, 1979. [11] H. Schultz and R. Shiflett, M & m sequences, College Mathematics Journal, 36 (2005), 191-198.  doi: 10.2307/30044851.

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