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Geometrical properties of the mean-median map
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom |
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 [
References:
[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. 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.
![]() |
[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. |
[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. Google Scholar |
[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. |
show all references
References:
[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. 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.
![]() |
[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. |
[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. Google Scholar |
[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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