Geometrical properties of the mean-median map

Abstract

We study the mean-median map as a dynamical system on the space of finite multisets 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 multiset [0, x, 1] is greater than 1.Date: May 7, 2019. 6 the largest being 571460 7 the set of all fractions in [0, 1] with denominator at most 5000

The Akiyama Mean-Median Map Has Unbounded Transit Time and Discontinuous Limit

“…x. Substituting this and x ℓ+1 = −(ℓ − 3)x into (7) and ( 8) gives x 2ℓ+1 = −(2ℓ − 3)x and x 2ℓ+2 = −(2ℓ − 2)x, extending the formula (6). Moreover, since x 2ℓ+2 < x 2ℓ+1 < x 4 , then M 2ℓ+2 = x ℓ+2 , x 4 = x 4 = M 2ℓ+1 , so, by part (ii) of Proposition 3, we have x n = x 4 = 2x − 1 for every n 2ℓ + 3.…”

“…The simultaneous occurrence of the unboundedness of the transit time and the discontinuity of the limit function is unsurprising. Indeed, in the original mmm we have pointed out that these will be two interrelated consequences if a local functional orbit is found to be divergent [7,Theorems 5.4 and 5.6]. While such divergence has not been found in the original mmm, we find it near x = 0 in the Akiyama mmm.…”

“…Case II: x ∈ 1 ℓ , 1 ℓ−1 . Substituting x 4 = 2x − 1 and x ℓ+1 = −(ℓ − 3)x into (7) and ( 8) gives (5). Moreover, we have…”

“…One of the most striking features of Figure 2 is the presence of symmetries, particularly around x = 1 2 . In this closing section, we briefly explain the symmetry near x = 1 2 in the light of what has been done for the original mmm [7]. As in [7], we now regard [x, 1], x ∈ (0, 1), as a set of univariate piecewise-affine continuous real functions [in this case Y 1 (x) = x and Y 2 (x) = 1]; we refer to such a set as a bundle [7, Section 2.2].…”

“…2 The middle number after sorting if n is odd, the mean of the middle pair otherwise. Despite intensive research effort [9,4,5,3,7,6,8,10], these terminating conjectures, as well as two additional conjectures to follow, are still open even in the case of smallest nontrivial initial sets: those of size three. The fact that the mmm commutes with elementwise affine transformations [4, Section 3] makes the orbit of every such set affine-equivalent to that of a univariate initial set [0, x, 1], for some real number x ∈ 1 2 , 2 3 which we call the initial condition.…”