On basic forbidden patterns of functions

TL;DR

This paper investigates the structure of forbidden permutation patterns in one-dimensional interval maps, especially the logistic map, providing bounds and conditions for their existence and characterization.

Contribution

It establishes that the logistic map has infinitely many basic forbidden patterns for certain parameters and offers bounds and conditions for forbidden patterns in piecewise monotone maps.

Findings

Logistic map has infinitely many basic forbidden patterns for 1<r<=4.

Provides a lower bound on the number of basic forbidden patterns of L_4.

Offers an upper bound on the length of the shortest forbidden pattern in piecewise monotone maps.

Abstract

The allowed patterns of a map on a one-dimensional interval are those permutations that are realized by the relative order of the elements in its orbits. The set of allowed patterns is completely determined by the minimal patterns that are not allowed. These are called basic forbidden patterns. In this paper we study basic forbidden patterns of several functions. We show that the logistic map L_r(x)=rx(1-x) and some generalizations have infinitely many of them for 1<r<=4, and we give a lower bound on the number of basic forbidden patterns of L_4 of each length. Next, we give an upper bound on the length of the shortest forbidden pattern of a piecewise monotone map. Finally, we provide some necessary conditions for a set of permutations to be the set of basic forbidden patterns of such a map.