On one of Birkhoff's theorems for backward limit points

TL;DR

This paper extends Birkhoff's classical theorem to describe the behavior of backward limit points in interval maps, generalizing previous results on omega limit points and non-wandering sets.

Contribution

It formulates a new theorem characterizing neighborhoods of various backward limit points for interval maps, building on and generalizing prior work by Birkhoff and Sharkovsky.

Findings

Provides a new theoretical framework for backward limit points

Generalizes Birkhoff's theorem to broader classes of points

Enhances understanding of interval map dynamics

Abstract

In 1927 George Birkhoff in his book Dynamical Systems presented a theorem that describes the behaviour of trajectories outside of a set of non-wandering points on an arbitrary compacta. Much later in 1960s Sharkovsky followed up on Birkhoff's work and published even stronger result, this time focusing on the set of omega limit points for interval maps. In this article we formulate similar statement for a neighbourhood of a set of different types of backward limit points for maps of the interval.