On one of Birkhoff's theorems for backward limit points

TL;DR

The paper addresses extending Birkhoff's forward-limit theorem to backward limit points for noninvertible interval maps by formulating and proving a backward-limit version using $α$-limit and special $α$-limit sets. It proves a uniform bound $M$ on how many points of any backward orbit branch can lie outside a neighborhood of $A(f)$ for onto interval maps, and identifies conditions under which $sα(x)=α(x)$ or the result extends to $SA(f)$ when $Rec(f)$ is closed. The work clarifies the relationship between backward-limit structures and maximal ω-limit sets, distinguishing periodic, basic, and solenoidal types, and discusses limitations in solenoidal contexts. Overall, it contributes to backward dynamics theory for noninvertible maps and connects backward-limit behavior to classical ω-limit structures in interval 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.

Figures (1)

• Figure 1: Graph of the map $f: \left[0, 1\right] \rightarrow \left[0, 1\right].$

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 7
• Remark 8
• Theorem 9
• proof