A note on Sarkovskii's Theorem in connected linearly ordered spaces

We prove that, for a connected linearly ordered space L, the following conditions are equivalent: (1) L satisfies Sarkovskii's Theorem, (2) there exist turbulent functions on L, and (3) there exists a compact subspace of L which satisfies Sarkovskii's Theorem. Our results are applied in two ways. Firstly, we show that there exist connected linearly ordered spaces without infinite minimal sets; secondly, for each cardinal number A of uncountable cofinality, we construct a connected linearly ordered space L such that: (1) L is a compact nonfirst countable space satisfying Sarkovskii's Theorem, (2) L admits a dense first countable subset, and (3) the density of L is A.