The bifurcation set as a topological invariant for one-dimensional dynamics

The assumption that a and b avoid the boundary points {0, 1} simply reduces certain technicalities and is not of any further importance. For an explicit study of general continuous maps on [0, 1] with interval holes of the form [0, t) and (t, 1] where t ∈ [0, 1], see [22].

For $\mathbb{I}=\mathbb{T}$, this is true for all segments. For $\mathbb{I}=\left[0,1\right]$, this is true for all but those lines in ${\mathcal{B}}_{f}$ with arbitrarily small first or second coordinate, see also the previous footnote.

As in theorem A (f), we consider the space of all continuous maps $f:\mathbb{I}\to \mathbb{I}$ equipped with the uniform topology, and the space of all non-empty closed subsets of Δ endowed with the Hausdorff metric.

More precisely, in the interval case, proposition 2.2 yields transitivity for all points except 0 and 1. Yet, denseness of periodic points for transitive maps implies minimality of f, see the remark before proposition 3.12. Further, recall that there are no minimal continuous maps on [0, 1], i.e., ${\mathcal{B}}_{f}$ is always non-empty for $\mathbb{I}=\left[0,1\right]$.

Note that formally speaking, as our present definition of ${\mathcal{B}}_{f}$ excludes points with coordinate entries equal to zero, we could not apply lemma 4.9. However, this issue is of a rather formal nature (see also the remark in the introduction) and will further not play a role in the discussion of the discontinuity of $s{\mapsto}{\mathcal{B}}_{{T}_{s}}$ as this discussion has to be carried out explicitly anyway.

Note that s₀ is necessarily different from $\sqrt{2}$, since ${T}_{\sqrt{2}}\left(0\right)$ coincides with the unique fixed point of ${T}_{\sqrt{2}}$ which gives that ${c}_{\sqrt{2}}$ is not periodic. Hence, ${\mathcal{I}}^{\prime }$ is always an interval which has a non-trivial intersection with $\left[\sqrt{2},2\right]$.