Abstract
For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some of) their endpoints. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.
Highlights
For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their endpoints
Given some dynamical system on a topological space and an open subset, it is natural to study the associated surviving set, that is, the collection of all points which never enter this subset under forward iteration
We propose to study the family of all interval holes representing distinct surviving dynamics as a source of topological invariants
Summary
Given some dynamical system on a topological space and an open subset (called hole in the following), it is natural to study the associated surviving set, that is, the collection of all points which never enter this subset under forward iteration. Notice further that the second part of point (b) implies that the bifurcation set is a collection of horizontal and vertical segments, while the first part of (b) gives-together with point (d)-that these segments can essentially be obtained by drawing a horizontal and vertical line from the double points to the diagonal[5] This observation emphasizes the importance of double points which is even more prominent due to their close relation to kneading sequences of expansive Lorentz maps (see the caption of figure 1) as well as of nice points introduced in [28] (see remark 3.3). (b) For s ∈ J some step is accumulated from below and s is a discontinuity point of s → BTs. We close the introduction noting that the bifurcation set itself is clearly not a dynamical invariant, we can introduce an induced invariant, see section 2.1.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
