# About strange invariant set of the Lozi mappings

Consider the Tent map: $$f_{μ}(x)=μx$$, if $$x<0.5$$ and $$f_{μ}(x)=μ(1-x)$$ if $$x≥0.5$$. In this page (https://en.wikipedia.org/wiki/Tent_map) it was stated that:

If $$μ$$ is greater than $$2$$ the map's Julia set becomes disconnected, and breaks up into a Cantor set within the interval $$[0,1]$$. The Julia set still contains an infinite number of both non-periodic and periodic points (including orbits for any orbit length) but almost every point within $$[0,1]$$ will now eventually diverge towards infinity.

This divergence of orbits was observed for the lozi map $$L_{a,b}(x,y)=(1-a|x|+y,bx)$$ when $$a=2-b/2$$

In this paper (https://link.springer.com/article/10.1007/BF02584740) the author say in page 118:

As is already the case for $$l₂$$ (here it is the Tent map above), the closure of the invariant "manifold" (see [11] for the use of this word for $$L_{a,b}$$) of the non trivial fixed point of $$L_{2-b/2,b}$$ is not a strange attractor since in any neighborhood of this closure, one can find points whose trajectory diverges. This closure appears however as a "strange invariant set," as well as $$[0, 1]$$ when one considers $$l₂$$ as a mapping from $$ℝ$$ to itself.

I want to prove this fact (in any neighborhood of this closure, one can find points whose trajectory diverges) for the Lozi map. However, I have no idea on how this is possible.