@article{1,
author = {Alexandre Miranda Trilles Junior},
affiliation={UJ},
authorurl={},
title={  Discontinuity of the accessible prime end rotation number in the Lozi family},

abstract={   The prime end rotation number of a non-separating attractor of an orientation-preserving plane homeomorphism 
describes how the accessible points of the attractor rotate. Our motivation comes from a question of Yorke about whether 
this number varies continuously with the parameters for Hénon attractors, and from a conjecture of Alligood and Yorke 
stating that in typical one-parameter families of area-contracting diffeomorphisms it can jump only on a discrete set of parameters.

In this talk we study both problems for the orientation-preserving Lozi family, a piecewise affine version of the Hénon family 
that is also a special case of the border-collision normal form. We show that for any Cantor set in the unit interval, whose 
measure can be anything less than one, there is a one-parameter subfamily of Lozi maps, indexed by the unit interval, along 
which the prime end rotation number of the spinning attractor equals 1/3 on the Cantor set and 1/2 elsewhere. The rotation 
number is therefore discontinuous exactly at the points of the Cantor set, and at each of them the spinning attractor is 
also discontinuous in the Hausdorff metric. These jumps come from a period-three orbit born in a border collision as a 
degenerate saddle. We will also discuss what this invariant does not see, such as infinitely many topological conjugacy 
classes with the same rotation number. This is joint work with J. P. Boroński, D.-S. Ou and N. Schuback.
},
year = {2026/27},
month = {10},
day={23}
}