Inducing, slopes, and conjugacy classes (Q1366858)

Let \(N=[a,b]\) be a closed interval and \(f:N\to N\) a continuous map. This map is a \(d\)-modal piecewise affine map, if there exist points \(a=a_0< a_1< \cdots <a_d< a_{d+1} = b\) such that \(f_i= f|_{[a_i,a_i+1]}\) is affine and \(Df_iDf_{i+1} <0\) \((Df_i\) is the derivative of \(f_i)\). The set \(F_d\) of \(d\)-modal piecewise affine maps can be considered as a subset of \(\mathbb{R}^{2d +2}\) and the subset \(E_d\) of such maps that are eventually expanding (there is an integer \(n\geq 1\) so that \(|Df^n |>1\) whenever this derivative is defined) is a submanifold of \(\mathbb{R}^{2d+ 2}\). The authors prove that if \(f\in E_d\), then the conjugacy class of \(f\) is contained in a codimension 1 submanifold of \(E_d\), and if the slopes of some essential branches of \(f\) are increased, the topological type changes. The proof of the above result is based on a study of the relation between induced Markov maps and ergodic theoretical behavior: the authors establish, among other results, that every element of \(E_d\) has the Markov property, and a piecewise affine map has the Markov property if and only if it does not have zero-dimensional absorbing sets.