Piecewise Linear Model for Tree Maps

Abstract

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

“…(1) and (2) are contained in [40,Theorem A.3]. (3) follows from Yoccoz's theorem on landing of rays at finitely renormalizable parameters (see [20] for the proof).…”

“…Our work is a generalization to tree maps of Milnor and Thurston's kneading theory [32] for interval maps. The general strategy is similar to [2], but our view is towards application to Hubbard trees. Moreover, since we are mostly interested in principal veins, we will treat in detail only the case of trees with particular topological types.…”

“…. , f p−1 Note that we take the left-hand side of the interval Ω W to be open, thus roots of maximal tuning windows are non-renormalizable 2. Let us describe the behavior of Hausdorff dimension with respect to the tuning operator:Proposition 9.3.…”

Kneading theory for tree maps

“…Recently, Baillif and de Carvalho [9] generalized the theorems of Parry [23] and Milnor-Thurston for tree maps, which state that every piecewise monotone interval map with positive topological entropy log(s) is semiconjugated to a piecewise linear interval map with slope ±s everywhere. Recently, Baillif and de Carvalho [9] generalized the theorems of Parry [23] and Milnor-Thurston for tree maps, which state that every piecewise monotone interval map with positive topological entropy log(s) is semiconjugated to a piecewise linear interval map with slope ±s everywhere.…”

Topological entropy of transitive dendrite maps

“…By [BC01], every P -Markov tree map f : T → T is semiconjugate to a P ′ -linear tree map f ′ : T ′ → T ′ with constant slope λ ′ = e h(f ) . The following trivial fact shows that if f is P -linear and transitive then the "constant slope" can be achieved by a simple change of the metric on T .…”