Article Contents
Article Contents

Forward triplets and topological entropy on trees

• * Corresponding author: David Juher

Work supported by grants MTM2017-86795-C3-1-P and 2017 SGR 1617. Lluís Alsedà acknowledges financial support from the Spanish Ministerio de Economía y Competitividad grant number MDM-2014-0445 within the "María de Maeztu" excellence program

• We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map $f$ has positive entropy if and only if some iterate $f^k$ has a periodic orbit with three aligned points consecutive in time, that is, a triplet $(a,b,c)$ such that $f^k(a) = b$, $f^k(b) = c$ and $b$ belongs to the interior of the unique interval connecting $a$ and $c$ (a forward triplet of $f^k$). We also prove a new criterion of entropy zero for simplicial $n$-periodic patterns $P$ based on the non existence of forward triplets of $f^k$ for any $1\le k<n$ inside $P$. Finally, we study the set $\mathcal{X}_n$ of all $n$-periodic patterns $P$ that have a forward triplet inside $P$. For any $n$, we define a pattern that attains the minimum entropy in $\mathcal{X}_n$ and prove that this entropy is the unique real root in $(1,\infty)$ of the polynomial $x^n-2x-1$.

Mathematics Subject Classification: Primary: 37E15, 37E25.

 Citation:

• Figure 1.  Two non-homeomorphic trees $T$ and $S$, with 9-periodic orbits $P = \{x_i\}_{i = 1}^9$ and $Q = \{y_i\}_{i = 1}^9$ of two respective (unspecified) tree maps ${f}: {T} \longrightarrow {T}$ and ${g}: {S} \longrightarrow {S}$

Figure 2.  An interval model $(T,P,f)$ and the corresponding pattern, that can be identified with the permutation $(3,4,2,5,1)$

Figure 3.  Two 8-periodic simplicial patterns $\mathcal{P}$ and $\mathcal{Q}$

Figure 4.  A fully rotational 12-periodic pattern $\mathcal{P}$

Figure 5.  The pattern $\mathcal{Q}_6$

Figure 6.  Two models of the same pattern

Figure 7.  On the left, the canonical model $(T,P,f)$ of a 6-periodic pattern $\mathcal{P}$, for which $f(y) = y$. On the right, the Markov graph of $(T,P,f)$

Figure 8.  The two possible arrangements of the connected components $C_0,C_1,C_2$ in the proof of Lemma 4.2. The black points belong to $P$

Figure 9.  Topological induction

Figure 10.  A 15-periodic pattern with two discrete components and a $\pi$-reduced 5-pattern

Figure 11.  A 15-periodic pattern with two discrete components and a 3-division

Figure 12.  Two possible sequences $\mathcal{P}\ge\mathcal{Q}_1\ge\mathcal{Q}_2$ and $\mathcal{P}\ge\mathcal{R}_1\ge\mathcal{R}_2$ of pull outs leading to two (different) complete openings of $\mathcal{P}$ with respect to the forward triplet $(4,5,6)$

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