Topological and algebraic reducibility for patterns on trees

2013 Ergodic Theory and Dynamical Systems  
We extend the classical notion of block structure for periodic orbits of interval maps to the setting of tree maps and study the algebraic properties of the Markov matrix of a periodic tree pattern having a block structure. We also prove a formula which relates the topological entropy of a pattern having a block structure with that of the underlying periodic pattern obtained by collapsing each block to a point, and characterize the structure of the zero entropy patterns in terms of block
more » ... rms of block structures. Finally, we prove that an n-periodic pattern has zero (positive) entropy if and only if all n-periodic patterns obtained by considering the k-th iterate of the map on the invariant set have zero (respectively, positive) entropy, for each k relatively prime to n. 1991 Mathematics Subject Classification. Primary: 37E25. LLUÍS ALSEDÀ, DAVID JUHER AND FRANCESC MAÑOSAS point of valence different from 2 will be called a vertex of T and the set of vertices of T will be denoted by V (T ). Each point of valence 1 will be called an endpoint of T. The set of such points will be denoted by En(T ). The points in V (T ) \ En(T ) have valence greater than or equal to 3. They will be called the branching points of T . Also, the closure of a connected component of T \ V (T ) will be called an edge of T . Any tree which is a union of r ≥ 2 intervals whose intersection is a unique point y of valence r will be called an r-star, and y will be called its central point. Given any subset X of a topological space, we will denote by Int(X) and Cl(X) the interior and the closure of X, respectively. For a finite set P we will denote its cardinality by |P |. A triplet (T, P, f ) will be called a model if f : T −→ T is a tree map and P is a finite f -invariant set. In particular, if P is a periodic orbit of f and |P | = n then (T, P, f ) will be called an n-periodic model. Given X ⊂ T we will define the convex hull of X, denoted by X T or simply by X , as the smallest closed connected subset of T containing X. When X = {x, y} we will write x, y or [x, y] to denote X . The notations (x, y), (x, y] and [x, y) will be understood in the natural way. Let T be a tree and let P ⊂ T be a finite subset of T . The pair (T, P ) will be called a pointed tree. A set Q ⊂ P is said to be a discrete component of (T, P ) if either |Q| > 1 and there is a connected component C of T \ P such that Q = Cl(C) ∩ P , or |Q| = 1 and Q = P . We say that two pointed trees (T, P ) and (T ′ , P ′ ) are equivalent if there exists a bijection φ : P −→ P ′ which preserves discrete components. In this case, two discrete components C of (T, P ) and C ′ of (T ′ , P ′ ) will be called equivalent if C ′
doi:10.1017/etds.2013.52 fatcat:jov576h23jcixazkrfgfl7piwq