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Topological and algebraic reducibility for patterns on trees

LLUÍS ALSEDÀ, DAVID JUHER, FRANCESC MAÑOSAS

2013
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Ergodic Theory and Dynamical Systems
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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
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... 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