![]() ![]() ![]() ![]() A fundamental object that appearsin this context is the dimension group of a dynamical system, an algebraic con-cept imported from the study of C ∗ -algebras. So one could ask, what are the natural restrictionsthat arise from being strongly orbit equivalent. All these results make clear that entropy poses no restriction whatsoever to be-ing strongly orbit equivalent. Later, in a series of papers, Sugisaki generalizedthis result, showing among other things that for any c ∈ any Cantor mini-mal system is strongly orbit equivalent to a system whose topological entropy equals c. This was dramatically disproved byBoyle and Handelman, who showed that any Cantor minimal system (a systemwhere the phase space is a Cantor space) is strongly orbit equivalent to a zero en-tropy Cantor system. Whenthe strong orbit equivalence property was introduced in topological dynamics, itwas not clear its relationship with entropy, in particular, it was asked if entropywas preserved under strong orbit equivalence. They arestrongly orbit equivalent if they are orbit equivalent and the cocyle function isdiscontinuous at most in one point (see Section 2.7 for precise definitions). Two minimal topological dynamical systems are orbit equivalent if there is ahomeomorphism between the phase spaces sending orbits to orbits. As a consequence, we getthat any Choquet simplex can be realized as the set of invariant measures ofa minimal Toeplitz subshift whose complexity grows slower than p n. G, G +, u ) and anysequence of positive numbers ( p n ) n ∈ N such that lim n/p n = 0, there exist aminimal subshift whose dimension group is order isomorphic to ( G, G +, u ) andwhose complexity function grows slower than p n. This is doneby proving that for any simple dimension group with unit ( We show that within any strong orbit equivalent class, there existminimal subshifts with arbitrarily low superlinear complexity. PAULINA CECCHI BERNALES AND SEBASTI ´AN DONOSO S e p STRONG ORBIT EQUIVALENCE AND SUPERLINEARCOMPLEXITY ![]()
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