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Toeplitz subshift conjugacy invariant full#
linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson–Furstenberg boundary. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. a closed shift invariant (X) X set) is minimalif it has no proper non empty subshift. Automorphism of classical minimal systems A subshift X AZ,(i.e. doi: 10.3934/dcds.2018068 13 Yoshikazu Katayama, Colin E. Automorphisms of low complexity subshifts 2. Even though these results allow us to slowly understand the group of automorphisms of low complexity subshifts, the complete picture is still unclear, even for particular classes of subshifts. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. Interestingly, in the author provided a Toeplitz subshift with complexity p X (n) C n 1.757, whose automorphism group is not finitely generated. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. These ideas are applied to the Henon map to prove the existence of chaotic dynamics on an open set of parameter values. The zeta function is clearly an invariant of the conjugacy class of a map. We use this information to provide a characterization of invariant sets which admit a semi-conjugacy onto the space of sequences on K symbols with dynamics given by a subshift. The zeta function of a: X - X is defined by Cf(t) exp (E'7 (Nlit)/i), where Ni is the number of (isolated) points of X left fixed by fi. We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. A semi-conjugacy from a to b is such a map h, which is not required to be a homeomorphism. Toeplitz subshifts of arbitrary positive topological entropy on residually. This result provides a partial affirmative answer to a question asked by Sabok and Tsankov.Īs pointed Cantor minimal systems are represented by properly ordered Bratteli diagrams, we also establish that the Borel complexity of equivalence of properly ordered Bratteli diagrams is $\d$.Subshifts with slow complexity and simple groups with the Liouville property Subshifts with slow complexity and simple groups with the Liouville property for Z-actions, it is an important conjugacy invariant which has been studied. We prove that the topological conjugacy relation on Toeplitz subshifts with separated holes is a hyperfinite Borel equivalence relation. The other main result of this thesis concerns the topological conjugacy relation on Toeplitz subshifts. As a byproduct of our analysis, we also show that $\d$ is a lower bound for the topological conjugacy relation on Cantor minimal systems.
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We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $\newcommand\d\d$. This thesis is a contribution to the project of analyzing the Borel complexity of the topological conjugacy relation on various Cantor minimal systems. In recent years, the study of the Borel complexity of naturally occurring classification problems has been a major focus in descriptive set theory. Section 4, these results are applied to the class of Toeplitz sequences. The defense was very nice, with an extremely clear account of the main results, and the question session included a philosophical discussion on various matters connected with the dissertation, including the principle attributed to Gao that any collection of mathematical structures that has a natural Borel representation has a unique such representation up to Borel isomorphism, a principle that was presented as a Borel-equivalence-relation-theory analogue of the Church-Turing thesis.īurak Kaya | MathOverflow profile | ar$\chi$iv profileĪbstract. Theorem 1.2 (i) The subshift has a unique invariant Borel probability mea. The dissertation committee consisted of Simon Thomas, Gregory Cherlin, Grigor Sargsyan and myself, as the outside member. degree at Rutgers University under the supervision of Simon Thomas. Burak Kaya successfully defended his dissertation, “Cantor minimal systems from a descriptive perspective,” on March 24, 2016, earning his Ph.D.