Erratum to "Minimal sets of non-resonant torus homeomorphisms" (Fund. Math. 211 (2011), 41-76)
Ferry Kwakkel (2011)
Fundamenta Mathematicae
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Ferry Kwakkel (2011)
Fundamenta Mathematicae
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Tomasz Downarowicz (2011)
Colloquium Mathematicae
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We construct an example of two commuting homeomorphisms S, T of a compact metric space X such that the union of all minimal sets for S is disjoint from the union of all minimal sets for T. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of "doubly recurrent" points (see [F]). Not only are these points recurrent under both S and T, but they recur along the same sequence of powers. Our...
Ferry Kwakkel (2011)
Fundamenta Mathematicae
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As was known to H. Poincaré, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms, that is, torus homeomorphisms isotopic to the identity for which the...
Dusa McDuff (1981)
Annales de l'institut Fourier
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Necessary conditions are found for a Cantor subset of the circle to be minimal for some -diffeomorphism. These conditions are not satisfied by the usual ternary Cantor set.
Sergiĭ Kolyada, L'ubomír Snoha, Sergeĭ Trofimchuk (2001)
Fundamenta Mathematicae
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For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and share with A those topological properties which describe how large a set is. Using...
Jason Gait (1972)
Compositio Mathematica
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Grzegorz Lewicki (1993)
Collectanea Mathematica
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Dariusz Tywoniuk (2012)
Colloquium Mathematicae
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We construct a continuous non-invertible minimal transformation of an arbitrary solenoid. Since solenoids, as all other compact monothetic groups, also admit minimal homeomorphisms, our result allows one to classify solenoids among continua admitting both invertible and non-invertible continuous minimal maps.
Henk Bruin, Sergiǐ Kolyada, L'ubomír Snoha (2003)
Colloquium Mathematicae
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We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.
Walter H. Gottschalk (1964)
Annales de l'institut Fourier
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Wojciech Hyb (1978)
Colloquium Mathematicae
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Joseph Auslander, Brindell Horelick (1970)
Compositio Mathematica
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Khalil, R. (2002)
Rendiconti del Seminario Matematico
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