On the minimal overlap problem of Erdös
L Moser (1959)
Acta Arithmetica
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L Moser (1959)
Acta Arithmetica
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H. Fredricksen, E. J. Ionascu, F. Luca, P. Stănică (2008)
Acta Arithmetica
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Asit Baran-Raha (1972)
Colloquium Mathematicae
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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.
J. Marshall Ash, A. Eduardo Gatto, Stephen Vági (1990)
Colloquium Mathematicae
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Pierre Michel (1975)
Publications mathématiques et informatique de Rennes
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Walter H. Gottschalk (1964)
Annales de l'institut Fourier
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Aikawa, Hiroaki (1993)
Annales Academiae Scientiarum Fennicae. Series A I. Mathematica
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Zaslavski, Alexander J. (2002)
Abstract and Applied Analysis
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Ferry Kwakkel (2011)
Fundamenta Mathematicae
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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.
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...
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...
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.
Khalil, R. (2002)
Rendiconti del Seminario Matematico
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Basilio Messano, Antonio Zitarosa (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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We introduce generalized dynamical systems (including both dynamical systems and discrete dynamical systems) and give the notion of minimal set of a generalized dynamical system. Then we prove a generalization of the classical G.D. Birkhoff theorem about minimal sets of a dynamical system and some propositions about generalized discrete dynamical systems.
Wojciech Hyb (1978)
Colloquium Mathematicae
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