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Metric Entropy of Nonautonomous Dynamical Systems

Christoph Kawan (2014)

Nonautonomous Dynamical Systems

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn, μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion...

Minimal actions of homeomorphism groups

Yonatan Gutman (2008)

Fundamenta Mathematicae

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on Φ ⊂ 2 2 X , the space of maximal chains in 2 X , equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on U H o m e o ( X ) , the universal minimal space of Homeo(X), is not transitive (improving...

Minimal models for ℤ d -actions

Bartosz Frej, Agata Kwaśnicka (2008)

Colloquium Mathematicae

We prove that on a metrizable, compact, zero-dimensional space every ℤ d -action with no periodic points is measurably isomorphic to a minimal ℤ d -action with the same, i.e. affinely homeomorphic, simplex of measures.

Minimal nonhomogeneous continua

Henk Bruin, Sergiǐ Kolyada, L'ubomír Snoha (2003)

Colloquium Mathematicae

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.

Minimal non-invertible transformations of solenoids

Dariusz Tywoniuk (2012)

Colloquium Mathematicae

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.

Minimal number of periodic points for smooth self-maps of S³

Grzegorz Graff, Jerzy Jezierski (2009)

Fundamenta Mathematicae

Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant D r m [ f ] , introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate D ³ r [ f ] for all self-maps of S³.

Minimal periods of maps of rational exterior spaces

Grzegorz Graff (2000)

Fundamenta Mathematicae

The problem of description of the set Per(f) of all minimal periods of a self-map f:X → X is studied. If X is a rational exterior space (e.g. a compact Lie group) then there exists a description of the set of minimal periods analogous to that for a torus map given by Jiang and Llibre. Our approach is based on the Haibao formula for the Lefschetz number of a self-map of a rational exterior space.

Minimal, rigid foliations by curves on ℂ ℙ n

Frank Loray, Julio C. Rebelo (2003)

Journal of the European Mathematical Society

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂ ℙ n for every dimension n ≥ 2 and every degree d ≥ 2 . Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d , has a finite singular set and all the regular leaves are dense in the whole of ℂ ℙ n . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation...

Minimal sets of generalized dynamical systems

Basilio Messano, Antonio Zitarosa (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

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