Minimal Bratteli diagrams and the dimension groups of AF -algebras.
Minimal measures.
Minimal models for -actions
We prove that on a metrizable, compact, zero-dimensional space every -action with no periodic points is measurably isomorphic to a minimal -action with the same, i.e. affinely homeomorphic, simplex of measures.
Minimal models of foliated algebraic surfaces
Minimal nonhomogeneous continua
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
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
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 , 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 for all self-maps of S³.
Minimal periods of maps of rational exterior spaces
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 polynomials of some beta-numbers and Chebyshev polynomials
Minimal, rigid foliations by curves on
We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space for every dimension and every degree . Precisely, we construct a foliation which is induced by a homogeneous vector field of degree , has a finite singular set and all the regular leaves are dense in the whole of . Moreover, satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if is conjugate to another holomorphic foliation...
Minimal sequences and the Kadison-Singer problem.
Minimal sets of foliations on complex projective spaces
Minimal sets of generalized dynamical systems
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.
Minimal sets of non-resonant torus homeomorphisms
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 rotation...
Minimal surfaces via loop groups.
Minimal systems and distributionally scrambled sets
In this paper we investigate numerous constructions of minimal systems from the point of view of -chaos (but most of our results concern the particular cases of distributional chaos of type and ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results answer...
Minimal tori in S4.
Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers
Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the...
Minimizers of Dirichlet functionals on the -torus and the weak KAM theory
Minimizing convex functions by continuous descent methods.