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.

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ{\mathbb{P}}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. 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 $ℂ{\mathbb{P}}^n$. 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...

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.

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...

In this paper we investigate numerous constructions of minimal systems from the point of view of $({ℱ}_1,{ℱ}_2)$-chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$-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...

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...

For continuous maps of an interval into itself we consider cycles (periodic orbits) that are non-reducible in the sense that there is no non-trivial partition into blocks of consecutive points permuted by the map. Among them we identify the miror ones. They are those whose existence does not imply existence of other non-reducible cycles of the same period. Moreover, we find minor patterns of a given period with minimal entropy.

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if $f_{{\lambda }_0}$ is critically finite with non-degenerate critical point $c_1\left({\lambda }_0\right),...,c_n\left({\lambda }_0\right)$ such that $f_{{\lambda }_0}^{k_i}\left(c_i\left({\lambda }_0\right)\right)=p_i\left({\lambda }_0\right)$ are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31  $\lambda \mapsto (f_{\lambda }^{k_1}\left(c_1\left(\lambda \right)\right)-p_1\left(\lambda \right),...,f_{\lambda }^{k_{d-2}}\left(c_{d-2}\left(\lambda \right)\right)-p_{d-2}\left(\lambda \right))$ is a local diffeomorphism...

A mixed formulation is given for elastic problems. Existence and uniqueness of the discretized problem are given for conformal continuous interpolations for the stress tensor components and for the components of the displacement vector. A counterpart of the problem is discussed in the case of an even-dimensional Euclidean space with an associated Hamiltonian vector field and the Poisson structure. For conformal interpolations of the same order the question remains open.

A countable group Γ has the Haagerup approximation property if and only if the mixing actions are dense in the space of all actions of Γ.