In the important paper on impulsive systems [K1] several notions are introduced and several properties of these systems are shown. In particular, the function ϕ which describes "the time of reaching impulse points" is considered; this function has many important applications. In [K1] the continuity of this function is investigated. However, contrary to the theorem stated there, the function ϕ need not be continuous under the assumptions given in the theorem. Suitable examples are shown in this paper....

We compare four different notions of chaos in zero-dimensional systems (subshifts). We provide examples showing that in that case positive topological entropy does not imply strong chaos, strong chaos does not imply complicated dynamics at all, and ω-chaos does not imply Li-Yorke chaos.

Several results on stability in impulsive dynamical systems are proved. The first main result gives equivalent conditions for stability of a compact set. In particular, a generalization of Ura's theorem to the case of impulsive systems is shown. The second main theorem says that under some additional assumptions every component of a stable set is stable. Also, several examples indicating possible complicated phenomena in impulsive systems are presented.

We show that all periods of periodic points forced by a pattern for interval maps are preserved for high-dimensional maps if the multidimensional perturbation is small. We also show that if an interval map has a fixed point associated with a homoclinic-like orbit then any small multidimensional perturbation has periodic points of all periods.

This paper is concerned with strong chain recurrence introduced by Easton. We investigate the depth of the transfinite sequence of nested, closed invariant sets obtained by iterating the process of taking strong chain recurrent points, which is a related form of the central sequence due to Birkhoff. We also note the existence of a Lyapunov function which is decreasing off the strong chain recurrent set. As an application, we give a necessary and sufficient condition for the coincidence of the strong...

Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give...

We investigate the nonautonomous periodic system of ODE’s of the form $\dot{x}=\overrightarrow{v}\left(x\right)+r_p\left(t\right)(\overrightarrow{w}\left(x\right)-\overrightarrow{v}\left(x\right))$, where $r_p\left(t\right)$ is a $2p$-periodic function defined by $r_p\left(t\right)=0$ for $t\in \langle 0,p\rangle$, $r_p\left(t\right)=1$ for $t\in (p,2p)$ and the vector fields $\overrightarrow{v}$ and $\overrightarrow{w}$ are related by an involutive diffeomorphism.

A solution of the Feigenbaum functional equation is called a Feigenbaum map. We investigate the likely limit set (i.e. the maximal attractor in the sense of Milnor) of a non-unimodal Feigenbaum map, prove that it is a minimal set that attracts almost all points, and then estimate its Hausdorff dimension. Finally, for every s ∈ (0,1), we construct a non-unimodal Feigenbaum map with a likely limit set whose Hausdorff dimension is s.

Let $f_s$ and $f_t$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_s$ and $f_t$ are periodic and the inverse limit spaces $(I,f_s)$ and $(I,f_t)$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

We introduce the cohomological Conley type index theory for multivalued flows generated by vector fields which are compact and convex-valued perturbations of some linear operators.

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...

If $f:[0,1]\to ℝ$ is strictly increasing and continuous define $T_{f}x=f\left(x\right)(mod1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\epsilon$-close to $T_f$, if $\tilde{T}x=\tilde{f}\left(x\right)(mod1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to ℝ$ with $\parallel \tilde{f}{-f\parallel }_{\infty }<\epsilon$. It is proved that the topological pressure $p(T_f,g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to ℝ$, if and only if $0$ is...

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F=F_{[l,r]}$, l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v⃗}\left(\Phi \right)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max\left(\right|z_l|,|z_r\left|\right)logA$ if $z_l{z}_r\ge 0$ and by $|z_r-z_l|$ in the opposite case, where $z_l=x+ly$, $z_r=x+ry$. We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

Let $K$ be a local field, $a\in K$ and $\varphi :x\to x^p+a$ where $p$ denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system $(K,\varphi )$ are cycles and describe the cycles of this system.

We show that for a finitely generated group of C² circle diffeomorphisms, the entropy of the action equals the entropy of the restriction of the action to the non-wandering set.

We prove some results concerning the entropy of Darboux (and almost continuous) functions. We first generalize some theorems valid for continuous functions, and then we study properties which are specific to Darboux functions. Finally, we give theorems on approximating almost continuous functions by functions with infinite entropy.

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_G$. We consider the map ${\tau }_{\phi }$ defined on X × G by ${\tau }_{\phi }:(x,g)\mapsto (\tau x,\phi \left(x\right)g)$ and the cocycle ${\left(\phi ₙ\right)}_{n\in \mathbb{Z}}$ generated by φ. Using a characterization of the ergodic invariant measures for ${\tau }_{\phi }$, we give the form of the ergodic decomposition of $\mu \left(dx\right)\otimes m_G\left(dg\right)$ or more generally of the ${\tau }_{\phi }$-invariant measures ${\mu }_{\chi }\left(dx\right)\otimes \chi \left(g\right)m_G\left(dg\right)$, where ${\mu }_{\chi }\left(dx\right)$ is χ∘φ-conformal for an exponential χ on G.