The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking...

We extend the recent results from the class $𝒞(I,I)$ of continuous maps of the interval to the class $𝒞(𝕊,𝕊)$ of continuous maps of the circle. Among others, we give a characterization of $\omega$-limit sets and give a characterization of sets of transitive points for these maps.

We prove that two $C^3$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1+\alpha }$ conjugate for some $\alpha >0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty }$ critical circle maps with the same rotation number that are not $C^{1+\beta }$ conjugate for any $\beta >0$. The class of rotation numbers for which such examples exist contains...

Článek je pokračováním našich dřívějších stejnojmenných příspěvků, v nichž jsme vyšetřovali možnosti aplikace různých variant Šarkovského věty o koexistenci periodických bodů a orbit pro intervalová zobrazení na diferenciální rovnice a inkluze. I tentokrát se budeme zabývat stejným problémem, avšak pro zobrazení na kružnici. Na rozdíl od intervalových zobrazení zde totiž mj. nemusí periodické orbity implikovat existenci pevných bodů, což představuje největší překážku. Na druhé straně lze takto rozšířit...

We prove the C¹-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $𝕋$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $𝕋$, and Bohr if it is recurrent for all products of rotations on $𝕋$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which...

Soit $r\ge 1$ un réel. Ici, on étudie les homéomorphismes du cercle qui sont de classe $P$$C^r$ par morceaux et de nombres de rotation irrationnels. On caractérise ceux qui sont $C^r$ par morceaux conjugués à des $C^r$-difféomorphismes. Comme conséquence, on obtient un critère de conjugaison...

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${𝕊}^1$, that is, families $ℱ=\{F^v{𝕊}^1⟶{𝕊}^{1}v\in V\}$ of homeomorphisms such that $$F^{v_1}\circ F^{v_2}=F^{v_1+v_2},v_1,v_2\in V,$$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathrm{c}ardV>1$) abelian group).

A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$, $h_T\left(f\right)$, is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_T\left(f\right)=0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric...