Related papers: Dense chaos for continuous interval maps
Based on newly discovered properties of the shift map (Theorem 1), we believe that chaos should involve not only nearby points can diverge apart but also faraway points can get close to each other. Therefore, we propose to call a continuous…
In this paper, various chaotic properties and their relationships for interval maps are discussed. It is shown that the proximal relation is an equivalence relation for any zero entropy interval map. The structure of the set of…
It is known that the topological entropy of a continuous interval map $f$ is positive if and only if the type of $f$ for Sharkovskii's order is $2^d p$ for some odd integer $p\ge 3$ and some $d\ge 0$; and in this case the topological…
We prove that the set of maps which exhibit distributional chaos of type 1 (DC1) is $C^0$-dense in the space of continuous self-maps of given any compact topological manifold (possibly with boundary).
A random chaotic interval map with noise which causes coarse-graining induces a finite-state Markov chain. For a map topologically conjugate to a piecewise-linear map with the Lebesgue measure being ergodic, we prove that the Shannon…
We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. We show, among others, that if $f$ is chaotic in…
We give a hierarchy of many-parameter families of maps of the interval [0,1] with an invariant measure and using the measure, we calculate Kolmogorov--Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps…
We give a unified proof of the existence of turbulence for some classes of continuous interval maps which include, among other things, maps with periodic points of odd periods > 1, some maps with dense chain recurrent points and densely…
Let $\phi(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $\phi(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \infty]$ onto itself which also maps the set of irrational points…
We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called P-Lipschitz maps (where P is a finite invariant set) we give an upper…
Maps $f,g\colon I\to I$ are called strongly commuting if $f\circ g^{-1}=g^{-1}\circ f$. We show that strongly commuting, piecewise monotone maps $f,g$ can be decomposed into a finite number of invariant intervals (or period 2 intervals) on…
Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…
We give hierarchy of one-parameter family F(a,x) of maps of the interval [0,1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps…
In their celebrated "Period three implies chaos" paper, Li and Yorke proved that if a continuous interval map f has a period 3 point then there is an uncountable scrambled set S on which f has very complicated dynamics. One question arises…
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic…
We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirb\'ak, \v{L}. Snoha, V. \v{S}pitalsk\'y [Ergodic…
A continuous map $f$ on a compact metric space $X$ induces in a natural way the map $\tilde f$ on the hyperspace $\mathcal K(X)$ of all closed non-empty subsets of $X$. We study the question of transmission of chaos between $f$ and $\tilde…
Our context is Filippov systems defined on two-dimensional manifolds having a finite number of tangency points. We prove that topological transitivity is a necessary and sufficient condition for the occurrence of non-deterministic chaos…
We give a definition of chaos for a continuous self-map of a general topological space. This definition coincides with the Devanney definition for chaos when the topological space happens to be a metric space. We show that in a uniform…
In this paper, we first prove that the topological entropy of induced map of any distal homeomorphism of a compact metric space is null. Then we consider induced map $2^f$ of an arbitrary pointwise periodic homeomorphism $f:X\to X$ of a…