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### Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Fundamenta Mathematicae

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${\left({X}_{i}\right)}_{i=1}^{\infty }$ and a sequence of continuous maps ${\left({f}_{i}\right)}_{i=1}^{\infty }$, ${f}_{i}:{X}_{i}\to {X}_{i+1}$, is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of ${f}_{n}○...○{f}_{2}○{f}_{1}$. As an application we construct a large class of smooth triangular maps of the square of type ${2}^{\infty }$ and positive...

### Topological size of scrambled sets

Colloquium Mathematicae

A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has $limin{f}_{n\to \infty }d\left(fⁿ\left(x\right),fⁿ\left(y\right)\right)=0$ and $limsu{p}_{n\to \infty }d\left(fⁿ\left(x\right),fⁿ\left(y\right)\right)>0$, d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled...

### Minimal nonhomogeneous continua

Colloquium Mathematicae

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.

### Noninvertible minimal maps

Fundamenta Mathematicae

For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and ${f}^{-1}\left(A\right)$ share with A those topological properties which describe how large a set is. Using these results...

### All solenoids of piecewise smooth maps are period doubling

Fundamenta Mathematicae

We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if ${p}_{1}<...<{p}_{n}$ is a periodic orbit of a continuous map f then there is a union set ${q}_{1},...,{q}_{n-1}$ of some periodic orbits of f such that ${p}_{i}<{q}_{i}<{p}_{i+1}$ for any i.

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