Detecting topological transitivity of piecewise monotone interval maps

Through kneading theory, developed by Milnor and Thurston, we present an algorithm which enables us to detect the topological transitivity of a relevant class of piecewise monotone interval maps.     

Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps

We prove that for continuous maps on the interval, the existence of an -cycle implies the existence of points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.

     

Absolutely continuous invariant measures for expanding piecewise linear maps

We prove the existence of absolutely continuous invariant measures for arbitrary expanding piecewise linear maps on bounded polyhedral domains in Euclidean spaces ℝ ^d . Oblatum 6-V-1999 & 8-VI-2000?Published online: 11 October 2000     

Isochronicity of a class of piecewise continuous oscillators

Motivated by a classical pendulum clock model suggested by Andrade in 1920, we study the equation and prove that for a nonlinear analytic the origin is never an isochronous focus or an isochronous center.

     

Waterman classes and piecewise monotone approximations

We obtain a necessary and sufficient condition for a class of functions with a given estimate of the decreasing rate of their piecewise monotone approximations to be embedded into the Waterman class (the class of functions with bounded Λ-variation).     

Cutting sets of continuous functions on the unit interval

For a function $f:[0,1]\to \mathbb{R}$, we consider the set $E\left(f\right)$ of points at which $f$ cuts the real axis. Given $f:[0,1]\to \mathbb{R}$ and a Cantor set $D?[0,1]$ with $\{0,1\}?D$, we obtain conditions equivalent to the conjunction $f\in C[0,1]$ (or $f\in C^{\infty }[0,1]$) and $D?E\left(f\right)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E\left(f\right)$ is a closed nowhere dense subset of $f^{?1}\left[\left\{0\right\}\right]$. Additionally, if $Int{f}^{?1}\left[\left\{0\right\}\right]=0?$, each $x\in \{0,1\}\cap E\left(f\right)$ is an accumulation point of $E\left(f\right)$. Our main result states that, for a closed nowhere dense set $F?[0,1]$ with each $x\in \{0,1\}\cap F$ being an accumulation point of $F$, there exists $f\in C^{\infty }[0,1]$ such that $F=E\left(f\right)=f^{?1}\left[\left\{0\right\}\right]$.     

Schauder-type expansions of continuous functions on the unit interval

Let the space of continuous functions on [0, 1] which vanish at 0 be denoted by C. It will be shown that for any complete orthonormal set of functions {α_i(s)} of bounded variation and such that α_i(1) = 0, there is a simply described linear combination of the continuous functions {∝₀^tα_i(s) ds} which converges uniformly to x(t) for almost all x ε C (“almost all” in the sense of Wiener measure).     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Seven kinds of monotone maps

Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalized convexity properties of the underlying function. In this way, pure first-order characterizations of various types of generalized convex functions are obtained.     

Characterizations of generalized monotone maps

This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.     

Equivalent conditions involving common fixed points for maps on the unit interval

Let be a continuous self-map of the unit interval . Equivalent conditions are given to ensure that has a common fixed point with every continuous map that commutes with on a suitable subset of . This extends a recent result of Gerald Jungck.

     

An exact formula for the measure dimensions associated with a class of piecewise linear maps

An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.Communicated by Michael F. Barnsley.