scholarly journals Constant slope maps on the extended real line

2017 ◽  
Vol 38 (8) ◽  
pp. 3145-3169 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Real Line ◽  
Half Line ◽  
Piecewise Monotone ◽  
Bounded Interval ◽  
Interval Map ◽  
Piecewise Continuous ◽  
Constant Slope

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps

1994 ◽  
Vol 14 (4) ◽  
pp. 621-632 ◽  
Author(s):  
V. Baladi ◽  
D. Ruelle
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Constant Weight ◽  
Piecewise Constant ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Interval Map ◽  
Piecewise Continuous

AbstractWe consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

scholarly journals No semiconjugacy to a map of constant slope

2014 ◽  
Vol 36 (3) ◽  
pp. 875-889 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Sufficient Conditions ◽  
Sufficient Criterion ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Piecewise Continuous ◽  
Necessary And Sufficient ◽  
Constant Slope

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 323-332 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Real Line ◽  
Positive Measure ◽  
Schwarzian Derivative ◽  
Large Range ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Central Piece ◽  
Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

scholarly journals The Wiener-Pitt Phenomenon on the Half-Line

1962 ◽  
Vol 13 (1) ◽  
pp. 37-38 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Real Line ◽  
Half Line ◽  
Stieltjes Transform ◽  
The Real

It has been well known for many years (2) that if Fμ(t) is the Fourier-Stieltjes transform of a bounded measure μ on the real line R, which is bounded away from zero, it does not follow that [Fμ(t)]−1 is also the Fourier-Stieltjes transform of a measure. It seems of interest (as was remarked, in conversation, by J. D. Weston) to consider measures on the half-line R+ = [0, ∞[, instead of on R.

PIECEWISE LINEAR MODEL FOR TREE MAPS

2001 ◽  
Vol 11 (12) ◽  
pp. 3163-3169 ◽  
Author(s):  
MATHIEU BAILLIF ◽  
ANDRÉ DE CARVALHO
Keyword(s):  
Linear Model ◽  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Monotone Map ◽  
Piecewise Linear Model ◽  
Constant Slope

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

scholarly journals On spectra of upper Sergeev frequencies of linear differential equations

2019 ◽  
pp. 28-32
Author(s):  
Aliaksei S. Vaidzelevich
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Linear Differential Equation ◽  
Fourth Order ◽  
Real Line ◽  
Half Line ◽  
Integer Number ◽  
Order Linear Differential Equation ◽  
Third Order ◽  
Linear Differential

It is known that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order greater than two with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended real line. Moreover, for the spectra of upper frequencies of third-order equations this result was inverted under the assumption that the spectra contain zero. In the present paper we obtain an inversion of the above statement for equations of the fourth order and higher. Namely, for an arbitrary zero-containing Suslin subset S on the non-negative half-line of the extended real line and a positive integer number n greater than three a n order linear differential equation is constructed, which spectra of the upper Sergeev frequencies of zeros, signs, and roots coincide with the set S.

One-Dimensional Semiconjugacy Revisited

2003 ◽  
Vol 13 (07) ◽  
pp. 1657-1663 ◽  
Author(s):  
J. F. Alves ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Positive Topological Entropy ◽  
Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

Construction of wavelets and framelets on a bounded interval

2018 ◽  
Vol 16 (06) ◽  
pp. 807-849 ◽  
Author(s):  
Bin Han ◽  
Michelle Michelle
Keyword(s):  
Signal Processing ◽  
Boundary Conditions ◽  
Real Line ◽  
Construction Method ◽  
The Real ◽  
Bounded Interval ◽  
Orthogonal Wavelets ◽  
Simple Boundary

Many problems in applications are defined on a bounded interval. Therefore, wavelets and framelets on a bounded interval are of importance in both theory and application. There is a great deal of effort in the literature on constructing various wavelets on a bounded interval and exploring their applications in areas such as numerical mathematics and signal processing. However, many papers on this topic mainly deal with individual examples which often have many boundary wavelets with complicated structures. In this paper, we shall propose a method for constructing wavelets and framelets in [Formula: see text] from symmetric wavelets and framelets on the real line. The constructed wavelets and framelets in [Formula: see text] often have a few simple boundary wavelets/framelets with the additional flexibility to satisfy various desired boundary conditions. To illustrate our construction method, from several spline refinable vector functions, we present several examples of (bi)orthogonal wavelets and spline tight framelets in [Formula: see text] with very simple boundary wavelets/framelets.

Distribution of the smallest visited point in a greedy walk on the line

2016 ◽  
Vol 53 (3) ◽  
pp. 880-887
Author(s):  
Katja Gabrysch
Keyword(s):  
Poisson Process ◽  
Real Line ◽  
Half Line ◽  
The Real ◽  
Independence Properties

AbstractWe consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

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