scholarly journals Thermodynamic formalism for transient dynamics on the real line

Nonlinearity ◽  
2022 ◽  
Vol 35 (2) ◽  
pp. 1093-1118
Author(s):  
M Gröger ◽  
J Jaerisch ◽  
M Kesseböhmer
Keyword(s):  
Real Line ◽  
Thermodynamic Formalism ◽  
Dimensional Model ◽  
Limit Sets ◽  
New Formalism ◽  
One Dimensional ◽  
The Real ◽  
Transient Behaviour ◽  
Interval Maps ◽  
Full Dimension

Abstract We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic Z -extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the full dimension spectrum with respect to α-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (02) ◽  
pp. 427-435 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Real Line ◽  
Random Variables ◽  
Random Interval ◽  
Space Filling ◽  
One Dimensional ◽  
The Real ◽  
Fixed Constant ◽  
Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx , the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (03) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

The Unruh effect without space–time

2020 ◽  
Vol 17 (04) ◽  
pp. 2050057
Author(s):  
Michele Arzano
Keyword(s):  
Real Line ◽  
Thermal Effects ◽  
Thermal State ◽  
Space Time ◽  
Unruh Effect ◽  
Affine Transformations ◽  
One Dimensional ◽  
The Real ◽  
Simple Quantum ◽  
The One

We show how the characteristic thermal effects found for a quantum field in space–time geometries admitting a causal horizon can be found in a simple quantum system living on the real line. The analysis we present is essentially group theoretic in nature: a thermal state emerges naturally when comparing representations of the group of affine transformations of the real line. The freedom in the choice of different notions of translation generators is the key to the one-dimensional Unruh effect we describe.

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

A one dimensional random space-filling problem

1968 ◽  
Vol 5 (2) ◽  
pp. 427-435 ◽  
Author(s):  
John P. Mullooly
Keyword(s):  
Asymptotic Behavior ◽  
Finite Number ◽  
Real Line ◽  
Random Variables ◽  
Random Interval ◽  
Space Filling ◽  
One Dimensional ◽  
The Real ◽  
Fixed Constant ◽  
Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx, the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (3) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

On absolutely nonmeasurable homomorphisms of commutative groups

2013 ◽  
Vol 50 (3) ◽  
pp. 287-295
Author(s):  
A. Kharazishvili
Keyword(s):  
Real Line ◽  
Quotient Group ◽  
Torsion Subgroup ◽  
Dimensional Torus ◽  
One Dimensional ◽  
The Real ◽  
The One

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.

scholarly journals The one-dimensional heat equation in the Alexiewicz norm

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Heat Equation ◽  
Real Line ◽  
Distributional Derivative ◽  
One Dimensional ◽  
The Real ◽  
Space Of Distributions ◽  
The One

AbstractA distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ

scholarly journals The Classical Continuum without Points

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Current Practice ◽  
Classical Analysis ◽  
One Dimensional ◽  
The Real ◽  
Actual Infinity ◽  
Archimedean Property ◽  
Weak Set Theory

This chapter develops a “semi-Aristotelian” account of a one-dimensional continuum. Unlike Aristotle, it makes significant use of actual infinity, in line with current practice. Like Aristotle, this account does not recognize points, at least not as parts of regions in the space. The formal background is classical mereology together with a weak set theory. The chapter proves an Archimedean property, and establishes an isomorphism with the Dedekind–Cantor structure of the real line. It also compares the present framework to other point-free accounts, establishing consistency relative to classical analysis.

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