On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

1979 ◽  
Vol 34 (3) ◽  
pp. 213-237 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Positive Entropy ◽  
Piecewise Monotonic

ENTROPY OF EQUILIBRIUM MEASURES OF CONTINUOUS PIECEWISE MONOTONIC MAPS

2004 ◽  
Vol 04 (01) ◽  
pp. 85-94 ◽  
Author(s):  
JÉRÔME BUZZI
Keyword(s):  
Dynamical Systems ◽  
Positive Entropy ◽  
Unique Equilibrium ◽  
Mixing Properties ◽  
Equilibrium Measures ◽  
Thermodynamical Formalism ◽  
Transfer Operators ◽  
Piecewise Monotonic

Considering the thermodynamical formalism of dynamical systems, P. Walters showed that for β-transformations all Lipschitz weights define quasi-compact transfer operators and therefore unique equilibrium measures which additionally have positive entropy and good mixing properties. In this note we generalize this to continuous piecewise monotonic maps of the interval. The case of piecewise monotonic maps with discontinuities remains open.

Model sets with positive entropy in Euclidean cut and project schemes

2019 ◽  
Vol 52 (5) ◽  
pp. 1073-1106 ◽  
Author(s):  
Tobias JÄGER ◽  
Daniel LENZ ◽  
Christian OERTEL
Keyword(s):  
Positive Entropy ◽  
Model Sets

scholarly journals The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms

2020 ◽  
pp. 1-26
Author(s):  
SNIR BEN OVADIA
Keyword(s):  
Symbolic Dynamics ◽  
Full Measure ◽  
Probability Measures ◽  
Positive Entropy ◽  
Surface Diffeomorphisms ◽  
Srb Measures ◽  
Hyperbolic Diffeomorphisms ◽  
Set Of Points ◽  
Math Phys ◽  
Homoclinic Classes

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

scholarly journals The choice of a thermodynamic formulation dramatically affects modelled chemical zoning in minerals

2021 ◽  
Author(s):  
Lucie Tajcmanova ◽  
Yury Podladchikov ◽  
Evangelos Moulas
Keyword(s):  
Mantle Convection ◽  
Diffusion Flux ◽  
Chemical Diffusion ◽  
Irreversible Processes ◽  
Dynamic Processes ◽  
Positive Entropy ◽  
Admissibility Condition ◽  
Chemical Zoning ◽  
Natural Processes ◽  
The Earth

<p>Quantifying natural processes that shape our planet is a key to understanding the geological observations. Many phenomena in the Earth are not in thermodynamic equilibrium. Cooling of the Earth, mantle convection, mountain building are examples of dynamic processes that evolve in time and space and are driven by gradients. During those irreversible processes, entropy is produced. In petrology, several thermodynamic approaches have been suggested to quantify systems under chemical and mechanical gradients. Yet, their thermodynamic admissibility has not been investigated in detail. Here, we focus on a fundamental, though not yet unequivocally answered, question: which thermodynamic formulation for petrological systems under gradients is appropriate – mass or molar?  We provide a comparison of both thermodynamic formulations for chemical diffusion flux, applying the positive entropy production principle as a necessary admissibility condition. Furthermore, we show that the inappropriate solution has dramatic consequences for understanding the key processes in petrology, such as chemical diffusion in the presence of stress gradients.</p>

scholarly journals Peak Estimation of Univariate Spectraby the Best L1 Piecewise Monotonic Approximation Method

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Magnetic Resonance ◽  
Least Squares ◽  
Approximation Method ◽  
Nmr Spectra ◽  
Inorganic Compound ◽  
Piecewise Monotonic ◽  
Inherent Problem ◽  
New Algorithms ◽  
Nuclear Magnetic

We consider applications of the best L1 piecewise monotonic approximation method for the peak estimation of three sets of up to 2500 measurements of Raman, Infrared and Nuclear Magnetic Resonance (NMR)spectra. Peak estimation is an inherent problem of spectroscopy. The location of peaks and their intensities arethe signature of a sample of an organic or an inorganic compound. The diversity and the complexity of our measurements makes it a difficult test of the effectiveness of the method. We find that the method identifies efficientlypeaks and we compare to the results obtained by the analogous least squares calculations. These results havemany similarities and occasionally considerable differences due to both properties of the norms employed in theoptimization calculations and nature of the spectra. Our results may be helpful to subject analysts as part of theinformation on which decisions will be made for estimating peaks in sequences of spectra and to the developmentof new algorithms that are particularly suitable for peak estimation calculations.

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