Erratum to ``Multifractal spectra of Birkhoff averages for a piecewise monotone interval map" (Fund. Math. 208 (2010), 95–121)

2011 ◽  
Vol 214 (2) ◽  
pp. 199-200
Author(s):  
Franz Hofbauer
Keyword(s):  
Piecewise Monotone ◽  
Interval Map ◽  
Multifractal Spectra

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.

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.

Study of the structural degradation of steel 15Kh5M upon continuous duty

2020 ◽  
Vol 86 (1) ◽  
pp. 38-43
Author(s):  
Vladimir A. Kim ◽  
Valeriya V. Lysenko ◽  
Anna A. Afanaseva ◽  
Khasan I. Turkmenov
Keyword(s):  
Grain Boundaries ◽  
Structural Changes ◽  
Structural Transformations ◽  
Structural Degradation ◽  
Structural Arrangement ◽  
Dispersed Particles ◽  
Complex Process ◽  
Multifractal Spectra ◽  
Steel 15Kh5m ◽  
Complex Indicators

Structural degradation of the material upon long-term thermal and force impacts is a complex process which includes migration of the grain boundaries, diffusion of the active elements of the external and technological environment, hydrogen embrittlement, aging, grain boundary corrosion and other mechanisms. Application of the fractal and multifractal formalism to the description of microstructures opens up wide opportunities for quantitative assessment of the structural arrangement of the material, clarifies and reveals new aspects of the known mechanisms of structural transformations. Multifractal parameterization allows us to study the processes of structural degradation from the images of microstructures and identify structural changes that are hardly distinguishable visually. Any quantitative structural indicator can be used to calculate the multifractal spectra of the microstructure, but the most preferable is that provides the maximum range of variation in the numerical values of the multifractal components. The results of studying structural degradation of steel 15Kh5M upon continuous duty are presented. It is shown that structural degradation of steel during operation under high temperatures and stresses is accompanied by enlargement of the microstructural objects, broadening of the grain boundaries and allocation of the dispersed particles which are represented as point objects in the images. The processes of structural degradation lead to an increase in the range of changes in the components of the multifractal spectra. High values of complex indicators of structural arrangement indicate to an increase in heterogeneity and randomness at the micro-scale level, but at the same time, to manifestation of the ordered combinations of individual submicrostructures. Those structural transformations adapt the material to external impacts and provide the highest reliability and fracture resistance of the material.

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