scholarly journals Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

2021 ◽  
pp. 1-37
Author(s):  
WAEL BAHSOUN ◽  
CARLANGELO LIVERANI
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Absolute Continuity ◽  
Hyperbolic Systems ◽  
Transfer Operator ◽  
Statistical Properties ◽  
Functions Of Bounded Variation ◽  
Expanding Maps ◽  
New Information ◽  
Piecewise Expanding Maps

Abstract Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

scholarly journals The Dolgopyat inequality in bounded variation for non-Markov maps

2017 ◽  
Vol 18 (02) ◽  
pp. 1850006
Author(s):  
Henk Bruin ◽  
Dalia Terhesiu
Keyword(s):  
Bounded Variation ◽  
Transfer Operator ◽  
Regularity Conditions ◽  
Functions Of Bounded Variation ◽  
Interval Map ◽  
Roof Function

Let [Formula: see text] be a (non-Markov) countably piecewise expanding interval map satisfying certain regularity conditions, and [Formula: see text] the corresponding transfer operator. We prove the Dolgopyat inequality for the twisted operator [Formula: see text] acting on the space BV of functions of bounded variation, where [Formula: see text] is a piecewise [Formula: see text] roof function.

scholarly journals Inducing schemes for multi-dimensional piecewise expanding maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Peyman Eslami
Keyword(s):  
Dynamical Systems ◽  
Statistical Properties ◽  
Expanding Maps ◽  
Return Times ◽  
Return Map ◽  
Exponential Tails ◽  
Inducing Schemes ◽  
Piecewise Expanding Maps

<p style='text-indent:20px;'>We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.</p>

Weak Gibbs measures for certain non-hyperbolic systems

2000 ◽  
Vol 20 (5) ◽  
pp. 1495-1518 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Gibbs Measures ◽  
Periodic Points ◽  
Statistical Properties ◽  
Equilibrium States ◽  
Pointwise Dimension ◽  
Absolutely Continuous ◽  
Higher Dimensional

We study a weak Gibbs property of equilibrium states for potentials of weak bounded variation and for maps admitting indifferent periodic points. We further establish statistical properties of the weak Gibbs measures and bounds of their pointwise dimension. We apply our results to higher-dimensional maps (which are not necessarily conformal) with indifferent periodic points and show that their absolutely continuous finite invariant measures are weak Gibbs measures.

scholarly journals Robustness of ergodic properties of non-autonomous piecewise expanding maps

2017 ◽  
Vol 39 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
MATTEO TANZI ◽  
TIAGO PEREIRA ◽  
SEBASTIAN VAN STRIEN
Keyword(s):  
Invariant Mass Distribution ◽  
Hyperbolic Systems ◽  
Memory Loss ◽  
Time Dependent ◽  
Usual Sense ◽  
Expanding Maps ◽  
Ergodic Properties ◽  
Stochastically Stable ◽  
Autonomous Dynamical Systems ◽  
Piecewise Expanding Maps

Recently, there has been an increasing interest in non-autonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map$F$results in statistical behaviour close to that of$F$. We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of time-dependent perturbations (arbitrarily—rather than randomly—chosen at each step) close to a given map$F$remains close to the invariant mass distribution of$F$. Moreover, for almost every point, Birkhoff averages along trajectories do not fluctuate wildly. This result complements recent results on memory loss for non-autonomous dynamical systems.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

scholarly journals Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 990
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Functions Of Bounded Variation ◽  
Functional Applications ◽  
Čebyšev Functional

In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided.

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