Hyperbolic Measure Research Articles

Shadowing for nonuniformly hyperbolic maps in Hilbert spaces

We prove a shadowing lemma for nonuniformly hyperbolic maps in Hilbert spaces. As applications, we firstly prove that the positive Lyapunov exponents of a hyperbolic ergodic measure µ can be approximated by positive Lyapunov exponents of atomic measures on hyperbolic periodic orbits; secondly, give an upper estimation of metric entropy by using the exponential growth rate of the number of such periodic points that their atomic measures approximate µ and their positive Lyapunov exponents approximate the positive Lyapunov exponents of µ.

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

Data Envelopment Analysis and hyperbolic efficiency measures: Extending applications and possibilities for between-group comparisons

Data Envelopment Analysis (DEA) is a widely used tool to estimate relative efficiencies. However, this methodology encounters issues when between-group comparisons with variable returns of scale (VRS) are of interest. In this specification efficiency scores can be undefined when the projection of the efficiency estimation of an observation from one group does not have an intersection with the frontier of the other group. To address this issue, a hyperbolic efficiency estimation can be used. Up to now, the literature lacks consideration of weight restrictions and non-discretionary variables in the hyperbolic DEA method. Weight restrictions allow redefining the production possibility set (PPS), while non-discretionary variables adjust the model’s orientation. Considering these factors, it is possible to incorporate prior knowledge, substitution effects, and increase discriminatory power. This paper introduces the respective mathematical formulations and the practical implementation in statistical software. The estimation methods have been combined in an R-package called hyperbolicDEA in order to facilitate the application of hyperbolic DEA with weight restrictions, non-discretionary variables, and additional functionalities. An empirical example of fish farms illustrates the advantages of the introduced methodologies and the functionalities of the R-package.

Lyapunov exponents and hyperbolic periodic orbits for endomorphisms

We show that Lyapunov exponents of hyperbolic ergodic measures of C2-endomorphisms are approximated by the Lyapunov exponents of hyperbolic measures on periodic orbits.

Hyperbolic Sine Similarity Measure of SVNS for Open-Pit Mine Slope Stability Classification and Assessment

Hyperbolic Sine Similarity Measure of SVNS for Open-Pit Mine Slope Stability Classification and Assessment

Dimension approximation in smooth dynamical systems

AbstractFor a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ , then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$ . The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $ .

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.

MAGDM approach using the hyperbolic sine similarity measure of confidence neutrosophic number credibility sets in a fuzzy credibility multi-valued circumstance

Group decision-making is that individuals collectively make a choice from a set of alternatives. Then, in complex decision-making problems, the decision-making process is no longer subject to a single individual, but group decision-making. Hence, the decision reliability and credibility of the collective evaluation information become more critical. However, current decision-making methods lack the confidence level and credibility measure of group evaluation information. To ensure the confidence level and credibility measure of small-scale group decision-making problems, the aim of this paper is to propose a Multi-Attribute Group Decision-Making (MAGDM) approach using a hyperbolic sine similarity measure between Confidence Neutrosophic Number Credibility Sets (CNNCSs) in the circumstance of Fuzzy Credibility Multi-Valued Sets (FCMVSs). To achieve this aim, this paper contains the following works. First, we present FCMVS to represent the mixed information of fuzzy sequences and credibility degree sequences with different and/or identical fuzzy values. Second, according to the normal distribution and confidence level of fuzzy values and credibility degrees in FCMVS, FCMVS is transformed into CNNCS to avoid the operational issue between different fuzzy sequence lengths in FCMVSs and to ensure the confidence neutrosophic numbers/confidence intervals of fuzzy values and credibility degrees. Third, a hyperbolic sine similarity measure of CNNCSs is established in the circumstance of FCMVSs. Fourth, a MAGDM approach is developed based on the weighted hyperbolic sine similarity measure in the circumstance of FCMVSs. Fifth, the proposed MAGDM approach is applied to an actual example of the equipment supplier choice problem to illustrate the efficiency and rationality of the proposed MAGDM approach in a FCMVS circumstance. In general, this study reveals new contributions in the representation, transformation method, and similarity measure of small-scale group assessment information, as well as the proposed MAGDM method subject to the normal distribution and confidence levels in small-scale MAGDM scenarios.

A Faber–Krahn inequality for Wavelet transforms

AbstractFor some special window functions , we prove that, over all sets of fixed hyperbolic measure , those for which the Wavelet transform with window concentrates optimally are exactly the discs with respect to the pseudo‐hyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler in Abreu and Dörfler (Inverse Problems 28 (2012) 16). Our techniques make use of a framework recently developed by Nicola and Tilli in Nicola and Tilli (Invent. Math. 230 (2022) 1–30), but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis.

Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

AbstractWe extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$ . We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

Development of entropy measure for selecting highly sensitive WPT band to identify defective components of an axial piston pump

This paper proposes a novel tangent hyperbolic fuzzy entropy measure-based method for determining the highly (most) sensitive frequency band to easily identify defective components in an axial piston pump. The validation of proposed scheme was done by experimental and simulation analysis. In the proposed method, initially, the raw vibration signal is decomposed into 16 frequency bands. Second, the energy of the WPT band is computed. Third, lower bounds of energy readings at the 4th level of decomposition are extracted, and thereafter they are normalised and finally converted into the forms of fuzzy sets. Following that, the proposed measure's value is computed. The proposed measure is analogous to the quantity of available information. The WPT4 band with the highest measure value will have the most information and can be designated as sensitive. Finally, envelope demodulation of the sensitive band is computed to identify the defects by evaluating the magnitude of vibration at fundamental frequencies. The superiority of the proposed sensitive band selection technique has been justified by comparing the findings with the widely used Fast-Kurtogram, Protugram, and Spectral-Kurtosis methods. The authenticity of the proclaimed tangent hyperbolic fuzzy entropy measure is accomplished with the existing Deluca and Termini fuzzy entropy measures. The latter entropy measure exhibited some inconsequential and unspecified results and established the superiority of our defect identification methodology.

A fractal uncertainty principle for Bergman spaces and analytic wavelets

Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined as a union of annuli that are equidistributed in the hyperbolic measure. The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a two-sided bound for the norm of a localization operator involving the fractal dimension log⁡2/log⁡3 in the exponent. As in the STFT case and in Dyatlov's fractal uncertainty principle, the (hyperbolic) measure of the dilated iterates of the Cantor set in the disk tends to infinity, while the corresponding norm of the localization operator tends to zero.

Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

For a C1+α diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μ can be approximated by a sequence of hyperbolic sets {Λn}n≥1. In this paper, we prove that the Hausdorff dimension for Λn on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of μ can be approximated by the Hausdorff dimension of Λn.To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of −ψs(⋅,fn) and the properties of the uniformly hyperbolic dynamical systems.

Correction to: Horseshoes for diffeomorphisms preserving hyperbolic measures

Correction to: Horseshoes for diffeomorphisms preserving hyperbolic measures

Bicomplex Version of Lebesgue’s Dominated Convergence Theorem and Hyperbolic Invariant Measure

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue’s dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.

A Brain Tumor Image Segmentation Method Based on Quantum Entanglement and Wormhole Behaved Particle Swarm Optimization.

PurposeAlthough classical techniques for image segmentation may work well for some images, they may perform poorly or not work at all for others. It often depends on the properties of the particular image segmentation task under study. The reliable segmentation of brain tumors in medical images represents a particularly challenging and essential task. For example, some brain tumors may exhibit complex so-called “bottle-neck” shapes which are essentially circles with long indistinct tapering tails, known as a “dual tail.” Such challenging conditions may not be readily segmented, particularly in the extended tail region or around the so-called “bottle-neck” area. In those cases, existing image segmentation techniques often fail to work well.MethodsExisting research on image segmentation using wormhole and entangle theory is first analyzed. Next, a random positioning search method that uses a quantum-behaved particle swarm optimization (QPSO) approach is improved by using a hyperbolic wormhole path measure for seeding and linking particles. Finally, our novel quantum and wormhole-behaved particle swarm optimization (QWPSO) is proposed.ResultsExperimental results show that our QWPSO algorithm can better cluster complex “dual tail” regions into groupings with greater adaptability than conventional QPSO. Experimental work also improves operational efficiency and segmentation accuracy compared with current competing reference methods.ConclusionOur QWPSO method appears extremely promising for isolating smeared/indistinct regions of complex shape typical of medical image segmentation tasks. The technique is especially advantageous for segmentation in the so-called “bottle-neck” and “dual tail”-shaped regions appearing in brain tumor images.

A novel hyperbolic intuitionistic fuzzy divergence measure based mammogram enhancement for visual elucidation of breast lesions

In this article, we have proposed a novel mammogram image enhancement approach for visual interpretation of breast masses using an adaptive intuitionistic fuzzy divergent measure under hyperbolic regularization. The proposed scheme pursuits to raise the underexposed and abnormal growth of cells such as breast masses, deformity tissues, nodules, and lumps in mammogram images. In the first step, a complementary image has been generated from the source mammogram to separate object and background area. Thereafter, both source and background images are mapped into Intuitionistic Fuzzy Set (IFS) plane under Hyperbolic Regularization (HR). A novel entropy based intuitionistic fuzzy divergence measure has been designed using hyperbolic function for the membership modification in this work. In addition, fuzzy ambiguity correction distance function has been obtained from the hesitant score vectors from both source and complementary images. Werner’s AND-OR has been utilized on both of them to generate modified membership function. Finally, we have achieved enhanced mammogram with visually improved breast masses through defuzzification process. It has been shown that the proposed method provides better result in comparison with the existing methods in terms of qualitative and higher score of image quantity measurement metrics such as SSIM (0.83), FQI (0.85), and IFQI (0.88).

MCGDM Approach Using the Weighted Hyperbolic Sine Similarity Measure of Neutrosophic (Indeterminate Fuzzy) Multivalued Sets for the Teaching Quality Assessment of Teachers

MCGDM Approach Using the Weighted Hyperbolic Sine Similarity Measure of Neutrosophic (Indeterminate Fuzzy) Multivalued Sets for the Teaching Quality Assessment of Teachers

Potential Theory on Gromov Hyperbolic Spaces

Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.