Ergodic Maps Research Articles

On the completely irregular set of maps with the shadowing property

We prove that the completely irregular set is Baire generic for every non-uniquely ergodic transitive continuous map which satisfies the shadowing property and acts on a compact metric space without isolated points. We also show that, under the previous assumptions, the orbit of every completely irregular point is dense. Afterwards, we analyze the connection between transitivity and the shadowing property, draw a few consequences of their joint action within the family of expansive homeomorphisms, and discuss several examples to test the scope of our results.

A family of distal functions and multipliers for strict ergodicity

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\Z)$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence in particular does not coincide with $\mathcal{W}$. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within $\mathcal{D}$ or in general, is necessarily a constant. An example of a metric, strictly ergodic, distal flow is constructed which admits a non-strictly ergodic $2$-fold minimal self-joining. It then follows that the enveloping group of this flow is not strictly ergodic (as a $T$-flow). Finally we show that the distal, strictly ergodic Heisenberg nil-flow is relatively disjoint over its largest equicontinuous factor from the universal Weyl flow $|\mathcal{W}|$.

A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics

AbstractFor smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller and Liverani [28, 33], which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results, as well as the differentiability of the variance in quenched central limit theorems (CLTs), for random dynamical systems (RDSs) that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasize that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite-dimensional multiplicative ergodic theory.

Naturally bounded relative motion for formation flying near triangular libration points

This paper proposes a numerical searching and ergodic mappings method to seek the naturally bounded relative orbits near the triangular libration points in the Earth-Moon system. An orbital energy saved numerical continuation method along one axis is proposed and the Poincaré map near the triangular libration points is obtained, based on which two characteristic parameters are defined to generate the naturally bounded relative motions, i.e., the crossing interval period and the reference angle. Then, a numerical searching method is employed to generate an ergodic mapping and a neural network is utilized to realize the transformation from the ergodic mappings to orbital states. By pairing the ergodic mappings of different initial orbital states, the naturally bounded relative motion realized can remain for a long time and does not diverge.

Identification of Nonlinear Discrete Systems From Probability Density Sequences

This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown nonlinear and ergodic map from causal sequences of probability density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all the map’s branches are monotonically increasing. The original solution constructs a matrix estimate of the map’s Frobenius-Perron operator, which governs the evolution of density functions under iteration of the map, from the density sequences. The one-dimensional map is reconstructed from this matrix. In contrast, the proposed solution constructs a higher-order matrix estimate of the Frobenius-Perron operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing map branches and enlarges the class of maps that can be successfully reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum mode characteristics, regardless of the unknown map’s canonicity or branch monotonicity. Numerical examples that illustrate the proposed solution’s characteristics are presented.

Some Results on Invariant Measures for 1-dimensional Maps

For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.

Integration in some new concept of ergodic functions and application to some epidemiological models

In a former work, we introduced a new class of functions, which we call measure pseudo ‐asymptotically omega periodic functions and give some properties of those functions as the topological properties of the space of such functions, the invariance by translation of this space, and the convolution product in this space. In this paper, we consider the same class of functions and give contributions as the stability by the integral operator; we present some examples and composition results. Then we apply the obtained results to investigate the existence of measure pseudo ‐asymptotically omega periodic mild solutions for some nonlinear infinite delay integral equations arising in epidemiology; exactly, we consider the nonlinear infinite delay integral equation: Then we apply these results to a finite delay integral equation when the delay is time dependent

Decay of entropy solutions of a scalar conservation law beyond a stationary ergodic setting

We prove the asymptotic decay of the Cauchy problem for a multidimensional scalar conservation law with initial data being a Besicovitch ergodic functions or the composition of stationary functions with stochastic deformations.

A chaos-based block cipher based on an enhanced logistic map and simultaneous confusion-diffusion operations

Over the years, there has been considerable interest in the area of chaos-based encryption due to the fact that cryptographic algorithms and chaotic maps share a wide-range of similar characteristics. The majority of chaos-based ciphers were designed for encrypting digital images and vast amounts of data. This paper proposes a new chaos-based block cipher algorithm (CBCA) based on an improved logistic chaotic map. To strengthen the performance of the classical logistic chaotic map, a new chaotification method based on a multiplicative inverse function is used which leads to improved properties such as ergodicity and entropy, both of which are desirable for cryptographic applications. The secret key is used to perturb the chaotic variables, and these perturbed variables are used to produce data sequences for the block cipher’s diffusion and confusion structures. The permutation (diffusion) structure is controlled by an ergodic chaotic map, while the substitution (confusion) structure is controlled by the chaotic points themselves. This new ergodicity-based diffusion approach provides higher security and efficiency. The proposed algorithm not only possess good statistical properties provided by the enhanced chaotic map, but it is also key sensitive and uniformly distributed. Statistical evaluation of the proposed algorithm was performed using text and images as inputs to depict its capability to secure various types of media. The performance of the proposed cipher is then compared to other recently proposed algorithms in literature. The performance comparison indicates that CBCA is efficient, secure and that it can be used to encrypt both small and large amounts of data.

On some classes of 1-Lipschitz measure-preserving ergodic functions on ℤp

We study some classes of [Formula: see text]-Lipschitz measure-preserving functions on the group of [Formula: see text]-adic integers [Formula: see text] and then establish necessary and sufficient conditions for their ergodicity by means of their van der Put coefficients.

Genericity of historic behavior for maps and flows

We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive non-hyperbolic diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.

Discrepancy properties and conjugacy classes of interval exchange transformations

Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange transformations with low-discrepancy orbits are known so far and only for n=2,3 intervals, there are criteria to completely characterize those interval exchange transformations. In this paper, it is shown that having low-discrepancy orbits is a conjugacy class invariant under composition of maps. To a certain extent, this approach allows us to distinguish interval exchange transformations with low-discrepancy orbits from those without. For n=4 intervals, the classification is almost complete with the only exceptional case having monodromy invariant rho = (4,3,2,1). This particular monodromy invariant is discussed in detail.

Mahler Coefficients of Some 3-adic Ergodic Functions

Mahler Coefficients of Some 3-adic Ergodic Functions

Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

<p style='text-indent:20px;'>We prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to constants, of the ergodic functions. The results are obtained by means of a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators.

Research on Image-Based Movement Accuracy Monitoring of Aerobics

The movement accuracy monitoring of aerobics is mostly performed through three-dimensional reconstruction of aerobic movements. The feature extraction of aerobics is based on the optimal classification decision function, which extracts all the features of aerobics and thus reduces the accuracy of aerobics monitoring. In order to extract the aerobic motion in the background with higher accuracy, a new image-based monitoring method is proposed. First, the Kinect depth image acquisition method is used to preprocess the image, and then Hog3D is used to extract aerobic movement features and analyze the extraction results. This new method solves the problem of video content classification in aerobics precision monitoring. The Adaboost method in probability statistics is used to identify the accuracy of aerobic motions. This paper uses probability function to link the postures of aerobics and forms an action sequence and its ergodic function to take the maximum value of an aerobic exercise. The accuracy of aerobics is monitored by using the method of level by level proportional example. The experimental results show that this method can effectively improve the accuracy of aerobic track monitoring, reduce the energy consumption of aerobic movement accuracy monitoring, and has good use value.

Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

AbstractWe obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.

Execution examination of chaotic S-box dependent on improved PSO algorithm

Achieving proper nonlinear properties and autocorrelation in the S-box structure is an open challenge in cryptography. Besides, there have been numerous articles on the optimization of S-box, using two types of fitness functions for optimization. This study investigated both types of functions and compares their performance. In addition, this study used ergodic chaotic maps. First, the performance of particle swarm optimization (PSO) was improved using these maps. Then, the new chaotic S-boxes were designed based on the ergodic maps. After that, the improved PSO was used for optimization to obtain the best S-boxes. This optimization was performed once by selecting nonlinearity as a fitness function. At the second optimization, the entropy source was selected as a fitness function for optimization by examining the P-value of the mono-test frequency. Finally, the related results for the introduced chaotic S-boxes were compared to the optimized chaotic S-boxes with two types of fitness functions. The introduced S-boxes were safe due to the use of ergodic maps with high keyspace length. Furthermore, the simulation performance was analyzed and compared with other relevant approaches.

µ pseudo rotating-periodic solutions for differential equations

In this article, we combine rotating periodic functions with \(\mu\) ergodic functions to obtain a new class of functions called \(\mu\) pseudo rotating periodic functions. Then we study the existence and uniqueness of \(\mu\) pseudo rotating periodic solutions for linear systems, semi-linear systems, and non-linear systems by exponential dichotomy.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/43/abstr.html

Semi-Poisson nearest-neighbor distance statistics for a 2D dissipative map

Typical pseudotrajectories of 2D ergodic maps are known to possess Wignerian nearest-neighbor distance distributions (NNDDs). In the case of 2D chaotic dissipative maps, bounded aperiodic pseudotrajectories typically evolve on planar strange attractors. In this microarticle, the hypothesis that such pseudotrajectories should possess NNDDs that are intermediate between the Poisson and Wigner distributions is put forward and a rare example for which the intermediate distribution can be clearly identified is presented. In particular, it is demonstrated numerically that typical pseudotrajectories evolving in the strange attractor of the standard 2D Ikeda map possess semi-Poissionian NNDDs.

Geometric and measure-theoretic shrinking targets in dynamical systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.