Ergodic Maps Research Articles

The uniform laws of large numbers for the tent map

The tent map is an ergodic map defined on the unit interval. This paper considers the asymptotic behaviors of the various processes generated by the tent map. We get the uniform versions of law of large numbers for the tent map. An application to Monte Carlo integration is provided.

The large deviations theorem and sensitivity

Let X be a compact metric space and f : X → X be a continuous map. In this paper, we prove that if f is a topologically strongly ergodic map, then f is sensitively dependent on initial conditions. Moreover, we investigate the relationships between the large deviations theorem and sensitivity, and show that if f satisfies the large deviations theorem, then f is sensitively dependent on initial conditions if and only if f is neither minimal nor equicontinuous.

Constructing ergodic exponential maps with dense post-singular orbits

AbstractWe construct ergodic exponential maps fλ(z)=λez such that the forward orbit of the origin is dense in $\Bbb C$.

Measured topological orbit and Kakutani equivalence

Suppose $X$ and $Y$ are Polish spaces each endowed with Borel probability measures $\mu$ and $\nu$. We call these Polish probability spaces. We say a map $\phi$ is a <i>nearly continuous</i> if there are measurable subsets $X_0\subseteq X$ and $Y_0\subseteq Y$, each of full measure, and $\phi:X_0\to Y_0$ is measure-preserving and continuous in the relative topologies on these subsets. We show that this is a natural context to study morphisms between ergodic homeomorphisms of Polish probability spaces. In previous work such maps have been called <i>almost continuous</i> or <i>finitary</i>. We propose the name <i>measured topological dynamics</i> for this area of study. Suppose one has measure-preserving and ergodic maps $T$ and $S$ acting on $X$ and $Y$ respectively. Suppose $\phi$ is a measure-preserving bijection defined between subsets of full measure on these two spaces. Our main result is that such a $\phi$ can always be <i>regularized</i> in the following sense. Both $T$ and $S$ have full groups ($FG(T)$ and $FG(S)$) consisting of those measurable bijections that carry a point to a point on the same orbit. We will show that there exists $f\in FG(T)$ and $h\in FG(S)$ so that $h\phi f$ is nearly continuous. This comes close to giving an alternate proof of the result of del Junco and Şahin, that any two measure-preserving ergodic homeomorphisms of nonatomic Polish probability spaces are continuously orbit equivalent on invariant $G_\delta$ subsets of full measure. One says $T$ and $S$ are evenly Kakutani equivalent if one has an orbit equivalence $\phi$ which restricted to some subset is a conjugacy of the induced maps. Our main result implies that any such measurable Kakutani equivalence can be regularized to a Kakutani equivalence that is nearly continuous. We describe a natural nearly continuous analogue of Kakutani equivalence and prove it strictly stronger than Kakutani equivalence. To do this we introduce a concept of nearly unique ergodicity.

Waiting for long excursions and close visits to neutral fixed points of null-recurrent ergodic maps

Waiting for long excursions and close visits to neutral fixed points of null-recurrent ergodic maps

SOLVABLE THREE-DIMENSIONAL RATIONAL CHAOTIC MAP DEFINED BY JACOBIAN ELLIPTIC FUNCTIONS

Cryptanalysis needs a lot of pseudo-random numbers. In particular, a sequence of independent and identically distributed (i.i.d.) binary random variables plays an important role in modern digital communication systems. Sufficient conditions have been recently provided for a class of ergodic maps of an interval onto itself: R1 → R1 and its associated binary function to generate a sequence of i.i.d. random variables. In order to get more i.i.d. binary random vectors, Jacobian elliptic Chebyshev rational map, its derivative and second derivative which define a Jacobian elliptic space curve have been introduced. Using duplication formula gives three-dimensional real-valued sequences on the space curve onto itself: R3 → R3. This also defines three projective onto mappings, represented in the form of rational functions of xn, yn, zn. These maps generate a three-dimensional sequence of i.i.d. random vectors.

Minimal-work principle and its limits for classical systems

The minimal-work principle asserts that work done on a thermally isolated equilibrium system is minimal for the slowest (adiabatic) realization of a given process. This principle, one of the formulations of the second law, is operationally well defined for any finite (few particle) Hamiltonian system. Within classical Hamiltonian mechanics, we show that the principle is valid for a system of which the observable of work is an ergodic function. For nonergodic systems the principle may or may not hold, depending on additional conditions. Examples displaying the limits of the principle are presented and their direct experimental realizations are discussed.

Families of ergodic and exact one-dimensional maps

We give a parametrized family of rational interval maps of degree two, each ergodic, exact and preserving a measure equivalent to a Lebesgue measure. The family includes the unique quadratic Chebyshev polynomial as its only polynomial map. We extend the family to other settings on the circle and real line. We also give numerical approximations to the entropy of the equivalent invariant measure and the Hausdorff dimension of the singular measure of maximal entropy.

The generalized Lyapunov theorem and its application to quantum channels

We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the ‘Lyapunov direct method’. First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in open quantum systems and quantum information, namely quantum channels. In this context, we also discuss the relations between mixing and ergodicity (i.e. the property that there exists only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.

The large deviations theorem and ergodicity☆

Abstract In this paper, some relationships between stochastic and topological properties of dynamical systems are studied. For a continuous map f from a compact metric space X into itself, we show that if f satisfies the large deviations theorem then it is topologically ergodic. Moreover, we introduce the topologically strong ergodicity, and prove that if f is a topologically strongly ergodic map satisfying the large deviations theorem then it is sensitively dependent on initial conditions.

The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems

Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy rates—measures of new disorder per new observed value—are equal for ergodic finite-alphabet information sources (discrete-time stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n-dimensional intervals. This result generalizes a previous one for piecewise monotone interval maps on the real line [C. Bandt, G. Keller, B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002) 1595–1602.] at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits. The case of non-ergodic finite-alphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic non-discrete information sources when entropy is replaced by differential entropy in the usual way.

COARSE-GRAINED OBSERVATION OF DISCRETIZED MAPS

We investigate why discretized versions fN of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization fN on a coarse M × M grid, with N = cM, c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M × M incidence matrix AM associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix BN,M associated with the aforementioned coarse grid, if [Formula: see text], where deg (f) denotes the degree of f. We show how coarse-grained registration can be used as a "digital" definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and fN. Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.

CLASSIFICATION OF CHAOTIC SEQUENCES WITH OPEN-LOOP ESTIMATOR — OPTIMAL DESIGN FOR NOISY ENVIRONMENTS

In this paper, an approach based on ergodic properties for classifying chaotic sequences is given. It is particularly robust due to the open-loop structure of the detector. Unlike previous works in this direction, such as those concerning the inverse system approach, the detector is not uniquely determined by the transmitter (identical or subsystem thereof), but instead depends on the measurement noise model. The classification is optimized under such imperfect observation conditions. The method is introduced in general for the case of chaotic sequences generated by ergodic maps, and a special case is analyzed in detail to illustrate the method. This special example resorts to Tchebychev maps and some additional symmetries to make up a simple signaling scheme which is low in complexity on both transmitter and receiver sides, while at the same time relatively robust, due to the open-loop structure of the detector.

Generalized Esclangon-Landau Results and Applications to Linear Difference-differential Systems in Banach Spaces

For solutions y of a linear difference-differential system on a halfline or the reals it is shown that if y and f belong to a class 𝒜 of Banach-space valued functions, then all derivatives y (j) are in 𝒜 too, The main assumption on is here: If and all differences for then for with Examples are the almost periodic asymptotic ap, almost automorphic, Eberlein weakly ap, Stepanoff ap, pseudo ap, various classes of ergodic functions, weighted L p , C k , uniformly continuous functions, O(w). The assumption can be weakened to y an element of some “mean” class where for Also proofs of various results on mean classes needed here and of for the mentioned above are supplied. guenzler@math.uni-kiel.de

Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl–Bohr–Amerio–Kadets theorem states that the indefinite integral y= Pφ of an almost periodic (ap) φ : R→X is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c 0. This is here generalized in several directions: Instead of R it holds also for φ defined only on a half-line J , instead of ap functions abstract classes A with suitable properties are admissible, φ∈ A can be weakened to φ in some “mean” class M q+1 A , then Pφ∈ M q A ; here MA contains all f∈ L 1 loc with (1/h)∫ 0 hf(·+s) ds in A for all h>0 (usually A⊂ MA⊂ M 2 A⊂⋯ strictly); furthermore, instead of boundedness of y mean boundedness, y in some M kL ∞ , or in M k E , E= ergodic functions, suffices. The Loomis–Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ( t+ h)− Ψ( t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” Δ A in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr–Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P( D) y= φ of degree m with ap φ is extended similarly for φ∈ M q+m A ; then y∈ M q A provided, for example, y is in some M kU with U= L ∞ or is totally ergodic and, for the half-line, Re λ⩾0 for all eigenvalues P( λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Re λ>0, there exists a unique solution y growing not too fast; this y is in M q A if φ∈ M q+m A , for quite general A .

Information sources using chaotic dynamics

A sequence of binary random variables has found significant applications in modem digital communication systems. For such sequences, several kinds of linear feedback shift register sequences have been proposed. It is, however, well known in probability theory that the Bernoulli shift is a fundamental theoretic model of a sequence of independent identically distributed (i.i.d.) binary random variables. In this paper after reviewing fundamental subjects of chaotic dynamics, in particular a close relationship between information sources and Markov chains, we give the generation method of sequences of i.i.d. binary random variables using chaotic dynamics. Such a generation method is given as a sufficient condition composed of simple symmetric properties for some class of ergodic maps. Furthermore, we give the applications of such sequences: (1) to running-key sequences for stream cipher systems and (2) to a color image communication system through code-division multiple access channels and its extended version, a digital watermarking system. In addition, the performance of spread spectrum codes generated by a Markov chain is theoretically evaluated in asynchronous direct-sequence/code-division multiple access systems.

Series Criteria for Growth Rates of Partial Maxima of Iterated Ergodic Map Values

Birkhoff's well-known ergodic theorem states that the simple averages of a sequence of real (integrable) function values on successive iterates of a measure-preserving mapping T converge a.s. to the conditional expected value of the function conditioned on the invariant sigma-field. If the mapping is in addition ergodic, then the limit is simply the unconditional expected value: $$\frac{1}{n}\sum\limits_{k = 0}^{n - 1} {f \circ T^k \to \int_\Omega {f\;dP,{ a}{.s as }n \to \infty } } { (0}{.1)}$$ In this article, we discuss the analogous result for sequences of partial maxima: given a measurable f, if T is measure-preserving and ergodic then $$M_n = \mathop {\max }\limits_{k\; \leqslant \;n} f \circ T^k \uparrow {ess}\;\sup f,{ a}{.s as }n \to \infty {(0}{.2)}$$ Series criteria are provided which characterize the a.s. maximal and minimal growth rates of the sequence of partial maxima.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller–Reuter–Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

Chaotic synchronization of coupled ergodic maps.

With few exceptions, studies of chaotic synchronization have focused on dissipative chaos. Though less well known, chaotic systems that lack dissipation may also synchronize. Motivated by an application in communication systems, we couple a family of ergodic maps on the N-torus and study the global stability of the synchronous state. While most trajectories synchronize at some time, there is a measure zero set that never synchronizes. We give explicit examples of these asynchronous orbits in dimensions two and four. On more typical trajectories, the synchronization error reaches arbitrarily small values and, in practice, converges. In dimension two we derive bounds on the average synchronization time for trajectories resulting from randomly chosen initial conditions. Numerical experiments suggest similar bounds exist in higher dimensions as well. Adding noise to the coupling signal destroys the invariance of the synchronous state and causes typical trajectories to desynchronize. We propose a modification of the standard coupling scheme that corrects this problem resulting in robust synchronization in the presence of noise.