Elton, J. (1987), An Ergodic Theorem for Iterated Maps, Journal of Ergodic Theory and Dynamical Systems 7, 481--488.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= w Zn (X n ) 8n 0; where U(X ) is the uniform distribution on X . n=0 is of course an homogeneous Markov chain with state space X and its transition matrix P = p ij ) ij is de ned by p ij : P[X n 1 = x j jX n = x i ] k:w k (x i ) x j p k ; 8x i ; x j 2 X : By Elton s theorem ([5]) we know that m 1 Xn ;x = x ; a.s. 8x 2 X ; 3 where ( x ) x2X is the only invariant probability measure for the transition matrix P. Indeed one could easily see that A is a closed recurrent class for n=0 , and that X n A contains only transient states (see [11] The result ....

Elton J. (1987) An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, p. 481-488.

....by Isaac [38] Isaac proved the special case of Theorem 6, when (X; d) is assumed to be compact, condition (19) holds, and the p i s are assumed to be Lipschitzcontinuous. Question: Can Theorem 6 be proved if the p 0 i s satisfy Harris condition (14) The following result was proved in [23]. Theorem 7. Elton (1987) Assume the conditions of Theorem 6. Let fZ n (x)g denote the associated Feller chain and let denotes its unique stationary probability measure. Then lim n 1 P n Gamma1 k=0 f(Z k (x) n = Z fd; for all f 2 C(X) a:s: 20) for any x 2 X. Remark 15. In the case ....

J. H. Elton. An ergodic theorem for iterated maps. Ergodic Theory Dynam. Systems, 7(4):481--488, 1987.

....it, for example, as I = #N = 1, 2, N . For an IFS, one examines the behaviour of the sequence f i 1 f i 2 f i n x for any initial point x # X and any code sequence i 1 i 2 # I # . In the past 15 years, IFS theory has been a very active area of research in fractal geometry [73, 8, 29, 83, 84, 48, 9] and has found applications in diverse areas such as mathematical finance, signal processing, computer graphics, image compression, learning automata, neural nets, statistical physics and real number computation [11, 12, 7, 10, 22, 83, 84, 15, 13, 44] A simple example of an IFS can be constructed ....

....P(i n = k) p k (1 # k # N ) for all n # 1. Let x 0 # X , and put x n 1 = f i n (x n ) for all n # 0. Then Elton s ergodic theorem states that the time average of any real valued continuous function is the same as its phase average with respect to the invariant measure #: Theorem 8.1. [48] Let g : X # R be a continuous function and suppose x 0 # X . Then, for almost all sequences i 1 , i 2 , lim k## 1 N N 1 # n=0 g(x n ) # g(x) d#(x) Therefore, the free energy density is given by f(#) lim N## 1 N N 1 # n=0 B(# n ) # B d#. 434 ABBAS ....

J. Elton, An ergodic theorem for iterated maps, Journal of Ergodic Theory and Dynamical Systems, vol. 7 (1987), pp. 481--487.

....With an IFS W = fw k ; k = 1; Ng we must associate a set of real numbers fp k ; k = 1; Ng such that 0 p k 1 for all k, and P N k=1 = 1. choose x 0 2 R n for i = 1; 2; choose k with probability p k , and generate the point x i = w k (x i Gamma1 ) Elton s theorem [13] ensures that the set fx i g converges almost surely to A, regardless of the initial point x 0 . FIgure 2.1 illustrates the progress of the algorithm in constructing the Sierpinski triangle. 2.2.2 Observations This is a stochastic algorithm which models a Markov process. The probabilities p k ....

.... the point x i = w k (x i Gamma1 ) define i by i (X q ) i Gamma 1) i i Gamma1 (X q ) 1 i for q = R(x i ) i (X q ) i Gamma 1) i i Gamma1 (X q ) for q 6= R(x i ) As i increases, i converges almost surely to the invariant measure, as a consequence of Elton s theorem [13]. A modification of the DIA, called the Constant Mass Algorithm [25] allows the invariant measure to be found deterministically: choose 0 2 B 1 (R n ) for i = 1; 2; define i = W; P ) i Gamma1 ) By Hutchinson s theorem [18] i converges to the invariant measure of (W; P ) The ....

Elton, J.: "An ergodic theorem for iterated maps" Journal of Ergodic Theory and Dynamical Systems 7:481-488 (1987)

....du d ecalage de Bernouilli. Nous commencerons dans la section 2 par rappeler les notions utiles sur les fonctions mesurables et l ergodicit e, afin de rendre la pr esentation aussi autonome que possible. 58 C. Tricot et R. Riedi Notons que ce th eor eme a d ej a et e d emontr e dans [2], r ef erence constamment cit ee par la suite sur cette question. Sans vouloir entrer dans une pol emique, disons que la d emonstration y est obscure et que les sp ecialistes pr ef erent l accepter sans v erification. Nous avons cru utile, a des fins p edagogiques, d en proposer une autre qui a ....

J. Elton, An Ergodic Theorem for Iterated Maps, Journal of Ergodic theory and Dynamical Systems, 7 (1987), 481-488.

.... individual approach of GP. 1 Introduction IFSs (Iterated Functions System) theory is an important topic in fractals, and provides powerful tools to investigate fractal sets. The action of systems of contractive maps to produce fractal sets has been considered by many authors (see for example [11, 2, 3, 7, 10]) and most fractal image compression techniques are based on IFSs [4, 12, 8] Non affine IFSs which we call Mixed IFSs, in order to emphasize 1 the fact that they are not anymore restricted to a composition of affine functions provide an interesting variety of shapes, whose practical ....

J. H. Elton, "An Ergodic Theorem for Iterated Maps," in Georgia Tech. preprint, 1986.

....(IFS) theory is an important topic in fractals, introduced by J. Hutchinson [25] These studies have provided powerful tools for the investigation of fractal sets, and the action of systems of contractive maps to produce fractal sets has been considered by numerous authors (see, for example, [4, 6, 14, 22]) A major challenge of both theoretical and practical interest is the resolution of the so called inverse problem [5, 40, 58, 59] Except for some particular cases, no exact solution is known. A lot of work has been done in this framework, and some solutions exist based on deterministic or ....

J. H. Elton. An ergodic theorem for iterated maps. In Georgia Tech. preprint, 1986.

....by J. Hutchinson [14] and the existence of a unique compact invariant set was proven. These studies have provided powerful tools for the investigation of fractal sets, and the action of systems of contractive maps to produce fractal sets has been considered by numerous authors (see for example [2, 3, 8, 12]) A major challenge of both theoretical and practical interest is the resolution of the so called inverse problem [20, 26, 25, 4] An exact solution can be found in some particular cases, but in general, no exact solution is known. From a computational viewpoint this problem may be formulated as ....

J. H. Elton. An ergodic theorem for iterated maps. Georgia Tech. preprint, 1986.

....sequence oe = foe 1 ; oe 2 ; the orbit x n is dense on the attractor A of the IFS w. As such, the Chaos Game can be used to generate computer approximations of A. However, it also provides approximations to the invariant measure as a consequence of the following ergodic theorem for IFS [7]: For almost all code sequences oe = oe 1 ; oe 2 ; lim n 1 1 n 1 n X k=0 f(x k ) Z X f(x) x) 65) for all continuous (and simple) functions f : X R. By setting f(x) I S (x) in Eq. 65) for an S X, we obtain (S) lim n 1 1 n 1 n X k=0 I S (x k ) 66) In ....

J. Elton, An ergodic theorem for iterated maps, Erg. Th. Dyn. Sys. 7, 481-488 (1987).

....the assumptions that (1) the OE i maps are contractive and (2) the sets w i (X) nonoverlapping, it follows that u n u(x n ) Let I k denote the characteristic function of B k . Then at the nth stage of this chaos game, S k n 1 n n X m=1 I k (xm ) u(x m ) 10) From Elton s Theorem [5], in the limit n 1 the right hand side of the above expression becomes R B k u(x)d (x) Corollary 1 Define the probabilities to be p i = m(X i ) P k m(X k ) Then lim n 1 1 n S k m(B k ) 1 m(B k ) Z B k u(x)d (x) u av (B k ) 11) the average value of u over B k . ....

J. Elton, An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, 481-488 (1987).

....Suppose i 1 ; i 2 ; is is a sequence of independent identically distributed random variables on f1; 2; Ng with probabilities P (i n = k) p k (1 k N ) for all n 1. Let i 1 2 X, and put i n 1 = f i n ( i n ) for all n 1. Then Elton s ergodic theorem states: Theorem 4. 5 [9] For almost all sequences i 1 ; i 2 ; and all 1 2 X, Z Z g(x) d (x) lim k 1 1 k k X n=1 g( i n ) It is also possible to deduce a two dimensional version of Elton s Theorem. Let h : X Theta X R be a continuous function. Let i 1 ; i 2 ; and j 1 ; j 2 ; be ....

J. Elton. An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481--487, 1987.

....we have L Z f d = R Z f d = lim m 1 S x (f; m ) for any x 2 X. If f satisfies a Lipschitz condition, then, for any ffl 0, we can obtain a finite algorithm to estimate R f d up to ffl accuracy [11] The only other method for computing the integral is by Elton s ergodic theorem [12]: The time average of f with respect to the non deterministic dynamical system f 1 ; f 2 ; f N : X X , where at each stage in the orbit of a point the map f i is selected with probability p i , tends, with probability one, to its space average, i.e. to its integral. However, in this ....

J. Elton. An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481--487, 1987.

.... fixed point of the transition operator T : P 1 UI P 1 UI 7 1 2 ffi OE Gamma1 Gamma 1 2 ffi OE Gamma1 : In other words, F n0 T n ffi I where, for n 1, T n ffi I = 1 2 n X i 1 ;i 2 ; i n= Sigma ffi OE i 1 OE i 2 : OE i n I : Using Elton s Theorem [8], we can now prove the following. Theorem 5.1 The limiting distribution of the couplings of the marginalist and smooth models are given by the invariant measures of the corresponding IFSs with probabilities, the unique fixed points of the transition operators on the domain P 1 UI. As an ....

J. Elton. An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481--487, 1987.

....with an initial x 0 2 R T and then chooses recursively and independently, x n 2 fw 1 (x n Gamma1 ) w 2 (x n Gamma1 ) w P (x n Gamma1 )g for n = 1; 2; 3; 2:6) where the probability of selecting w i is p i . The sequence fx n g 1 n=1 will converge to the attractor of the IFS [17]. Thus, in each iteration, a map is selected at random and the point x n is transformed by that map. In the limit, the path of x n will trace out the attractor. The probabilities, p i , need to be assigned to each map before this algorithm can be used. Because these probabilities are used in ....

J. Elton, "An Ergodic Theorem for Iterated Maps," Ergodic Theorems and Dynamical Systems, no. 7, 1987.

....of wandering behavior. In this paper, an approach is presented that explicitly assumes the opposite that there are many correct actions for a given agent state. In doing so, we hope to gain access to analytical methods from Iterated Function Systems (Barnsley 1993) ergodic systems theory (Elton 1987) and the theory of impulsive differential equations (Lakshmikantham et al. 1989; Bainov and Simeonov 1989) which may allow qualitative analysis of emergent behaviors and the analytical synthesis of behavioral systems for autonomous agents. Many important applications of emergent behavior, such as ....

....i (L) then the attractor of the IFS formed from those mappings will be close to the original image. The goal of this effort is the generation and analysis of RIA like behavior in autonomous agents. A key result for the RIA (and our application) is a corollary of Elton s theorem (Barnsley 1993; Elton 1987). The sequence, x n , is generated by an RIA on an IFS with probabilities. The invariant measure for the IFS is . Corollary to Elton s Theorem (Elton 1987) Let B be a Borel subset of X and let (boundary of B = 0. Let N (B, n) number of points in x 0 , x 1 , x n # B, for ....

[Article contains additional citation context not shown here]

Elton, J. F. (1987). An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481--488.

....oe and for all x the limit limn 1=n P in f Gamma w oe n ffi w oe n Gamma1 ffi Delta Delta Delta ffi w oe 1 (x) Delta exists and is equal to R X f(z) d (z) where is the invariant measure of the IFS. This is the so called ergodic property for the IFS and was proved by Elton in [3]. However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplification to the proof. Let X be a compact metric space and fw i g L i=1 a collection of L contraction maps on X. Let fp i g be a collection of L probabilities (i.e. P i p i = 1) In ....

....will denote n (oe) by oe n . For ff n 2 Sigma n we denote by p ff n the product p ff n p ff n Gamma1 Delta Delta Delta p ff 1 . Furthermore, we denote by w ff n the composition w ff n ffi w ff n Gamma1 ffi Delta Delta Delta w ff 1 : The following theorem was proved by Elton in [3]. We provide a simplified proof of this result. Theorem 1 For any continuous function f on X and any x 2 X we have lim n 1 1=n X in f Gamma w oe i ffi w oe i Gamma1 ffi Delta Delta Delta ffi w oe 1 (x) Delta = Z X f(z) d (z) for P almost all address sequences oe 2 Sigma. Proof: ....

Elton, John, An Ergodic Theorem for Iterated Maps, Journal of Ergodic Theory and Dynamical Systems 7 (1987), 481-488.

....that all points determined henceforth in this algorithm will lie on the attractor) 2. Pick oe n 2 f1; Ng at random according to the probabilities fp i g N i=1 and set xn = w oe n (xn Gamma1 ) 3. Plot xn 4. If n is less than the maximum number of iterations, go to step 2. Elton [9] proved the following theorem, commonly known as Elton s Ergodic Theorem, related to the Chaos Game. Proposition 3 Let fX; fw i g N i=1 , fp i g N i=1 g be an IFSP with invariant measure . Then for every starting point x 0 2 X and almost every sequence foe ng generated by the Chaos Game ....

J. Elton, An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, 481-488 (1987).

.... Delta Delta ; Ng. For an IFS, one examines the behaviour of the sequence f i 1 f i 2 Delta Delta Delta f i n x for any initial point x 2 X and any code sequence i 1 i 2 Delta Delta Delta 2 I . In the past 15 years, IFS theory has been a very active area of research in fractal geometry [73, 7, 28, 83, 84, 47, 8] and has found applications in diverse areas such as mathematical finance, signal processing, computer graphics, image compression, learning automata, neural nets, statistical physics and real number computation [10, 11, 6, 9, 21, 83, 84, 14, 12, 43] A simple example of an IFS can be constructed ....

....probabilities P (i n = k) p k (1 k N) for all n 1. Let x 0 2 X , and put x n 1 = f i n (x n ) for all n 0. Then Elton s ergodic theorem states that the time average of any real valued continuous function is the same as its phase average with respect to the invariant measure : Theorem 8. 1 [47] Let g : X R be a continuous function and suppose x 0 2 X. Then, for almost all sequences i 1 ; i 2 ; lim k 1 1 N N Gamma1 X n=0 g(x n ) Z g(x) d (x) Therefore, the free energy density is given by f(fi) Gamma lim N 1 1 N N Gamma1 X n=0 B( n ) Gamma Z B d : A ....

J. Elton. An ergodic theorem for iterated maps. Journal of Ergodic Theory and Dynamical Systems, 7:481--487, 1987.

No context found.

Elton, J. (1987), An Ergodic Theorem for Iterated Maps, Journal of Ergodic Theory and Dynamical Systems 7, 481--488.

No context found.

J. H. Elton. An ergodic theorem for iterated maps. Ergodic Theory Dynam. Systems, 7(4):481-488, 1987.

No context found.

Elton J. (1987) An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, p. 481-488.

No context found.

Elton J. (1987) An ergodic theorem for iterated maps, J. Erg. Th. Dyn. Sys. 7, p. 481-488.

No context found.

John H. Elton. Ergodic theorem for iterated maps. Ergod. Th. & Dynam. Sys., 7:481--488, 1987.

No context found.

J. H. Elton. An ergodic theorem for iterated maps. Georgia Tech. preprint, 1986.

No context found.

Elton, J.H. (1987) An ergodic theorem for iterated maps, Ergod. Th. & Dynam. Sys., 7, 481-488.

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