DYNAMICS OF THE RADIX EXPANSION MAP

The chaotic dynamics ofthe map x) ()Ix+ a) (mud 1) are studied using Parry's fl- expansion. It is shown that for 0, the number ofperiodic points ofperiod n is0)

When 13 is not an integer, things are not quite so tidy.There is still an invariant Borel measure differentiable on [0,1 (Sinai, 1981), although it is zero almost everywhere.However, although the map may still be proved chaotic by demonstrating a conjugacy with a shift map, the space Ap on which this shift map acts is somewhat problematic.We will deal only with the case < ]3 < 2, although most of the results generalize straightforwardly.In this case, an analysis of A/ reveals that the number of periodic poims of period n is asymptotically proportional to #(n), the number ofbinary sequences sis2...s, so that: IfX is a random variable on R, with probability densityf so that supp(f) [0,1 ], let 9'f denote the probability density of 2').Then, we show that 9d2'[0.11(0)ff-n #(n) It follows that #(n) O(ff'), since lim VnZI0,11(0) exists by the ergodicity of the mapping (Sinai, 1981) This resul, is new and apparently cannot be obtained from standard methods such as kneading theory (Coilet and Eckmarm, 1980) From here on, we will assume < fl < 2, a >_ 0 2 INVARIANT MEASURES Where a 0, Parry (1960) has studied the invariant measure of , and shown that it is umque In general, any invariant measure of is characterized by a Frobenius-Perron operator W.
Next, let us derive a formula for the n'th iterate of / LEMMA2.v"f(x) ff".,f(ff"x-ff"a+al"'),where{a',"' -'"'} is the set of all .., expansions fl-ls +ff-2s2+...+ff-nsn so that s, {0,1} and ff-lx+/-lsl +ff-2s2+...+-nsn < ff-n+lx + flls2 +...+fl-n+lsn < lx+ff-s, _< PROOF.Lemma takes care of the case n l, so we may proceed by induction.Assume that a 0 and that the statement is true for n k 1.Then f(.X') __,-I[ -l(/r-Ix)+ -l(ff-lx.+/r-')] This shows that the statement is true for n k If" a 0, the lemma follows from the observation that, where /l is the operator corresponding to ) (/It + a) (rood ), and /is the operator corresponding to x) =,B (rood I), v,f() /f(x a).This lemma permits us to estimate asymptotically the number #(n) defined above.The problem of determining #(n) for arbitrary n is apparently unsolved and seems to be very difficult.

APPROXIMATION WITH INTERVAL MAPS
Computer simulations have played a large role in the development of the theory of chaotic dynamical systems.One way to simulate the probabilistic behavior of a chaotic map like is to approximate the map by a sequence of interval maps.
Givenameasure/sothatsupp(f) [0,1],letP.n) P(n)=(Pl(n).,P(nn))T ,.-where t/n /n P f(x)dx Let en be the n x n matrix defined by (nP(n)) t//f(x)dx PROOF.(9n) is the probability that, after k iterations of , the image of a point selected from a uniform distribution on [i-n,)is in [i-,) Because is ergodic, for any two open intervals Uand V there is some k so that the Lebesgue measure of v(U)V is nonzero THEOREM 2 n has a unique fixed point/3(n) PROOF Since ]lll l, P(p'n)<_ 1.And since .,(An)ty 1, is an eigenvalue f ATn with j=l corresponding eigenvector (1,.-., 1), and hence is an eigenvalue of A n and p(A n 1.According to a standard linear algebra result, Lemma 3 implies that the multiplicity of is Thus there is a unique eigenvector/3(n) corresponding to the eigenvalue 1.
Given this result, it is easy to see that these/3in) converge to the invariant measure.For any x [0,11, let In(x)denote the interval/i-2,) which containsx.LetS n denote the set of all step functions constant on each interval [i-1 t__ / Let .(n)be the element of S n naturally induced by L n 'n/ Then we have THEOREM 3. lim y(n) y.
These results show that the interval maps n are qualitatively faithful representations of W. The chaotic behavior of is necessarily absent from any discrete approximation, but the probabilistic implications ofthis chaos are accurately mirrored. 4. DYNAMICS DEFINITION 1.Where s Sl...sn... is a binary sequence, let Bp(s)=ff-lsl +ff-2s2+...ff-nsn+... Let F/ be the set ofall s= sl...s or s= sis2.., so that B(s) > Let A/ be the set of all s sis2.., so that slSk+l...Sk+m F, sksk+l...F, for any k and .
F/ is the set of all "forbidden subsequences", and A/ is the set of all sequences containing no forbidden subsequences.
LEMMA 4. For each x [0,1] there is a unique binary sequence s A/ so that B(s) x.PROOF Existence is clear, one forms an expansion B/ exactly as one forms an expansion in an integer base.To show uniqueness, assume x Bfl(SlS2... sn_ lSn+l--.and x Bfl(SlS2... sn_ 10s+ls+2... Then the second expansion is not in A/ Chaos may be defined in many different ways Here we will adopt the topological approach found in Devaney (1989) ii) its periodic points are dense, iii) it is sensitive to initial conditions LEMMA 5. [0,1] [0,1] is topologically conjugate to the shift map cr A/ -+ A/, where PROOF.Follows from Lemma 4 by standard arguments

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: