By A.N. Sharkovsky, S.F. Kolyada, A.G. Sivak, V.V. Fedorenko

maps whose topological entropy is the same as 0 (i.e., maps that experience basically cyeles of pe 2 riods 1,2,2 , ... ) are studied intimately and elassified. numerous topological points of the dynamics of unimodal maps are studied in Chap ter five. We learn the specified gains of the restricting habit of trajectories of gentle maps. specifically, for a few elasses of gentle maps, we identify theorems at the variety of sinks and examine the matter of lifestyles of wandering periods. In bankruptcy 6, for a large elass of maps, we end up that the majority issues (with admire to the Lebesgue degree) are attracted by means of a similar sink. Our recognition is especially all in favour of the matter of life of an invariant degree completely non-stop with admire to the Lebesgue degree. We additionally learn the matter of Lyapunov balance of dynamical platforms and ascertain the measures of repelling and attracting invariant units. the matter of balance of separate trajectories below perturbations of maps and the matter of structural balance of dynamical platforms as a complete are mentioned in Chap ter 7. In bankruptcy eight, we research one-parameter households of maps. We study bifurcations of periodic trajectories and houses of the set of bifurcation values of the parameter, in eluding common houses resembling Feigenbaum universality.

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**Example text**

It is equal to + 1, - 1, or 0, respectively, if {' increases, decreases, or has an extremum at the point x. 1 (on monotonicity). The map x ~ 81x) is monotone. Proof. Note that the map x ~ 8 f (x) is either nonincreasing or nondecreasing depending on the type of extremum (minimum or maximum) attained at the point e by the function f: I ~ I. Assume that e is the maximum point. If x' < x", then let n be the least integer for which 8n(x') "* 8ix"). For n = 0, we have x'::; e ::; x" and 8 0 (x');::: 40 Elements oJ Symbolic Dynamics Chapter 2 80(x").

For Cl (I, I), we have APB (j) ~ AP(j) (Fedorenko [4]) and n (f) = n(f) (Sharkovsky [2]). This enables us to conclude that n (f) ::J C (f). , contain a dense trajectory). Representations of this sort are usually called spectral decompositions. The spectral decomposition of the set of nonwandering points is the most popular object of investigations. As a rule, in terms of this decomposition, one can easily describe the typical behavior of the trajectories of the corresponding dynamical system. To explain this in detail, we consider quadratic mappings from the family ft,,, described in Section 1.

I=O This power series is called the dynamical coordinate of the point x. The lexicographic ordering and topology of coordinatewise convergence on {8f (x)} induce the lexicographie ordering and topology of coordinatewise convergence on the set {8(x)}. Moreover, the correspondence x ~ 8(x) remains monotone and, for any x E I, there exist 8(x+) The series vf = 8(r) = lim8(y) and 8(x-) y~x = lim8(y). y~x is called the kneading invariant of the map f We have chosen the series 8(r) but not 8(c) because 8(c) = O.