By Shlomo Sternberg

A pioneer within the box of dynamical structures created this contemporary one-semester advent to the topic for his periods at Harvard college. Its wide-ranging therapy covers one-dimensional dynamics, differential equations, random walks, iterated functionality structures, symbolic dynamics, and Markov chains. Supplementary fabrics supply various on-line elements, together with PowerPoint lecture slides and MATLAB workouts. 2010 version.

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**Sample text**

Write zj+1 = g(zj ). The second equation gives −αg(g(z2j ) = g(zj ) and we can substitute z2j = −zj /α from the first equation to get −αg(g(−zj /α)) = g(zj ). If we expect this to hold not just for zj but for all values of z we get the functional equation: −αg(g(−z/α)) = g(z). If we assume that g has a power series expansion, and we compute up to terms of second order in z, we get an approximate value for α. The rescaling is centered at the maximum, so in order to avoid notational complexity, let us shift this maximum (for the logistic family) to the origin by replacing x by y = x − 12 .

BIFURCATIONS. 3, graphs of y = x, y = Lµ (x), y = L)µ(2) (x). The graph of (2) Lµ has four points of intersection with the line y = x: the two (repelling) fixed points of Lµ and two points of period two. 7 µ = 3. Much of the analysis of the preceding case applies here. 3) for seeking points of period two now has a (double) root. But this root is 32 = 1 − µ1 which is the fixed point. So there is still no point of period two other than the fixed points. The iterates continue to spiral in, but now ever so slowly since Lµ ( 23 ) = −1.

In this chapter we explain these ideas, and also describe what happens to the period √ there orbits when we decrease µ from slightly above the critical value 1 + 8 to slightly below it. Throughout this chapter, f will denote a continuous function on the reals whose domain of definition is assumed to include the given intervals in the various statements. 1 Period 3 implies all periods. 1. If I = [a, b] is a compact interval and I ⊂ f (I) then f has a fixed point in I. Proof. For some c, d ∈ I we have f (c) = a, f (d) = b.