p63

lim sup etc.

lim inf n→∞ a_n implies lim n→∞ inf{a_m : m ≥ n} . That is, the largest number not larger than a_m (m ≥ n) is regarded as a function of n and its limit in the n → ∞ limit is lim inf n→∞ a_n . lim sup can be understood parallelly by replacing inf with sup.

Scrambled set, some additional explanation

`lim inf is zero’ in (B) implies that any two orbits starting from two arbitrary point in R can come indefinitely close if you wait for a sufficiently long time. However, due to (A), these two trajectories separate again. On the other hand, `lim sum is positive’ implies that any orbit starting from $R$ never falls into a periodic orbit.

The relation t o Sarkovskii’s theorem

Theorem 2.1 cannot be proved from Li-Yorke’s theorem (even if Sarkovskii’s theorem is used):

A. N. ˇSarkovskii, “Coexistence of cycles of a continuous map of a line into itself,” Ukr. Mat. Z.  16 , 61-71 (1964) [See P. ˇStefan, “A theorem of ˇSarkovski on the existence of periodic orbits of continuous endomorphisms of the real line,” Commun. Math. Phys.  54 , 237 (1977) as well]．M. Misiurevic, “Remarks on Sharkovsky’s theorem,” Am. Math. Month. Nov 1997, p846 is a good summary.

Feigenbaum’s critical point

M. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys.  19 , 25–52 (1978). The analogy to critical phenomena and that scaling laws hold are obvious, if one knows Sinai’s thermodynamic formalism for dynamical systems: Ya. G. Sinai, “Gibbs measure in ergodic theory,” Russ. Math. Surveys, 166(4), 21 (1972). That some detailed information can be obtained if we assume that the system is defined by a differentiable map may be of some interest. 

 At his critical point, the topological entropy (the sup of the Kolmogorov-Sinai entropy allowed to the metric dynamical systems possible for the given dynamical system) is zero, so there is no chaos in our sense. However, there are examples having scrambled sets: J. Smıtal, “Chaotic functions with zero topological entropy,” Trans. Amer. Math. Soc.,  297 , 269 (1986). A good summary of entropy zero dynamical system is found in V. V. Fedorenko, A. N. ˇSarkovskii and J. Smital, “Characterization s of weakly chaotic maps of the interval,” Proc. Amer. Math. Soc.,  110 , 141-148 (1990).

Another characterization of chaos

The author happened to find the following paper:

Pat Touhey, ``Yet Another Defilnition of Chaos,’’ Am Math Month  104 , 411 (1997).

Given a metric space X and a continuous mapping f: X -> X, we say that f is chaotic on X if given U and V, non-empty open subsets of X, there exists a periodic point p \in U and a non-negative integer k such that f^k( p) \in V, that is, every pair of non-empty open subsets of X shares a periodic orbit.

 The paper proves that this and Devaney’s definition are equivalent. 

Footnote 66 additional comment

V. J. Lopez, “Paradoxical functions on the interval,” Proc. Amer. Math. Soc.,  120 , 465 (1994) proves the following: If a map  f from an interval I to I is expansive, then the dynamical system cannot have a measure positive scrambled set. However, if in I x I , x and y are both in the scrambled set, then the totality Ch(f) of {x, y} is always measurable. Furthermore, if f is expansive and its derivative is piecewisely Lipshitz, then Ch(f) has a positive measure.