One Dimensional Dynamics
— Welington De Melo, Sebastian van Strien
Chapter 5. Ergodic Properties and Invariant Measures.
1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.
A distortion result for unimodal maps with recurrence
Given a unimodal map , we say that an interval is symmetric if where is so that and if . Furthermore, for each symmetric interval let
for let be the minimal positive integer with and let
We call the Poincare map or transfer map to and the transfer time of to . The distortion result states that one can fined a sequence of symmetric neighbourhoods of the turning point such that the Poincare maps to these intervals have a distortion which is universally bounded:
Theorem 1.1. Let be a unimodal map with one non-flat critical point with negative Schwarzian derivative and without attracting periodic points. Then there exists and a sequence os symmetric intervals around the turning point which shrink to such that contains a scaled neighbourhood of and such that the following properties hold.
1. The transfer time on each component of is constant.
2. Let be a component of the domain of the transfer map to which does not intersect . Then there exists an interval such that is monotone, and . Here is the transfer time on , i.e., .
Corollary. There exists such that
1. for each component of not intersecting , the transfer map to sends diffeomorphically onto and the distortion of on is bounded from above by .
2. on each component of which is contained in , the map can be written as where the distortion of is universally bounded by .
As before, we say that is ergodic with respect to the Lebesgue measure if each completely invariant set (Here is called completely invariant if ) has either zero or full Lebesgue measure. An alternative way to define this notation of ergodicity goes as follows: is ergodic if for each two forward invariant sets and such that has Lebesgue measure zero, at most one of these sets has positive Lebesgue measure. (Here is called forward invariant if .)
Theorem 1.2 (Blokh and Lyubich). Let be a unimodal map with a non-flat critical point with negative Schwarzian derivative and without an attracting periodic points. Then is ergodic with respect to the Lebesgue measure.
Theorem 1.3. Let be a unimodal map with a non-flat critical point with negative Schwarzian derivative. Then has a unique attractor , for almost all and either consists of intervals or has Lebesgue measure zero. Furthermore, one has the following:
1. if has an attracting periodic orbit then is this periodic orbit;
2. if is infinitely often renormalizable then is the attracting Cantor set (in which case it is called a solenoidal attractor);
3. is only finitely often renormalizable then either
(a) coincides with the union of the transitive intervals, or,
(b) is a Cantor set and equal to .
If is not a minimal set then is as in case 3.a and each closed forward invariant set either contains intervals or has Lebesgue measure zero. Moreover, if does not contain intervals, then has Lebesgue measure zero.
Remark. Here a forward invariant set is said to be minimal if the closure of the forward orbit of a point in is always equal to . The attractors in case 3.b is called a non-renormalizable attracting Cantor set, or absorbing Cantor attractor or wild Cantor attractor. Such an attractor really exists which is proven in [BKNS], and one has the following strange phenomenon: there exist many orbits which are dense in some finite union of intervals and yet almost all points tend to a minimal Cantor set of Lebesgue measure zero (this Cantor set is ). The Fibonacci map is non-renormalizable and for which is a Cantor set. It was shown by Lyubich and Milnor that the quadratic map with this dynamics has no absorbing Cantor attractors. More generally, Jakobson and Swiatek proved that maps with negative Schwarzian derivative and which are close to the map do not have such Cantor attractors. Moreover, Lyubich has shown that these absorbing Cantor attractors can not exist if the critical point is quadratic. However, Bruin, Keller, Nowicki and Van Strien showed that the absorbing Cantor attractors exist for Fibonacci maps when the critical order is sufficiently large enough.
Theorem (Lyubich). If is unimodal, has a quadratic critical point, has negative Schwarzian derivative and has no periodic attractors, then each closed forward invariant set which has positive Lebesgue measure contains an interval.
The next result, which is due to Martens (1990), shows that if these absorbing Cantor attractors do not exist then one has a lot of ‘expansion’. Let not be in the pre orbit of and define to be the maximal interval on which is monotone. Let and be the components of and define be the minimum of the length of and .
Theorem 1.4 (Martens). Let be a unimodal map with negative Schwarian derivative whose critical point is non-flat. Then the following three properties are equivalent.
1. has no absorbing Cantor attractor;
2. for almost all ;
3. there exist neighbourhoods of with such that for almost every there exists a positive integer and an interval neighbourhood of such that is monotone, and .