Ergodic Properties

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 $f$, we say that an interval $U$ is symmetric if $\tau(U)=U$ where $\tau:[-1,1]\rightarrow [-1,1]$ is so that $f(\tau(x))=f(x)$ and $\tau(x)\neq x$ if $x\neq c$. Furthermore, for each symmetric interval $U$ let

$D_{U}=\{x: \text{ there exists } k>0 \text{ with } f^{k}(x)\in U\};$

for $x\in D_{U}$ let $k(x,U)$ be the minimal positive integer with $f^{k}(x)\in U$ and let

$R_{U}(x)=f^{k(x,U)}(x).$

We call $R_{U}: D_{U}\rightarrow U$ the Poincare map or transfer map to $U$ and $k(x,U)$ the transfer time of $x$ to $U$. 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 $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with one non-flat critical point with negative Schwarzian derivative and without attracting periodic points. Then there exists $\rho>0$ and a sequence os symmetric intervals $U_{n}\subseteq V_{n}$ around the turning point which shrink to $c$ such that $V_{n}$ contains a $\rho-$scaled neighbourhood of $U_{n}$ and such that the following properties hold.

1. The transfer time on each component of $D_{U_{n}}$ is constant.

2. Let $I_{n}$ be a component of the domain $D_{U_{n}}$ of the transfer map to $U_{n}$ which does not intersect $U_{n}$. Then there exists an interval $T_{n}\supseteq I_{n}$ such that $f^{k}|T_{n}$ is monotone, $f^{k}(T_{n})\supseteq V_{n}$ and $f^{k}(I_{n})=U_{n}$. Here $k$ is the transfer time on $I_{n}$, i.e., $R_{U_{n}}|I_{n}=f^{k}$.

Corollary. There exists $K<\infty$ such that

1. for each component $I_{n}$ of $D_{U_{n}}$ not intersecting $U_{n}$, the transfer map $R_{U_{n}}$ to $U_{n}$ sends $I_{n}$ diffeomorphically onto $U_{n}$ and the distortion of $R_{U_{n}}$ on $I_{n}$ is bounded from above by $K$.

2. on each component $I_{n}$ of $D_{U_{n}}$ which is contained in $U_{n}$, the map $R_{U_{n}}:I_{n}\rightarrow U_{n}$ can be written as $(f^{k(n)-1}|f(I_{n}))\circ f|I_{n}$ where the distortion of $f^{k(n)}|f(I_{n})$ is universally bounded by $K$.

As before, we say that $f$ is ergodic with respect to the Lebesgue measure if each completely invariant set $X$ (Here $X$ is called completely invariant if $f^{-1}(X)=X$) has either zero or full Lebesgue measure. An alternative way to define this notation of ergodicity goes as follows: $f$ is ergodic if for each two forward invariant sets $X$ and $Y$ such that $X\cap Y$ has Lebesgue measure zero, at most one of these sets has positive Lebesgue measure. (Here $X$ is called forward invariant if $f(X)\subseteq X$.)

Theorem 1.2 (Blokh and Lyubich). Let $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with a non-flat critical point with negative Schwarzian derivative and without an attracting periodic points. Then $f$ is ergodic with respect to the Lebesgue measure.

Theorem 1.3.  Let $f:[-1,1]\rightarrow [-1,1]$ be a unimodal map with a non-flat critical point with negative Schwarzian derivative. Then $f$ has a unique attractor $A$, $\omega(x)=A$ for almost all $x$ and $A$ either consists of intervals or has Lebesgue measure zero. Furthermore, one has the following:

1. if $f$ has an attracting periodic orbit then $A$ is this periodic orbit;

2. if $f$ is infinitely often renormalizable then $A$ is the attracting Cantor set $\omega(c)$ (in which case it is called a solenoidal attractor);

3. $f$ is only finitely often renormalizable then either

(a) $A$ coincides with the union of the transitive intervals, or,

(b) $A$ is a Cantor set and equal to $\omega(c)$.

If $\omega(c)$ is not a minimal set then $f$ is as in case 3.a and each closed forward invariant set either contains intervals or has Lebesgue measure zero. Moreover, if $\omega(c)$ does not contain intervals, then $\omega(c)$ has Lebesgue measure zero.

Remark. Here a forward invariant set $X$ is said to be minimal if the closure of the forward orbit of a point in $X$ is always equal to $X$. 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 $\omega(c)$). The Fibonacci map is non-renormalizable and for which $\omega(c)$ 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 $f(x)=4x(1-x)$ 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 $\ell$ is sufficiently large enough.

Theorem (Lyubich). If $f:[-1,1]\rightarrow [-1,1]$ is $C^{3}$ unimodal, has a quadratic critical point, has negative Schwarzian derivative and has no periodic attractors, then each closed forward invariant set $K$ 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 $x$ not be in the pre orbit of $c$ and define $T_{n}(x)$ to be the maximal interval on which $f^{n}|T_{n}(x)$ is monotone. Let $R_{n}(x)$ and $L_{n}(x)$ be the components of $T_{n}\setminus x$ and define $r_{n}(x)$ be the minimum of the length of $f^{n}(R_{n}(x))$ and $f^{n}(L_{n}(x))$.

Theorem 1.4 (Martens). Let $f$ be a $C^{3}$ unimodal map with negative Schwarian derivative whose critical point is non-flat. Then the following three properties are equivalent.

1. $f$ has no absorbing Cantor attractor;

2. $\limsup_{n\rightarrow \infty} r_{n}(x)>0$ for almost all $x$;

3. there exist neighbourhoods $U\subseteq V$ of $c$ with $cl(U)\subseteq int(V)$ such that for almost every $x$ there exists a positive integer $m$ and an interval neighbourhood $T$ of $x$ such that $f^{m}|T$ is monotone, $f^{m}(T)\supseteq V$ and $f^{m}(x)\in U$.