On the Statistical Attractors and Attracting Cantor Sets for Piecewise Smooth Maps

Abstract

We discuss some aspects of the asymptotic behavior of the forward orbits of most points for piecewise smooth maps of the interval, specially for contracting Lorenz maps. We are particularly interested in the statistical aspects of most orbits, the existence of statistical attractors and how most orbits in the basin of attraction of an attracting Cantor sets approach the attractor.

Appendix

A homterval is an open interval $I=(a,b)$ such that $f^n|_I$ is a homeomorphism for $n\ge 1$. This is equivalent to assume that $I\cap \mathcal {O}_f^-(\mathcal {C}_f)=\emptyset$. A homterval I is called a wandering interval if $I\cap {\mathbb B}_0(f)=\emptyset$ and $f^j(I)\cap f^k(I)=\emptyset$ for all $1\le j<k$, where ${\mathbb B}_0(f)$ the union of the basins of attraction of all periodic-like attractors.

Lemma 3.9

(Homterval Lemma, see [28]) Let $I=(a,b)$ be a homterval of f. If I is not a wandering interval, then $I\subset {\mathbb B}_0(f)\cup \mathcal {O}_f^{-}({\text {Per}}(f))$. Furthermore, if f is $C^3$ with $Sf<0$, and I is not a wandering interval, then the set $I\setminus {\mathbb B}_0$ has at most one point.

Lemma 3.10

(Koebe’s Lemma [28]) For every $\varepsilon >0$, $\exists K>0$ such that the following holds: Let M, T be intervals in ${\mathbb R}$ with $M\subset T$ and denote respectively by L and R the left and right components of $T\setminus M$. If $f:T\rightarrow f(T)\subset {\mathbb R}$ is a $C^3$ diffeomorphism with negative Schwarzian derivative and

$$|f(L)|\ge \varepsilon |f(M)| \text { and } |f(R)|\ge \varepsilon |f(M)|,$$

then $\frac{|Df(x)|}{|Df(y)|}\le K$ for $x,y\in M$.

Proposition 3.11

[See Proposition 3 in [10] and Proposition 2 of [38]] Let $\mathcal {C}_f\subset (0,1)$ be a finite set. If $f:[0,1]\setminus \mathcal {C}_f\rightarrow [0,1]$ is a $C^2$ non-flat local diffeomorphism, then there exists a function $O(\varepsilon )$ with $O(\varepsilon )\rightarrow 0$ as $\varepsilon \searrow 0$ with the following property: Let $J\subset T$ be an interval, R, L the connected components of $T\setminus J$ and $\delta :=\min \{|R|/|J|,|L|/|J|\}$. Let n be an integer and $T_0 \supset J_0$ intervals such that $f^n|_{T_0}$ is a diffeomorphism, $f^n(T_0)=T$ and $f^n(J_0)=J$. If $\varepsilon :=\max \{|f^j(T_0)|\,;\,0\le j\le n\}$, then

• (1) $\big |\frac{Df^n(x)}{Df^n(y)}\big |\le \big (\frac{1+\delta }{\delta }\big )^2 e^{O(\varepsilon )\sum _{i=0}^{n-1}|f^i(J_0)|}$, for all $x,y\in J_0$;

• (2) $\exists \delta '>0$ depending only on $\varepsilon$ and $\sum _{i=0}^{n-1}|f^i(J_0)|$ such that, for all $x,y\in J_0$ and $1\le j\le n$, we have $\big |\frac{Df^j(x)}{Df^j(y)}\big |\le \big (\frac{1+\delta '}{\delta '}\big )^2 e^{O(\varepsilon )\sum _{i=0}^{n-1}|f^i(J_0)|}$;

• (3) $\frac{|Df^n(x)|}{|Df^n(y)|}\le \exp \big (\frac{2}{\delta |J|}|f^n(x)-f^n(y)|+O(\varepsilon )\sum _{i=0}^{n-1}|f^i(x)-f^j(y)|\big )$ for every $x,y\in J_0$.

Proposition 3.12

(Proposition 3.4 for maps with negative Schwarzian derivative) Let U be an open subset of an interval (a, b) and $F:U\rightarrow (a,b)$ be a $C^3$ local diffeomorphism with negative Schwarzian derivative and such that $F(P)=(a,b)$ for every connected component of U. If ${\mathbb B}(F)$ denotes the union of all wandering intervals of F with the basin of attraction of all periodic-like attractors of F, then

$${\text {Leb}}\bigg (\bigcap _{n\ge 0}F^{-n}((a,b))\setminus {\mathbb B}(F)\bigg )\text { is either }0\text { or }b-a.$$

Furthermore, if ${\text {Leb}}\big (\bigcap _{n\ge 0}F^{-n}((a,b))\setminus {\mathbb B}(F)\big )=b-a$ then ${\text {Leb}}|_{(a,b)}$ is F strongly ergodic and $\omega _F(x)=[a,b]$ for almost every $x\in (a,b)$.   $\square$

Proof

Similar to the proof of Proposition 3.4. Essentially, one only need to replace Proposition 3.11 by Koebe’s Lemma (Lemma 3.10).

Proof of Proposition 2.5 As the attractor A is a cycle of intervals, we get from Theorem 1 that $\omega _f(x)=A$ for almost every $x\in [0,1]$. In particular, this forbids f to have wandering intervals and periodic attractors. As $\overline{\mathcal {O}_f^+(c_-)\cup \mathcal {O}_f^+(c_+)}$ has empty interior, we can consider a connected component $J\ne \emptyset$ of $A\setminus ( \overline{\mathcal {O}_f^+(c_-)\cup \mathcal {O}_f^+(c_+)})$. As $\overline{\mathcal {O}_f^+(c_-)\cup \mathcal {O}_f^+(c_+)}\ne A$, A is not a minimal set and so it cannot be a Cherry attractor. Hence, it follows from Theorem D of [8] that A is a chaotic cycle of intervals. In particular, this implies that $\overline{{\text {Per}}(f)\cap A}=A$. Thus, choose $q,p\in {\text {Per}}(f)\cap {\text {interior}}(J)$ with $p<q$. Let I be any connected component of $(p,q)\setminus \overline{\mathcal {O}_f^+(c_-)\cup \mathcal {O}_f^+(c_+)}$. Note that I is a nice interval, that is, $\mathcal {O}_f^+(\partial I)\cap I=\emptyset$. Let $I^*=\{x\in \ I\,;\,\mathcal {O}_f^+(f(x))\cap I\ne \emptyset \}$ and $F:I^*\rightarrow I$ the first return map, by f, to I. As $\omega _f(x)\supset I$ for almost every $x\in [0,1]$ and ${\text {Leb}}\circ f^{-1}\ll {\text {Leb}}$, we get that ${\text {Leb}}(I)={\text {Leb}}(I^*)={\text {Leb}}(\bigcap _{n\ge 0}F^{-1}(I^*))$. Thus, it follows from Proposition 3.12 that ${\text {Leb}}|_I$ is strongly ergodic with respect to F, in particular, F is ergodic.

The ergodicity of ${\text {Leb}}|_I$ with respect to f implies that ${\text {Leb}}$ is also ergodic with respect to f. Indeed, if $V=f^{-1}(V)$ and ${\text {Leb}}(V)>0$ then, as ${\text {Leb}}\circ f^{-1}\ll {\text {Leb}}$ and $\mathcal {O}_f^+(x)\cap I\ne \emptyset$ for ${\text {Leb}}$ almost all $x\in [0,1]$, we get that ${\text {Leb}}(V\cap I)>0$. As F is the first return map to I, it follows that $F^{-1}(V\cap I)=V\cap I$. Thus, by the ergodicity with respect to F, ${\text {Leb}}(V\cap I)={\text {Leb}}(I)$. That is, ${\text {Leb}}(I\setminus V)=0$. Using that ${\text {Leb}}\circ f^{-1}\ll {\text {Leb}}$, we get that ${\text {Leb}}([0,1]\setminus V)=0$, proving that ${\text {Leb}}$ is ergodic with respect to f.