Two questions on one-dimensional dynamical systems

Question

(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) that the dynamical system $([0,1),T_{\beta})$ is mixing with topological entropy $\log(\beta)$. Now I wonder if this System is Devaney chaotic, means that in addition to transitivity periodic orbits are dense. If $\beta$ is an integer this is well known. In the literature I found the result that the system $([0,1),T_{\beta})$ is sofic if $\beta$ is a Pisot number, an algebraic integer with all its conjugates insides the unite circle. From this we get a dense set of periodic orbits. Is this known for any other $\beta>1$?

(2) For $\mu\in (0,4]$ let $f_{\mu}:[0,1]\to[0,1]$ be given by $f_{\mu}(x)=\mu x(1-x)$. I know the celebrated result that there is a set $B$ of positiv Lebesgue measure such that $([0,1],f_{\mu})$ has an chaotic attractor for $\mu\in B$. If I am not mistaken chaos means here topological and not Devaney chaos. My question is for which algebraic numbers $\mu<4$ the system $([0,1],f_{\mu})$ has a Devaney chaotic subsystem? Perhaps this is a difficult question, is anything known?\

---

Question (1) is answered. Concerning (2) I understand now that we have Devaney chaos of $([0,1],f_{\mu})$ for all $\mu\in B$. Are there any explicitly known (algebraic) numbers in $B$?

Answer 1

For continuous interval maps, Devaney's notion chaos is equivalent to the existence of a transitive orbit: that implies both the density of periodic points and the sensitivity on initial conditions. See Proposition 7.2 here for example. Now concerning the second question, Jakobson's celebrated result says that maps in the quadratic family admit an absolutely continuous ergodic invariant measure for a subset of positive Lebesgue measure of parameters. The fact that one can add ergodicity is explicitly mentioned in Yoccoz's proof of Jakobson's theorem. Theorem 1.1 there even specifies the support of that measure as the interval between the critical value and its image (the theorem is formulated for the quadratic family $x\mapsto x^2+c$ which is conjugate to the logistic family $\{f_\mu\}_\mu$). Now the ergodicity implies that there exists a point $x$ (in fact a subset of positive Lebesgue measure of such points) whose orbit is dense in the interval which is the support of that measure. For $\mu$ belonging to a subset of positive Lebesgue measure of parameters, $f_\mu$ has the property just mentioned.

As for first family, $T_\beta$ is not continuous. Thus it is not possible to immediately deduce Devaney chaos even from transitivity and the density of periodic points. So I think one should rely on the literature on $\beta$-expansions.

• As mentioned by Anthony Quas, $T_\beta$ admits an ergodic absolutely continuous invariant measure. Again, this fact implies transitivity.
• Regarding the density of periodic points the OP asks about, the paper β-Expansions and symbolic dynamics outlines some of the known results. In particular, there is a semi-conjugacy from a one-sided shift space onto $T_\beta$ (Proposition 2.1). The paper mentions positive solutions of $\beta^{n+1}=\beta^n+1$ ($n$ large enough) and $\beta^4=3\beta^3+2\beta^2+3$ as examples where the subshift is of finite type while $\beta$ is neither Pisot nor Salem (nevertheless it should be Perron). This can help with questions about density of periodic points and topological mixing (which is stronger than topological transitivity) because those questions are well-understood for subshifts of finite type.
• Finally, the sensitive dependence of $T_\beta$ on initial conditions can be proved directly. Consider $x,y\in [0,1)$ with $|x-y|<\frac{1}{\beta}$. I claim there exists $n\in\Bbb{N}$ with $\left|T_\beta^{\circ n}(x)-T_\beta^{\circ n}(y)\right|\geq\frac{1}{\beta+1}$. If $x,y$ belong to the same monotonic piece of $T_\beta$, the distance between
  $T_\beta^{\circ n}(x),T_\beta^{\circ n}(y)$ is multiplied by $\beta>1$ as $n$ turns into $n+1$ unless $T_\beta^{\circ n}(x),T_\beta^{\circ n}(y)$ are in different monotonic parts of $T_\beta$. If those parts are not adjacent, then the distance between them will be larger than $\frac{1}{\beta}$. It only remains to address the case that $x':=T_\beta^{\circ n}(x),y':=T_\beta^{\circ n}(y)$ are on different sides of a point of discontinuity $\frac{k}{\beta} \,(k\in\Bbb{N})$, say $x'<\frac{k}{\beta}\leq y'$. If $|x'-y'|>\frac{1}{\beta+1}$, we are done. Otherwise:

$$ \begin{split} &\left|T_\beta^{\circ (n+1)}(x)-T_\beta^{\circ (n+1)}(y)\right| =\left|T_\beta(x')-T_\beta(y')\right| =\left|(\beta x'-(k-1))-(\beta y'-k)\right|\\ &\geq 1-\beta|x'-y'|\geq 1-\beta\left(\frac{1}{\beta+1}\right)=\frac{1}{\beta+1}. \end{split} $$

Answer 2

I would like to put an input on question 1). In fact, every greedy $\beta$-transfomation $T_\beta$ is chaotic in the sense of Devaney. The result can be checked in Chaotic and topological properties of $\beta$-transformations by B. Li and Y.C. Chen.

@KhashF gave a proof for the sensitivity to initial conditions of the map. In order to check that the map is transitive (in fact mixing) and that the set of periodic points is dense, the easiest way to work with is to use the fact that the greedy $\beta$-transformation is semi-conjugated to the classical $\beta$-shift. In the symbolic case, showing both facts is reasonably straightforward.

To the best of my knowledge, I don't know if the existence of an absolutely continuous invariant measure with respect to the Lebesgue Measure for a piecewise-continuous map with a finite number of discontinuities would imply that the map is topologically transitive. It is an interesting question on its own.