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### Open Access Journals

DCDS

In this paper we investigate ``double rotations'',

*i.e.,*interval translation maps that when considered on the circle, have just two intervals of continuity. Using the induction procedure described by Suzuki et al., we show that Lebesgue a.e. double rotation is of finite type,*i.e.,*it reduces to an interval exchange transformation. However, the set of infinite type double rotations is shown to have Hausdorff dimension strictly between $2$ and $3$, and carries a natural induction-invariant measure. It is also shown that non-unique ergodicity of infinite type double rotations, although occurring, is a-typical with respect to every induction-invariant probability measure in parameter space.
DCDS

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$
with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $\sigma^R : K_s \to K_s$.

DCDS

We combine the technique of inducing with a method of

We prove the following dichotomy. Every absolutely continuous invariant measure is either $\sigma$-finite, or else it is infinite on every set of positive Lebesgue measure.

*Johnson boxes*and construct new examples of S-unimodal maps $\varphi$ which do not have a finite absolutely continuous invariant measure, but do have a $\sigma$-finite one which is infinite on every non-trivial interval.We prove the following dichotomy. Every absolutely continuous invariant measure is either $\sigma$-finite, or else it is infinite on every set of positive Lebesgue measure.

DCDS

For a point $x$ in the inverse limit space $X$ with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that $x$ is accessible. It follows that there are uncountably many non-equivalent planar embeddings of $X$.

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