Inducing and unique ergodicity of double rotations
Henk Bruin Gregory Clack
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4133-4147 doi: 10.3934/dcds.2012.32.4133
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.
keywords: renormalisation double rotation unique ergodicity. interval translation map Piecewise isometries
On isotopy and unimodal inverse limit spaces
Henk Bruin Sonja Štimac
Discrete & Continuous Dynamical Systems - A 2012, 32(4): 1245-1253 doi: 10.3934/dcds.2012.32.1245
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 $\sigma^R : K_s \to K_s$.
keywords: inverse limit space. Isotopy tent map
New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure
Jawad Al-Khal Henk Bruin Michael Jakobson
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 35-61 doi: 10.3934/dcds.2008.22.35
We combine the technique of inducing with a method of 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.
keywords: infinite measure. sigma-finite invariant measure interval maps
Uncountably many planar embeddings of unimodal inverse limit spaces
Ana Anušić Henk Bruin Jernej Činč
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2285-2300 doi: 10.3934/dcds.2017100

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 $X$.

keywords: Unimodal map inverse limit space planar embeddings

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