Abstract
In this paper we provide a characterization for a shift maximal sequence of 1's and 0's to be the kneading sequence for a unimodal map $f$ with $f|_{\omega(c)}$ topologically conjugate to an adding machine, where $c$ is the turning point of $f$. We show that the unimodal map $f$ has an embedded adding machine if and only if $\mathcal{K}(f)$ is a one-sided, non-periodic Toeplitz sequence with the finite time containment property. We then show the existence of unimodal maps with Toeplitz kneading sequences that do not have the finite time containment property.
Highlights
A great deal of literature is dedicated to the study of unimodal maps of an interval to itself
It is well-known that every infinitely renormalizable unimodal map has an embedded adding machine. It was shown by Block et al [5] that adding machines can be embedded in non-infinitely renormalizable unimodal maps, and they can be embedded in maps from the symmetric tent family
In this paper we provide a characterization for a unimodal map to have an embedded adding machine relying only on its kneading sequence
Summary
A great deal of literature is dedicated to the study of unimodal maps of an interval to itself (see, for example, [6, 7, 10]). In Theorem 4.7 we show that the unimodal map f is such that f |ω(c) is topologically conjugate to an adding machine if and only if the kneading sequence K(f ) is a shift maximal, non-periodic, one-sided Toeplitz sequence with the finite time containment property; this property will be defined in Definition 3.4 This characterization holds in both the infinitely renormalizable and non-infinitely renormalizable cases. We provide an example of a shift maximal, non-periodic Toeplitz sequence that does not have the finite time containment property In this example, the turning point for the associated unimodal map f will be regularly recurrent, but f |ω(c) will not be topologically conjugate to an adding machine
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
