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  • Itineraries of Tent Maps up to Orbit Length 34

    work
    Creator:
    Robert Buckley, Grace O'Brien, and Zoe Zhou
    Description:
    The purpose of the research is to better understand and approximate the Thurston Set. This project was computational in nature and Python was used to collect our data. The data set contains encoded itineraries that can be used to compute values that are elements of the Thurston Set. A visual approximation of the Thurston Set can be found here ( https://arxiv.org/abs/1402.2008), on the first page Thurston’s own paper. The data can also be used to study the distribution of superattracting beta values within the interval (1, 2] and to explore an analogous Mandelbrot-Julia Correspondence. This research was conducted through the Lab of Geometry at Michigan under the advisement of Harrison Bray during the Fall semester of 2019. , The Python 3.x scripts in this deposit are the exact versions used to created the *.txt files that are in the zip archive. As the project continues, any expansion to the work, such as further analysis or visualization scripts, will be posted to the project's GitHub  https://github.com/Tent-Maps-Team/Thurston-Set. Also, a user can reproduce our results and generate bigger datasets on machines with large amounts of memory. , and The data consists of zipper folders representing tent map itinerary orbit lengths. These orbit files can be used to create visualizations, create and explore conjectures such as refining proposed bounds on the Thurston Set and supporting an analogous Mandelbrot-Julia Correspondence. Within these zipped folders are .txt files in CSV format with the naming structure of xx_y of admissible itineraries up to the length indicated by the folder name where xx is the length of the encoded itineraries included. The txt's have a single column and each line(row) is an array representing an encoding of an itinerary. Some of the txt's have been split into multiple parts (whenever there are more than 200 MB of itinerary data) and these txt's have been numbered using the y after the underscore. As we exclude the degenerate tent map (where β = 1), we cannot have orbit length 1 or 2 and this is why the orbits start with length 3 (i.e. start with 3.zip).
    Keyword:
    Math, mathematics, tent maps, thurston, milnor, Milnor-Thurston, supperattracting, entropy, orbit, and itineraries
    Citation to related publication:
    Buckley R, O’Brien G, Zhou Z (2021). On Itineraries of Tent Maps. Forthcoming.
    Discipline:
    Other