DOI: 10.7302/k7ex-v245
title: Itineraries of Tent Maps up to Orbit Length 34

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.
;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).
description: At a high level, we first iterate over all possible itinerary sequences of length less than 34 and encode them as a list of the positions of 1’s in the original binary sequence to save memory. Then, we check which encoded itineraries correspond to a superattracting beta value using the Milnor-Thurston admissibility criterion (https://link.springer.com/content/pdf/10.1007/BFb0082847.pdf). Next, we ruled out the vast majority (not all) of redundant encoded itineraries (there are some redundant itineraries that would have required checking against all others, which was not possible due to computational resources). Finally, we saved the unique encoded itineraries into folders named for their length, as the encoding is meaningful only when paired with the length. The code can utilize multithreading to speed up this exhausting computation, it is currently commented out, though this will require vast stores of memory. All work was done in Python. The source code which generated the dataset is archived with the data. As we expand our work past the data set into further analysis and visualization you can find this work at https://github.com/Tent-Maps-Team/Thurston-Set.