Work Description

Date: 2020-11-05

Dataset Title: Itineraries of Tent Maps up to Orbit Length 34

Dataset Creators: Robert Buckley, Grace O’Brien, Zoe Zhou

Dataset Contact: Robert Buckley: robuckle@umich.edu, Grace O’Brien: graceob@umich.edu, Zoe Zhou: zoezhou@umich.edu

Key Points:
- Complete set (with minimal redundancies) of itineraries of tent maps up to length 34
- This data was used for a new and direct proof that a tent map of slope beta has a topological entropy log beta.

Research Overview:
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 (Thurston W. 2014; 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.

Methodology:
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 (Milnor J, Thurston W. 1988; 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.

Below is a simple example of the dataset in action using Python. The code prints all itineraries of orbit length less than a given number. (Note: Indexing of numbers starts at 0 not 1)

import os
rootdir = '???' #folder download where zipped itineraries archives were unpacked
def print_lists_less_than(n):
for subdir, dirs, file_names in os.walk(rootdir):
for file_name in file_names:
file = open(os.path.join(subdir, file_name))
for itinerary in file:
print('{} with length {}'.format(itinerary, subdir.rsplit('\\', 1)[-1]))
print_lists_less_than(8)

Date Coverage: 2019-09-03 - 2019-12-18

Instrument and/or Software specifications: N/A

Related publication(s):
Buckley R, O’Brien G, Zhou Z (2021). On Itineraries of Tent Maps. Forthcoming.

Use and Access:
This data set is made available under a Creative Commons Attribution-NonCommercial 4.0 International license (CC BY-NC 4.0).

To Cite the Data:
Buckley R, O’Brien G, Zhou Z (2021). Itineraries of Tent Maps up to Orbit Length 34 [Data set]. University of Michigan - Deep Blue Data. https://doi.org/10.7302/k7ex-v245.

Articles Referenced Above:
Thurston W. 2014 Feb 9. Entropy in Dimension One. arXiv:14022008 [math]. [accessed 2021 Jun 3]. http://arxiv.org/abs/1402.2008.
Milnor J, Thurston W. 1988. On iterated maps of the interval. In: Alexander JC, editor. Dynamical Systems. Berlin, Heidelberg: Springer. (Lecture Notes in Mathematics). p. 465–563. https://link.springer.com/chapter/10.1007/BFb0082847

Files contained here:

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).

Tent Maps Itineraries File Manifest

3.zip
2_0.txt
4.zip
2_0.txt
5.zip
2_0.txt
4_0.txt
6.zip
2_0.txt
4_0.txt
7.zip
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11.zip
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14.zip
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15.zip
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16.zip
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17.zip
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18.zip
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