{"id":{"repo_id":"sfasu","oai_identifier":"oai:scholarworks.sfasu.edu:etds-1598"},"canonical_url":"https://search.dev.ndltd.org/etd/sfasu/oai:scholarworks.sfasu.edu:etds-1598","repository":{"repo_id":"sfasu","name":"Stephen F. Austin State University","base_url":"https://scholarworks.sfasu.edu/do/oai/"},"display":{"title":"Exploring the Mandelbrot Set","abstract":"The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-<br />tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases<br />its usefulness in fields across the many realms of math—ranging from physics to nu-<br />merical methods and even biology. While typically defined in terms of its bounded<br />sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-<br />eterization of connectivity itself, specifically that of complex-valued rational maps<br />of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges<br />the worlds of intuition and theory with a series of self-contained arguments found in<br />published texts over the years since the Mandelbrot set’s conception—all to answer<br />one question: is the Mandelbrot set connected? That is, are there any pieces of the<br />Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the<br />now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives<br />into topologically grounded ideas and makes use of some functional analysis.","abstract_html":"The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-&lt;br /&gt;tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases&lt;br /&gt;its usefulness in fields across the many realms of math—ranging from physics to nu-&lt;br /&gt;merical methods and even biology. While typically defined in terms of its bounded&lt;br /&gt;sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-&lt;br /&gt;eterization of connectivity itself, specifically that of complex-valued rational maps&lt;br /&gt;of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges&lt;br /&gt;the worlds of intuition and theory with a series of self-contained arguments found in&lt;br /&gt;published texts over the years since the Mandelbrot set’s conception—all to answer&lt;br /&gt;one question: is the Mandelbrot set connected? That is, are there any pieces of the&lt;br /&gt;Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the&lt;br /&gt;now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives&lt;br /&gt;into topologically grounded ideas and makes use of some functional analysis.","abstract_has_math":false,"creators":["Shirley, James"],"institution":null,"degree_name":"Master of Science - Mathematical Sciences","degree_level":"Thesis","degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":["Jacob Pratscher","Roy Joe Harris","Clint Richardson"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-05-11T07:00:00Z","date_published":"2024-05-11T07:00:00Z","updated_at":"2026-07-24T04:30:45Z","subjects":["Mandelbrot","Complex Analysis","Analysis","Connected","Connectivity","Dynamical Systems"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.sfasu.edu/etds/546","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jacob Pratscher","Roy Joe Harris","Clint Richardson"]},{"key":"dc:creator","label":"Author","values":["Shirley, James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2024-05-28T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science - Mathematical Sciences"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mandelbrot","Complex Analysis","Analysis","Connected","Connectivity","Dynamical Systems"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.sfasu.edu/etds/546"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-<br />tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases<br />its usefulness in fields across the many realms of math—ranging from physics to nu-<br />merical methods and even biology. While typically defined in terms of its bounded<br />sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-<br />eterization of connectivity itself, specifically that of complex-valued rational maps<br />of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges<br />the worlds of intuition and theory with a series of self-contained arguments found in<br />published texts over the years since the Mandelbrot set’s conception—all to answer<br />one question: is the Mandelbrot set connected? That is, are there any pieces of the<br />Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the<br />now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives<br />into topologically grounded ideas and makes use of some functional analysis."]},{"key":"dc:title","label":"Title","values":["Exploring the Mandelbrot Set"]}]}],"canonical_facts":{"dc:contributor":["Jacob Pratscher","Roy Joe Harris","Clint Richardson"],"dc:creator":["Shirley, James"],"dc:date.available":["2024-05-28T07:00:00Z"],"dc:description.abstract":["The Mandelbrot set is a mathematical mystery. Finding its home somewhere be-<br />tween holomorphic dynamics and complex analysis, the Mandelbrot set showcases<br />its usefulness in fields across the many realms of math—ranging from physics to nu-<br />merical methods and even biology. While typically defined in terms of its bounded<br />sequences, this thesis intends to illuminate the Mandelbrot set as a type of param-<br />eterization of connectivity itself, specifically that of complex-valued rational maps<br />of the form z → z² + c. This fully illustrated guide to the Mandelbrot set merges<br />the worlds of intuition and theory with a series of self-contained arguments found in<br />published texts over the years since the Mandelbrot set’s conception—all to answer<br />one question: is the Mandelbrot set connected? That is, are there any pieces of the<br />Mandelbrot set just ‘hanging off’ ? To answer this, we will appeal to the proof by the<br />now-famous collaborators Adrien Douady and John Hubbard, whose work deep-dives<br />into topologically grounded ideas and makes use of some functional analysis."],"dc:identifier":["https://scholarworks.sfasu.edu/etds/546"],"dc:subject":["Mandelbrot","Complex Analysis","Analysis","Connected","Connectivity","Dynamical Systems"],"dc:title":["Exploring the Mandelbrot Set"],"thesis:degree_discipline":["Mathematics and Statistics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science - Mathematical Sciences"]},"updated_at":"2026-07-24T04:30:45Z"}