Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Algorithms. (1)
- Cantor Sets (1)
- Continued Fractions (1)
- Dynamical Systems (1)
- Dynamical systems (1)
-
- Dynamics (1)
- Euclidean buildings (1)
- Fractals (1)
- Group actions (1)
- Hausdorff dimension (1)
- Heisenberg group (1)
- Hyperbolic geometry (1)
- Interval Maps (1)
- Kleinian group (1)
- Lebesgue Measure Preserving Maps (1)
- Lie groups (1)
- Linear fractional transformations (1)
- P-adic (1)
- Serre trees (1)
- Thompson' Group F (1)
Articles 1 - 5 of 5
Full-Text Articles in Dynamical Systems
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Rose-Hulman Undergraduate Mathematics Journal
In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.
On Cantor Sets Defined By Generalized Continued Fractions, Danielle Hedvig, Masha Gorodetski
On Cantor Sets Defined By Generalized Continued Fractions, Danielle Hedvig, Masha Gorodetski
Rose-Hulman Undergraduate Mathematics Journal
We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex setting, allowing the coefficient's argument to be a multiple of \pi/2. The numerical experiments we conducted showed that in these settings the set of numbers formed by such continued fractions is a Cantor set for large …
Lebesgue Measure Preserving Thompson Monoid And Its Properties Of Decomposition And Generators, William Li
Lebesgue Measure Preserving Thompson Monoid And Its Properties Of Decomposition And Generators, William Li
Rose-Hulman Undergraduate Mathematics Journal
This paper defines the Lebesgue measure preserving Thompson monoid, denoted by G, which is modeled on the Thompson group F except that the elements of G preserve the Lebesgue measure and can be non-invertible. The paper shows that any element of the monoid G is the composition of a finite number of basic elements of the monoid G and the generators of the Thompson group F. However, unlike the Thompson group F, the monoid G is not finitely generated. The paper then defines equivalence classes of the monoid G, use them to construct a monoid H …
Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott
Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott
Rose-Hulman Undergraduate Mathematics Journal
Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only …
New Experimental Investigations For The 3𝑥+1 Problem: The Binary Projection Of The Collatz Map, Benjamin Bairrington, Aaron Okano
New Experimental Investigations For The 3𝑥+1 Problem: The Binary Projection Of The Collatz Map, Benjamin Bairrington, Aaron Okano
Rose-Hulman Undergraduate Mathematics Journal
The 3x + 1 Problem, or the Collatz Conjecture, was originally developed in the early 1930's. It has remained unsolved for over eighty years. Throughout its history, traditional methods of mathematical problem solving have only succeeded in proving heuristic properties of the mapping. Because the problem has proven to be so difficult to solve, many think it might be undecidable. In this paper we brie y follow the history of the 3x + 1 problem from its creation in the 1930's to the modern day. Its history is tied into the development of the Cosper Algorithm, which maps binary sequences …