8 research outputs found
Tischler graphs of critically fixed rational maps and their applications
A rational map on the Riemann
sphere is called critically fixed if each critical point
of is fixed under . In this article we study properties of a
combinatorial invariant, called Tischler graph, associated with such a map.
More precisely, we show that the Tischler graph of a critically fixed rational
map is always connected, establishing a conjecture made by Kevin Pilgrim. We
also discuss the relevance of this result for classical open problems in
holomorphic dynamics, such as combinatorial classification problem and global
curve attractor problem
Exponential growth of some iterated monodromy groups
Iterated monodromy groups of postcritically finite rational maps form a rich class of self‐similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non‐polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non‐renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth
Tischler graphs of critically fixed rational maps and their applications
A rational map on the Riemann sphere is called critically fixed if each critical point of is fixed under . In this article we study properties of a combinatorial invariant, called Tischler graph, associated with such a map. More precisely, we show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. We also discuss the relevance of this result for classical open problems in holomorphic dynamics, such as combinatorial classification problem and global curve attractor problem
Tischler graphs of critically fixed rational maps and their applications
A rational map on the Riemann sphere is called critically fixed if each critical point of is fixed under . In this article we study properties of a combinatorial invariant, called Tischler graph, associated with such a map. More precisely, we show that the Tischler graph of a critically fixed rational map is always connected, establishing a conjecture made by Kevin Pilgrim. We also discuss the relevance of this result for classical open problems in holomorphic dynamics, such as combinatorial classification problem and global curve attractor problem
Exponential growth of some iterated monodromy groups
Iterated monodromy groups of postcritically finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth
Eliminating Thurston obstructions and controlling dynamics on curves
Every Thurston map on a -sphere induces a pull-back operation on Jordan curves , where is the postcritical set of . Here the isotopy class (relative to ) only depends on the isotopy class . We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map can be seen as a fixed point of the pull-back operation. We show that if a Thurston map with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying -sphere and construct a new Thurston map for which this obstruction is eliminated. We prove that no other obstruction arises and so is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem
Eliminating Thurston obstructions and controlling dynamics on curves
Every Thurston map on a -sphere induces a pull-back operation on Jordan curves , where is the postcritical set of . Here the isotopy class (relative to ) only depends on the isotopy class . We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map can be seen as a fixed point of the pull-back operation. We show that if a Thurston map with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying -sphere and construct a new Thurston map for which this obstruction is eliminated. We prove that no other obstruction arises and so is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem