Dendrite (mathematics)

In mathematics, a dendrite is a certain type of topological space that may be characterized either as a locally connected dendroid or equivalently as a locally connected continuum that contains no simple closed curves.^[1]

Importance[edit]

Dendrites may be used to model certain types of Julia set.^[2] For example, if 0 is pre-periodic, but not periodic, under the function $f(z)=z^{2}+c$ , then the Julia set of $f$ is a dendrite: connected, without interior.^[3]

References[edit]

^ Whyburn, Gordon Thomas (1942), Analytic Topology, American Mathematical Society Colloquium Publications, vol. 28, New York: American Mathematical Society, p. 88, MR 0007095.
^ Carleson, Lennart; Gamelin, Theodore W. (1993), Complex Dynamics, Universitext, vol. 69, Springer, p. 94, ISBN 9780387979427.
^ Devaney, Robert L. (1989), An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity, Addison-Wesley Publishing Company, p. 294, MR 1046376.

See also[edit]

Misiurewicz point
Real tree, a related concept defined using metric spaces instead of topological spaces
Dendroid (topology) and unicoherent space, two more general types of tree-like topological space

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[1] Whyburn, Gordon Thomas (1942), Analytic Topology, American Mathematical Society Colloquium Publications, vol. 28, New York: American Mathematical Society, p. 88, MR 0007095.

[2] Carleson, Lennart; Gamelin, Theodore W. (1993), Complex Dynamics, Universitext, vol. 69, Springer, p. 94, ISBN 9780387979427.

[3] Devaney, Robert L. (1989), An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity, Addison-Wesley Publishing Company, p. 294, MR 1046376.

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