Planar finitely Suslinian compacta
HTML articles powered by AMS MathViewer
- by Alexander Blokh, Michał Misiurewicz and Lex Oversteegen
- Proc. Amer. Math. Soc. 135 (2007), 3755-3764
- DOI: https://doi.org/10.1090/S0002-9939-07-08953-8
- Published electronically: August 15, 2007
- PDF | Request permission
Abstract:
We show that a planar unshielded compact set $X$ is finitely Suslinian if and only if there exists a closed set $F\subset \mathbb {S}^1$ and a lamination $\sim$ of $F$ such that $F/\!\sim$ is homeomorphic to $X$. If $X$ is a continuum, the analogous statement follows from Carathéodory theory and is widely used in polynomial dynamics.References
- Alexander Blokh, Chris Cleveland, and MichałMisiurewicz, Expanding polymodials, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 253–270. MR 2090774
- Alexander Blokh, Chris Cleveland, and MichałMisiurewicz, Julia sets of expanding polymodials, Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1691–1718. MR 2183289, DOI 10.1017/S0143385705000210
- Alexander Blokh and Lex Oversteegen, Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc. 356 (2004), no. 1, 119–133. MR 2020026, DOI 10.1090/S0002-9947-03-03415-9
- Richard D. Bourgin and Peter L. Renz, Shortest paths in simply connected regions in $\textbf {R}^2$, Adv. Math. 76 (1989), no. 2, 260–295. MR 1013673, DOI 10.1016/0001-8708(89)90054-6
- Morton Brown, Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321–323. MR 315714
- Adrien Douady, Descriptions of compact sets in $\textbf {C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
- Paul Fabel, “Shortest” arcs in closed planar disks vary continuously with the boundary, Topology Appl. 95 (1999), no. 1, 75–83. MR 1691933, DOI 10.1016/S0166-8641(97)00275-7
- K. Kuratowski, Topology II, Academic Press, New York, 1968.
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- R. L. Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), no. 4, 416–428. MR 1501320, DOI 10.1090/S0002-9947-1925-1501320-8
- W. Thurston, The combinatorics of iterated rational maps, Preprint (1985).
Bibliographic Information
- Alexander Blokh
- Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
- MR Author ID: 196866
- Email: ablokh@math.uab.edu
- Michał Misiurewicz
- Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- Email: mmisiure@math.iupui.edu
- Lex Oversteegen
- Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- Received by editor(s): January 4, 2006
- Received by editor(s) in revised form: September 8, 2006
- Published electronically: August 15, 2007
- Additional Notes: The first author was partially supported by NSF grant DMS 0456748
The second author was partially supported by NSF grant DMS 0456526
The third author was partially supported by by NSF grant DMS 0405774 - Communicated by: Alexander N. Dranishnikov
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3755-3764
- MSC (2000): Primary 54F15, 54D05, 37F10
- DOI: https://doi.org/10.1090/S0002-9939-07-08953-8
- MathSciNet review: 2336592