Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces


Author: Alexander M. Blokh
Journal: Proc. Amer. Math. Soc. 143 (2015), 3985-4000
MSC (2010): Primary 37B45, 37E25, 54H20; Secondary 37C25, 37E05
DOI: https://doi.org/10.1090/S0002-9939-2015-12589-0
Published electronically: March 6, 2015
MathSciNet review: 3359587
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that self-mappings of uniquely arcwise connected locally arcwise connected spaces are pointwise-recurrent if and only if all their cutpoints are periodic while all endpoints are either periodic or belong to what we call ``topological weak adding machines''. We also introduce the notion of a ray complete uniquely arcwise connected locally arcwise connected space and show that for them the above ``topological weak adding machines'' coincide with classical adding machines (e.g., this holds if the entire space is compact).


References [Enhancements On Off] (What's this?)

  • [AEO07] Gerardo Acosta, Peyman Eslami, and Lex G. Oversteegen, On open maps between dendrites, Houston J. Math. 33 (2007), no. 3, 753-770. MR 2335734 (2008k:54014)
  • [ALM00] Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, Combinatorial dynamics and entropy in dimension one, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1807264 (2001j:37073)
  • [Bes88] Mladen Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), no. 1, 143-161. MR 932860 (89m:57011), https://doi.org/10.1215/S0012-7094-88-05607-4
  • [BFMOT11] Alexander M. Blokh, Robbert J. Fokkink, John C. Mayer, Lex G. Oversteegen, and E. D. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc. 224 (2013), no. 1053, xiv+97. MR 3087640, https://doi.org/10.1090/S0065-9266-2012-00671-X
  • [Blo80s] A. Blokh, On Dynamical Systems on One-Dimensional Branched Manifolds. 1, 2, 3 (in Russian), Theory of Functions, Functional Analysis and Applications, Kharkov, 46 (1986), 8-18; 47 (1986), 67-77; 48 (1987), 32-46.
  • [ES45] P. Erdös and A. H. Stone, Some remarks on almost periodic transformations, Bull. Amer. Math. Soc. 51 (1945), 126-130. MR 0011437 (6,165b)
  • [Got44] W. H. Gottschalk, Powers of homeomorphisms with almost periodic properties, Bull. Amer. Math. Soc. 50 (1944), 222-227. MR 0009850 (5,213c)
  • [HY88] John G. Hocking and Gail S. Young, Topology, 2nd ed., Dover Publications, Inc., New York, 1988. MR 1016814 (90h:54001)
  • [KP98] B. Kolev and M.-C. Pérouème, Recurrent surface homeomorphisms, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 1, 161-168. MR 1620528 (99j:58162), https://doi.org/10.1017/S0305004197002272
  • [MO90] John C. Mayer and Lex G. Oversteegen, A topological characterization of $ {\bf R}$-trees, Trans. Amer. Math. Soc. 320 (1990), no. 1, 395-415. MR 961626 (90k:54031), https://doi.org/10.2307/2001765
  • [MNO92] John C. Mayer, Jacek Nikiel, and Lex G. Oversteegen, Universal spaces for $ {\bf R}$-trees, Trans. Amer. Math. Soc. 334 (1992), no. 1, 411-432. MR 1081940 (93a:54034), https://doi.org/10.2307/2153989
  • [Mon37] Deane Montgomery, Pointwise Periodic Homeomorphisms, Amer. J. Math. 59 (1937), no. 1, 118-120. MR 1507223, https://doi.org/10.2307/2371565
  • [MS84] John W. Morgan and Peter B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), no. 3, 401-476. MR 769158 (86f:57011), https://doi.org/10.2307/1971082
  • [MS88] John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in $ 3$-manifolds, Ann. of Math. (2) 127 (1988), no. 2, 403-456. MR 932305 (89e:57010a), https://doi.org/10.2307/2007061
  • [MT89] Piotr Minc and W. R. R. Transue, Sarkovskiĭ's theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc. 315 (1989), no. 1, 173-188. MR 965302 (89m:54054), https://doi.org/10.2307/2001378
  • [Nad92] Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552 (93m:54002)
  • [Nag12] Issam Naghmouchi, Pointwise-recurrent dendrite maps, Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1115-1123. MR 3082541, https://doi.org/10.1017/S0143385712000296
  • [Nik89] Jacek Nikiel, Topologies on pseudo-trees and applications, Mem. Amer. Math. Soc. 82 (1989), no. 416, vi+116. MR 988352 (90e:54075), https://doi.org/10.1090/memo/0416
  • [OT90] Lex G. Oversteegen and E. D. Tymchatyn, Recurrent homeomorphisms on $ {\bf R}^2$ are periodic, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1083-1088. MR 1037216 (91k:54070), https://doi.org/10.2307/2047760
  • [Sha64] O. M. Šarkovskiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Z. 16 (1964), 61-71 (Russian, with English summary). MR 0159905 (28 #3121)
  • [Sha64a] O. M. Šarkovskiĭ, Fixed points and the center of a continuous mapping of the line into itself, Dopovidi Akad. Nauk Ukraïn. RSR 1964 (1964), 865-868 (Ukrainian, with Russian and English summaries). MR 0165178 (29 #2467)
  • [Sha66] A. N. Šarkovskiĭ, The behavior of the transformation in the neighborhood of an attracting set, Ukrain. Mat. Ž. 18 (1966), no. 2, 60-83 (Russian). MR 0212784 (35 #3649)
  • [Sha66a] A. N. Šarkovskiĭ, Partially ordered system of attracting sets, Dokl. Akad. Nauk SSSR 170 (1966), 1276-1278 (Russian). MR 0209413 (35 #311)
  • [Sha67] O. M. Šarkovskiĭ, On a theorem of G. D. Birkhoff, Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A 1967 (1967), 429-432 (Ukrainian, with Russian and English summaries). MR 0212781 (35 #3646)
  • [Sha68] A. N. Šarkovskiĭ, Attracting sets containing no cycles, Ukrain. Mat. Ž. 20 (1968), no. 1, 136-142 (Russian). MR 0225314 (37 #908)
  • [Thu88] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. MR 956596 (89k:57023), https://doi.org/10.1090/S0273-0979-1988-15685-6
  • [Wea72] Norris Weaver, Pointwise periodic homeomorphisms of continua, Ann. of Math. (2) 95 (1972), 83-85. MR 0293600 (45 #2677)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B45, 37E25, 54H20, 37C25, 37E05

Retrieve articles in all journals with MSC (2010): 37B45, 37E25, 54H20, 37C25, 37E05


Additional Information

Alexander M. Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: ablokh@math.uab.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12589-0
Keywords: Periodic points, recurrent points, pointwise-recurrent maps, uniquely arcwise connected space, locally arcwise connected space
Received by editor(s): August 25, 2013
Received by editor(s) in revised form: May 24, 2014
Published electronically: March 6, 2015
Additional Notes: The author was partially supported by NSF grant DMS–1201450
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society