Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness

Kościelniak, Piotr; Oprocha, Piotr; Tuncali, Murat

doi:10.1090/S0002-9939-2013-11768-5

Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness
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by Piotr Kościelniak, Piotr Oprocha and Murat Tuncali PDF

Proc. Amer. Math. Soc. 142 (2014), 681-694 Request permission

Abstract:

We provide a method of construction of topologically mixing maps $f$ on topological graph $G$ with the shadowing property and such that the inverse limit with $f$ as the single bonding map is a hereditarily indecomposable continuum. Additionally, $f$ can be obtained as an arbitrarily small perturbation of any given topologically exact map on $G$, and if $G$ is the unit circle, then $f$ is necessarily topologically exact.

References

Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, 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, DOI 10.1142/4205
Ll. Alsedà, M. A. del Río, and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl. 232 (1999), no. 2, 359–375. MR 1683124, DOI 10.1006/jmaa.1999.6277
Ll. Alsedà, M. A. del Río, and J. A. Rodríguez, Transitivity and dense periodicity for graph maps, J. Difference Equ. Appl. 9 (2003), no. 6, 577–598. MR 1978125, DOI 10.1080/1023619021000040515
Ll. Alsedà, M. A. Del Río, and J. A. Rodríguez, A survey on the relation between transitivity and dense periodicity for graph maps, J. Difference Equ. Appl. 9 (2003), no. 3-4, 281–288. Dedicated to Professor Alexander N. Sharkovsky on the occasion of his 65th birthday. MR 1990336, DOI 10.1080/1023619021000047725
Marcy Barge and Beverly Diamond, The dynamics of continuous maps of finite graphs through inverse limits, Trans. Amer. Math. Soc. 344 (1994), no. 2, 773–790. MR 1236222, DOI 10.1090/S0002-9947-1994-1236222-1
R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451
L. Block, J. Keesling, and V. V. Uspenskij, Inverse limits which are the pseudoarc, Houston J. Math. 26 (2000), no. 4, 629–638. MR 1823960
A. M. Blokh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat. Nauk 42 (1987), no. 5(257), 209–210 (Russian). MR 928783
Rufus Bowen, Topological entropy and axiom $\textrm {A}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR 0262459
Morton Brown, On the inverse limit of Euclidean $N$-spheres, Trans. Amer. Math. Soc. 96 (1960), 129–134. MR 119182, DOI 10.1090/S0002-9947-1960-0119182-3
Lawrence Fearnley, Classification of all hereditarily indecomposable circularly chainable continua, Trans. Amer. Math. Soc. 168 (1972), 387–401. MR 296903, DOI 10.1090/S0002-9947-1972-0296903-9
George W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J. 31 (1964), 421–425. MR 166766
K. Kawamura, H. M. Tuncali, and E. D. Tymchatyn, Hereditarily indecomposable inverse limits of graphs, Fund. Math. 185 (2005), no. 3, 195–210. MR 2161403, DOI 10.4064/fm185-3-1
Piotr Kościelniak and Piotr Oprocha, Shadowing, entropy and a homeomorphism of the pseudoarc, Proc. Amer. Math. Soc. 138 (2010), no. 3, 1047–1057. MR 2566570, DOI 10.1090/S0002-9939-09-10162-4
D. P. Kuykendall, Irreducibility and indecomposability in inverse limits, Fund. Math. 80 (1973), no. 3, 265–270. MR 326684, DOI 10.4064/fm-80-3-265-270
D. Kwietniak, G. Harańczyk and P. Oprocha, Topological entropy and specification property for mixing graph maps, preprint, 2011.
Dominik Kwietniak and Piotr Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33 (2007), no. 1, 76–86. MR 2301847, DOI 10.1016/j.chaos.2005.12.033
Sergio Macías, Topics on continua, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2147759, DOI 10.1201/9781420026535
Lorenzo Sadun, Topology of tiling spaces, University Lecture Series, vol. 46, American Mathematical Society, Providence, RI, 2008. MR 2446623, DOI 10.1090/ulect/046
Piotr Minc and W. R. R. Transue, A transitive map on $[0,1]$ whose inverse limit is the pseudoarc, Proc. Amer. Math. Soc. 111 (1991), no. 4, 1165–1170. MR 1042271, DOI 10.1090/S0002-9939-1991-1042271-1
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059
Sam B. Nadler Jr., Continuum theory and graph theory: disconnection numbers, J. London Math. Soc. (2) 47 (1993), no. 1, 167–181. MR 1200986, DOI 10.1112/jlms/s2-47.1.167