Exact maps of the pseudo-arc
Keywords:
topological entropy, inverse limit space, pseudo-arc, topologically exactAbstract
In this paper, we provide a method of constructing a topologically exact map of the pseudo-arc using diagonal factor maps on the inverse limit space. Additionally, we give some criteria that guarantee when a topologically exact map of the pseudo-arc has infinite entropy.
References
R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742.
R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J . Math. 1(1951), 43-51.
R. H. Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer.
Math. Soc. 71 (1951), 267-273.
R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663.
Boroński, Činč, Oprocha, Beyond 0 and $infty$ : On the Barge Entropy Conjecture, The 53rd Annual Spring Topology Conference, Birmingham, AL March 14-16, 2019, preprint 2021, https://arxiv.org/pdf/2105.11133.pdf.
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414.
R. Bowen, Topological entropy and axiom A, in: Proc. Sympos. Pure Math. XIV (1970), 23-41.
M. Brown, On the inverse limit of Euclideann-spheres, Trans. Amer. Math. Soc. 96 (1960), 129--134.
E. Dinaburg, Relationship between topological entropy and metric entropy, Doklady Akademii Nauk SSSR. 170: 19.(1970).
T. Drwiega, P. Oprocha, Topologically Mixing Maps and the Pseudoarc, Ukr Math J 66 (2014), 197-208.
G. W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J. 31 (1964), 421-425.
L. Hoehn, C. Mouron, Hierarchies of chaotic maps on continua, Ergodic Theory and Dynamical Systems, 34(6) (2014), 1897-1913.
J.P. Huneke, Mountain Climbing, Transactions of the American Mathematical Society, 139 (1969), 383-391.
H. Kato, Chaos of continuum-wise expansive homeomorphisms and dynamical properties of sensitive maps of graphs, Pacific J. Math. 175 (1996), no. 1, 93116.
H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), no. 3, 576-598.
H. Kato, C. G. Mouron, Hereditarily indecomposable compacta do not admit expansive homeomorphisms, Proc. Amer. Math. Soc., 136 (10) (2008), 3689-3696.
J. Kennedy, A Transitive Homeomorphism on the Pseudoarc Which Is Semiconjugate to the Tent Map, Trans. Amer. Math. Soc., 326, no. 2 (1991), 773-793.
B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
P. Kościelniak, P. Oprocha, and M. Tuncali, Hereditarily indecomposable inverse limits of graphs: shadowing, mixing and exactness, Proc. Amer. Math. Soc., 142 (2014), 681-694.
W. Lewis, The Pseudo-Arc, Bol. Soc. Mat. Mexicana, 5 (1999), 25-77.
W. Lewis, P. Minc, Drawing the pseudo-arc, Houston J. Math. 36 (2010), 905-934.
J. Milnor Dynamics in one complex variable, volume no. 160, Princeton University Press, Princeton, 3rd ed edition, 2006.
P. Minc, W. R. R. Transue, A Transitive Map on [0,1] Whose Inverse Limit Is the Pseudoarc. Proc. A.M.S, 111 no. 4 (1991), 1165-1170.
J. Mioduszewski, Mappings of inverse limits, Colloquium Mathematicum 10 (1963), 39-44.
M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
E. E. Moise, An indecomposable plane continuum which is homeomorphic to each of itsnondegenerate subcontinua, Trans. Amer. Math. Soc. 63 (1948), 581-594.
E. E. Moise, A note on the pseudo-arc, Trans. Amer. Math. Soc. 67 (1949), 57-58.
C. Mouron, Entropy of shift maps of the pseudo-arc, Topology Appl. 159 (2012), 34-39.
C. Mouron, A chainable continuum that admits a homeomorphism with entropy of arbitrary value, Houston J. Math. 35 (4) (2009), 1079-1090.
S. B. Nadler, Jr. Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158. Marcel Dekker, Inc., New York, 1992. xiv +328
R. P. Vernon, Concerning Preservation of Indecomposability upon Taking a Preimage Under $z \rightarrow z^n$, Top. Proc. 31, no. 1 (2007), 331-348.
P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79 (1982), Springer-Verlag, New York-Berlin.
X. Ye, Topological entropy of the induced maps of the inverse limits with bonding maps, Topology Appl. 67 (1995), 113-118.