May  2017, 37(5): 2285-2300. doi: 10.3934/dcds.2017100

Uncountably many planar embeddings of unimodal inverse limit spaces

1. 

Faculty of Electrical Engineering and Computing, Unska 3,10000 Zagreb, Croatia

2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

* Corresponding author

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: AA was supported in part by Croatian Science Foundation under the project IP-2014-09-2285. HB and JČ were supported by the FWF stand-alone project P25975-N25. We gratefully acknowledge the support of the bilateral grant Strange Attractors and Inverse Limit Spaces, Österreichische Austauschdienst (OeAD) -Ministry of Science, Education and Sport of the Republic of Croatia (MZOS), project number HR 03/2014

For a point $x$ in the inverse limit space $X$ with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that $x$ is accessible. It follows that there are uncountably many non-equivalent planar embeddings of $X$.

Citation: Ana Anušić, Henk Bruin, Jernej Činč. Uncountably many planar embeddings of unimodal inverse limit spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2285-2300. doi: 10.3934/dcds.2017100
References:
[1]

M. Barge, Horseshoe maps and inverse limits, Pacific J. Math., 121 (1986), 29-39. doi: 10.2140/pjm.1986.121.29. Google Scholar

[2]

M. BargeK. Brucks and B. Diamond, Self-similarity of inverse limits of tent maps, Proc. Amer. Math. Soc., 124 (1996), 3563-3570. Google Scholar

[3]

M. BargeH. Bruin and S. Štimac, The Ingram conjecture, Geom. Topol., 16 (2012), 2481-2516. doi: 10.2140/gt.2012.16.2481. Google Scholar

[4]

M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity, 8 (1995), 29-42. doi: 10.1088/0951-7715/8/1/003. Google Scholar

[5]

R. H. Bing, Snake-like continua, Duke Math J., 18 (1951), 653-663. doi: 10.1215/S0012-7094-51-01857-1. Google Scholar

[6]

P. BoylandA. de Carvalho and T. Hall, Inverse limits as attractors in parametrized families, Bull. Lond. Math. Soc., 45 (2013), 1075-1085. Google Scholar

[7]

B. Brechner, On stable homeomorphisms and imbeddings of the pseudo-arc, Illinois Journal of Mathematics, 22 (1978), 630-661. Google Scholar

[8]

K. Brucks and B. Diamond, A symbolic representation of inverse limit spaces for a class of unimodal maps, Continuum Theory and Dynamical Systems, Lecture Notes in Pure Appl. Math., 170 (1995), 207-226. Google Scholar

[9]

K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246. Google Scholar

[10]

H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., 96 (1999), 191-208. doi: 10.1016/S0166-8641(98)00054-6. Google Scholar

[11]

C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann., 73 (1913), 323-370. Google Scholar

[12]

P. Cvitanović, Periodic orbits as the skeleton of classical and quantum chaos, Phys. D., 51 (1991), 138-151. Google Scholar

[13]

P. CvitanovićG. Gunaratne and I. Procaccia, Topological and metric properties of Hénon-type strange attractors, Phys. Rev. A(3), 38 (1988), 1503-1520. Google Scholar

[14]

W. Dȩbski and E. Tymchatyn, A note on accessible composants in Knaster continua, Houston J. Math., 19 (1993), 435-442. Google Scholar

[15]

D. Iliadis, An investigation of plane continua via Carathéodory prime ends, Dokl. Akad. Nauk SSSR, 204 (1972), 828-832. Google Scholar

[16]

W. T. Ingram and W. S. Mahavier, "Inverse Limits: From Continua to Chaos", Developments in Mathematics, vol. 25, Springer, New York, 2012. doi: 10.1007/978-1-4614-1797-2. Google Scholar

[17]

W. S. Mahavier, Embeddings of simple indecomposable continua in the plane, Topology Proc., 14 (1989), 131-140. Google Scholar

[18]

J. C. Mayer, Inequivalent embeddings and prime ends, Topology Proc., 8 (1983), 99-159. Google Scholar

[19]

S. Mazurkiewicz, Un théorème sur l'accessibilité des continus indécomposables, Fund, Math, 14 (1929), 271-276. Google Scholar

[20]

V. Mendoza, Proof of the pruning front conjecture for certain Hénon parameters, Nonlinearity, 26 (2013), 679-690. doi: 10.1088/0951-7715/26/3/679. Google Scholar

[21]

J. Milnor and W. Thurston, On Iterated maps of the interval, in "Dynamical Systems" (College Park, MD, 1986-87), Lecture Notes in Math. , 1342, Springer, Berlin, 1988. , 465–563. doi: 10.1007/BFb0082847. Google Scholar

[22]

J. R. Munkres, Topology Second Edition, Prentice-Hall, Inc. , Englewood Cliffs, New Jersey, 1975. Google Scholar

[23]

S. B. Nadler, Continuum Theory: An Introduction, Marcel Dekker, Inc. , New York, 1992. Google Scholar

[24]

S. P. Schwartz, Some Planar Embeddings of Chainable Continua Can be Expressed as Inverse Limit Spaces, PhD. Thesis, Montana State University, 1992. Google Scholar

[25]

R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487. doi: 10.1016/0040-9383(67)90005-5. Google Scholar

[26]

R. F. Williams, Classification of one dimensional attractors, in "Global Analysis" (Proc. Sympos. Pure Math. , Vol. XIV, Berkeley, Calif. 1968), Amer. Math. Soc. , Providence, R. I. , (1970), 341–361. doi: 10.1090/pspum/014/0266227. Google Scholar

show all references

References:
[1]

M. Barge, Horseshoe maps and inverse limits, Pacific J. Math., 121 (1986), 29-39. doi: 10.2140/pjm.1986.121.29. Google Scholar

[2]

M. BargeK. Brucks and B. Diamond, Self-similarity of inverse limits of tent maps, Proc. Amer. Math. Soc., 124 (1996), 3563-3570. Google Scholar

[3]

M. BargeH. Bruin and S. Štimac, The Ingram conjecture, Geom. Topol., 16 (2012), 2481-2516. doi: 10.2140/gt.2012.16.2481. Google Scholar

[4]

M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity, 8 (1995), 29-42. doi: 10.1088/0951-7715/8/1/003. Google Scholar

[5]

R. H. Bing, Snake-like continua, Duke Math J., 18 (1951), 653-663. doi: 10.1215/S0012-7094-51-01857-1. Google Scholar

[6]

P. BoylandA. de Carvalho and T. Hall, Inverse limits as attractors in parametrized families, Bull. Lond. Math. Soc., 45 (2013), 1075-1085. Google Scholar

[7]

B. Brechner, On stable homeomorphisms and imbeddings of the pseudo-arc, Illinois Journal of Mathematics, 22 (1978), 630-661. Google Scholar

[8]

K. Brucks and B. Diamond, A symbolic representation of inverse limit spaces for a class of unimodal maps, Continuum Theory and Dynamical Systems, Lecture Notes in Pure Appl. Math., 170 (1995), 207-226. Google Scholar

[9]

K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246. Google Scholar

[10]

H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., 96 (1999), 191-208. doi: 10.1016/S0166-8641(98)00054-6. Google Scholar

[11]

C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann., 73 (1913), 323-370. Google Scholar

[12]

P. Cvitanović, Periodic orbits as the skeleton of classical and quantum chaos, Phys. D., 51 (1991), 138-151. Google Scholar

[13]

P. CvitanovićG. Gunaratne and I. Procaccia, Topological and metric properties of Hénon-type strange attractors, Phys. Rev. A(3), 38 (1988), 1503-1520. Google Scholar

[14]

W. Dȩbski and E. Tymchatyn, A note on accessible composants in Knaster continua, Houston J. Math., 19 (1993), 435-442. Google Scholar

[15]

D. Iliadis, An investigation of plane continua via Carathéodory prime ends, Dokl. Akad. Nauk SSSR, 204 (1972), 828-832. Google Scholar

[16]

W. T. Ingram and W. S. Mahavier, "Inverse Limits: From Continua to Chaos", Developments in Mathematics, vol. 25, Springer, New York, 2012. doi: 10.1007/978-1-4614-1797-2. Google Scholar

[17]

W. S. Mahavier, Embeddings of simple indecomposable continua in the plane, Topology Proc., 14 (1989), 131-140. Google Scholar

[18]

J. C. Mayer, Inequivalent embeddings and prime ends, Topology Proc., 8 (1983), 99-159. Google Scholar

[19]

S. Mazurkiewicz, Un théorème sur l'accessibilité des continus indécomposables, Fund, Math, 14 (1929), 271-276. Google Scholar

[20]

V. Mendoza, Proof of the pruning front conjecture for certain Hénon parameters, Nonlinearity, 26 (2013), 679-690. doi: 10.1088/0951-7715/26/3/679. Google Scholar

[21]

J. Milnor and W. Thurston, On Iterated maps of the interval, in "Dynamical Systems" (College Park, MD, 1986-87), Lecture Notes in Math. , 1342, Springer, Berlin, 1988. , 465–563. doi: 10.1007/BFb0082847. Google Scholar

[22]

J. R. Munkres, Topology Second Edition, Prentice-Hall, Inc. , Englewood Cliffs, New Jersey, 1975. Google Scholar

[23]

S. B. Nadler, Continuum Theory: An Introduction, Marcel Dekker, Inc. , New York, 1992. Google Scholar

[24]

S. P. Schwartz, Some Planar Embeddings of Chainable Continua Can be Expressed as Inverse Limit Spaces, PhD. Thesis, Montana State University, 1992. Google Scholar

[25]

R. F. Williams, One-dimensional non-wandering sets, Topology, 6 (1967), 473-487. doi: 10.1016/0040-9383(67)90005-5. Google Scholar

[26]

R. F. Williams, Classification of one dimensional attractors, in "Global Analysis" (Proc. Sympos. Pure Math. , Vol. XIV, Berkeley, Calif. 1968), Amer. Math. Soc. , Providence, R. I. , (1970), 341–361. doi: 10.1090/pspum/014/0266227. Google Scholar

Figure 1.  Example of two basic arcs having a boundary point in common.
Figure 2.  Coding the Cantor set with respect to $(a)$ $ L=\ldots 111.$ and $(b)$ $ L=\ldots 101.$
Figure 3.  Case Ⅰ in the proof of Proposition 3.
Figure 4.  Case Ⅱ in the proof of Proposition 3.
Figure 5.  The planar representation of an arc in $X$ with the corresponding kneading sequence $\nu=100110010\ldots$. The ordering on basic arcs is such that the basic arc coded by $ L=1^{\infty}.$ is the largest.
Figure 6.  The planar representation of the same arc as in Figure 5 in $X$ with the corresponding kneading sequence $\nu=100110010\ldots$. The ordering on basic arcs is such that the basic arc coded by $ L=(101)^{\infty}.$ is the largest.
Figure 7.  Set-up in Lemma 4.1.
Figure 8.  Sets constructed in the proof of Lemma 4.1.
Figure 9.  Point $a = (a_0, \psi_L(L))$ is accessible.
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