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November  2019, 39(11): 6241-6260. doi: 10.3934/dcds.2019272

Escape dynamics for interval maps

Carlos Correia Ramos 1,, , Nuno Martins 2, and  Paulo R. Pinto 2,

1. 

Centro de Investigação em Matemática e Aplicações, Dep. of Mathematics, Univ. de Évora, R. Romão Ramalho, 59, 7000-671 Évora, Portugal
2. 

Dep. of Mathematics, CAMGSD, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal

* Corresponding author

Received  October 2018 Revised  March 2019 Published  August 2019

Fund Project: Correia Ramos's work was partially supported by national funds through Fundação Nacional para a Ciência e a Tecnologia (FCT) of Portugal grant PEstOE/MAT/UI0117/2014 and the Centro de Investigação em Matemática e Aplicações of Univ. of Évora. The work of Martins and Pinto was partially supported by FCT/Portugal grant UID/MAT/04459/2013.

We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix $ \widehat{A}_f $ of an interval map $ f $, extending the traditional matrix $ A_f $ which considers the transition among the Markov subintervals. We show that the escape transition matrix is a topological conjugacy invariant. We then characterize the $ 0 $–$ 1 $ matrices that can be fabricated as escape transition matrices of Markov interval maps $ f $ with escape sets. This shows the richness of this class of interval maps.

Keywords: Open dynamical systems, transition matrix, interval map, symbolic dynamics, escape set.
Mathematics Subject Classification: Primary: 37B10; Secondary: 37E05.
Citation: Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto. Escape dynamics for interval maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6241-6260. doi: 10.3934/dcds.2019272
References:
[1]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst., 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

S. Bundfuss, T. Krüger and S. Troubetzkoy, Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory Dynam. Systems, 31 (2011), 1305-1323.  doi: 10.1017/S0143385710000556.  Google Scholar

[4]

L. Clark, The β-transformation with a hole, Discrete Contin. Dyn. Syst., 36 (2016), 1249-1269.  doi: 10.3934/dcds.2016.36.1249.  Google Scholar

[5]

C. Correia Ramos, N. Martins and P. R. Pinto, Interval maps from Cuntz-Krieger algebras, J. Math. Anal. Appl., 374 (2011), 347-354.  doi: 10.1016/j.jmaa.2010.09.045.  Google Scholar

[6]

C. Correia Ramos, N. Martins and P. R. Pinto, Toeplitz algebras arising from escape points of interval maps, Banach J. Math. Anal., 11 (2017), 536-553.  doi: 10.1215/17358787-2017-0005.  Google Scholar

[7]

C. Correia Ramos, N. Martins and P. R. Pinto, On graph algebras from interval maps, Ann. Funct. Anal., 10 (2019), 203-217.  doi: 10.1215/20088752-2018-0019.  Google Scholar

[8]

B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representations of numbers, Ann. Inst. H. Poincaré Sect. A (N.S.), 29 (1978), 305-356.   Google Scholar

[9]

K. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Ltd., Chichester, 1997.  Google Scholar

[10]

J. P. Lampreia, A. Rica da Silva and J. Sousa Ramos, Construction of 0–1 matrices associated to period-doubling processes, Stochastica, 9 (1985), 165-178.   Google Scholar

[11]

J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugal. Math., 54 (1997), 1-18.   Google Scholar

[12] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[13]

N. Sidorov, Arithmetic dynamics, Cambridge Univ. Press, Cambridge, 310 (2003), 145-189.  doi: 10.1017/CBO9780511546716.010.  Google Scholar

show all references

References:
[1]

R. Alcaraz Barrera, Topological and ergodic properties of symmetric sub-shifts, Discrete Contin. Dyn. Syst., 34 (2014), 4459-4486.  doi: 10.3934/dcds.2014.34.4459.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

S. Bundfuss, T. Krüger and S. Troubetzkoy, Topological and symbolic dynamics for hyperbolic systems with holes, Ergodic Theory Dynam. Systems, 31 (2011), 1305-1323.  doi: 10.1017/S0143385710000556.  Google Scholar

[4]

L. Clark, The β-transformation with a hole, Discrete Contin. Dyn. Syst., 36 (2016), 1249-1269.  doi: 10.3934/dcds.2016.36.1249.  Google Scholar

[5]

C. Correia Ramos, N. Martins and P. R. Pinto, Interval maps from Cuntz-Krieger algebras, J. Math. Anal. Appl., 374 (2011), 347-354.  doi: 10.1016/j.jmaa.2010.09.045.  Google Scholar

[6]

C. Correia Ramos, N. Martins and P. R. Pinto, Toeplitz algebras arising from escape points of interval maps, Banach J. Math. Anal., 11 (2017), 536-553.  doi: 10.1215/17358787-2017-0005.  Google Scholar

[7]

C. Correia Ramos, N. Martins and P. R. Pinto, On graph algebras from interval maps, Ann. Funct. Anal., 10 (2019), 203-217.  doi: 10.1215/20088752-2018-0019.  Google Scholar

[8]

B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representations of numbers, Ann. Inst. H. Poincaré Sect. A (N.S.), 29 (1978), 305-356.   Google Scholar

[9]

K. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Ltd., Chichester, 1997.  Google Scholar

[10]

J. P. Lampreia, A. Rica da Silva and J. Sousa Ramos, Construction of 0–1 matrices associated to period-doubling processes, Stochastica, 9 (1985), 165-178.   Google Scholar

[11]

J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugal. Math., 54 (1997), 1-18.   Google Scholar

[12] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[13]

N. Sidorov, Arithmetic dynamics, Cambridge Univ. Press, Cambridge, 310 (2003), 145-189.  doi: 10.1017/CBO9780511546716.010.  Google Scholar

Figure 1.  Graph of the function in Example 2.4
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Figure 2.  Graph of the function $ f $ in Example 2.9
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Figure 3.  Graph of the function $ g_1 $ in Example 2.9
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Figure 4.  Graph of the function $ \phi_1 $ in Example 2.9
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