Escape dynamics for interval maps

Carlos Correia Ramos; Nuno Martins; Paulo R. Pinto

doi:10.3934/dcds.2019272

Article Contents

2019, Volume 39, Issue 11: 6241-6260. Doi: 10.3934/dcds.2019272

This issue Previous Article Bowen entropy for fixed-point free flows Next Article The vorticity equation on a rotating sphere and the shallow fluid approximation

Escape dynamics for interval maps

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
^* Corresponding author

Received: September 30, 2018

Revised: February 28, 2019

Published: August 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 37E05.

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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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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.
[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.
[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.
[4]	L. Clark, The β-transformation with a hole, Discrete Contin. Dyn. Syst., 36 (2016), 1249-1269. doi: 10.3934/dcds.2016.36.1249.
[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.
[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.
[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.
[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.
[9]	K. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Ltd., Chichester, 1997.
[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.
[11]	J. P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugal. Math., 54 (1997), 1-18.
[12]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[13]	N. Sidorov, Arithmetic dynamics, Cambridge Univ. Press, Cambridge, 310 (2003), 145-189. doi: 10.1017/CBO9780511546716.010.