Figure 1.
Graph of the function in Example 2.4
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November
2019, 39(11): 6241-6260.
doi: 10.3934/dcds.2019272
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 |
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.
Mathematics Subject Classification:
Primary: 37B10; Secondary: 37E05.
Citation:
Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto. Escape dynamics for interval maps. Discrete and Continuous Dynamical Systems,
2019, 39
(11)
: 6241-6260.
doi: 10.3934/dcds.2019272
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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. |
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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. |
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. |
| [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. |
Figure 2.
Graph of the function $ f $ in Example 2.9
Figure 3.
Graph of the function $ g_1 $ in Example 2.9
Figure 4.
Graph of the function $ \phi_1 $ in Example 2.9
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