Bowen-Franks groups of reducible bimodal subshifts of finite type. (English) Zbl 1104.37007
Here, bimodal maps are considered. It is assumed that the critical points are periodic. Hence, using a suitable partition into intervals, the bimodal map has the Markov property, and therefore it is conjugate to a subshift of finite type. From the transition matrix one can calculate the Bowen-Franks group of the map.
In [N. Martins, R. Severino and J. Sousa Ramos, J. Difference Equ. Appl. 9, 423–433 (2003; Zbl 1018.37008)], an explicit formula for the Bowen-Franks group involving the bimodal kneading pair of the map has been derived. However, if the period of the critical points is large, long calculations are required to determine the Bowen-Franks group.
Given two bimodal kneading pairs the *-product of these pairs was introduced in the paper of J. P. Lampreia, R. Severino and J. Sousa Ramos [Grazer Math. Ber. 346, 245–254 (2004; Zbl 1061.37008)]. This product is again a bimodal kneading pair.
The main result of the present paper provides a formula to calculate the Bowen-Franks group, if the kneading pair of the map can be written as the *-product of two kneading pairs.
In [N. Martins, R. Severino and J. Sousa Ramos, J. Difference Equ. Appl. 9, 423–433 (2003; Zbl 1018.37008)], an explicit formula for the Bowen-Franks group involving the bimodal kneading pair of the map has been derived. However, if the period of the critical points is large, long calculations are required to determine the Bowen-Franks group.
Given two bimodal kneading pairs the *-product of these pairs was introduced in the paper of J. P. Lampreia, R. Severino and J. Sousa Ramos [Grazer Math. Ber. 346, 245–254 (2004; Zbl 1061.37008)]. This product is again a bimodal kneading pair.
The main result of the present paper provides a formula to calculate the Bowen-Franks group, if the kneading pair of the map can be written as the *-product of two kneading pairs.
Reviewer: Peter Raith (Wien)
MSC:
| 37B10 | Symbolic dynamics |
| 37E05 | Dynamical systems involving maps of the interval |
| 20K25 | Direct sums, direct products, etc. for abelian groups |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15B36 | Matrices of integers |
Keywords:
bimodal map; subshift of finite type; flow equivalence; Bowen-Franks group; kneading theoryReferences:
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