Sofic shifts via Conley index theory: computing lower bounds on recurrent dynamics for maps. (English) Zbl 1436.37018
The authors provide an algorithm using discrete Conley index theory to construct sofic shifts that are topologically semiconjugate to discrete-time systems governed by maps. The technique is applied to several examples including the two-dimensional Hénon map, the three-dimensional LPA map (a Larvae-Pupae-Adult model describing the evolution of a population of flour beetles), and the Kot-Schaffer map [M. Kot and W. M. Schaffer, Math. Biosci. 80, 109–136 (1986; Zbl 0595.92011)].
Lower bounds on the topological entropy are obtained: for the Hénon map this is at least 0,4555, for the LPA is at least 0,1203 and for K-S map is 0,2406.
Several computational problems are discussed and the presented algorithm is better than the one from the previous paper by the present authors and R. Trevino [SIAM J. Appl. Dyn. Syst. 7, No. 4, 1477–1506 (2008; Zbl 1171.37011)].
It is worth to say that the approach used here is based on the works [A. Szymczak, Fundam. Math. 148, No. 1, 71–90 (1995; Zbl 0855.54051); J. Kwapisz, Ergodic Theory Dyn. Syst. 24, No. 4, 1173–1197 (2004; Zbl 1071.37010)].
Lower bounds on the topological entropy are obtained: for the Hénon map this is at least 0,4555, for the LPA is at least 0,1203 and for K-S map is 0,2406.
Several computational problems are discussed and the presented algorithm is better than the one from the previous paper by the present authors and R. Trevino [SIAM J. Appl. Dyn. Syst. 7, No. 4, 1477–1506 (2008; Zbl 1171.37011)].
It is worth to say that the approach used here is based on the works [A. Szymczak, Fundam. Math. 148, No. 1, 71–90 (1995; Zbl 0855.54051); J. Kwapisz, Ergodic Theory Dyn. Syst. 24, No. 4, 1173–1197 (2004; Zbl 1071.37010)].
Reviewer: Zdzisław Dzedzej (Gdansk)
MSC:
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 37B10 | Symbolic dynamics |
| 37B40 | Topological entropy |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
Keywords:
topological entropy; symbolic dynamics; Conley index; Hénon map; Kot-Schaffer map; computer-assisted proofReferences:
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