On the class of square Petrie matrices induced by cyclic permutations
Let n≥2 be an integer and let P={1,2,…,n,n+1}. Let Zp denote the finite field {0,1,2,…,p−1}, where p≥2 is a prime. Then every map σ on P determines a real n×n Petrie matrix Aσ which is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta func...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204309026 |
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| Summary: | Let n≥2 be an integer and let P={1,2,…,n,n+1}. Let
Zp denote the finite field {0,1,2,…,p−1},
where p≥2 is a prime. Then every map σ on P
determines a real n×n Petrie matrix Aσ which is
known to contain information on the dynamical properties such as
topological entropy and the Artin-Mazur zeta function of the
linearization of σ. In this paper, we show that if
σ is a cyclic permutation on P, then all such
matrices Aσ are similar to one another over Z2 (but
not over Zp for any prime p≥3) and their characteristic
polynomials over Z2 are all equal to ∑k=0nxk. As a
consequence, we obtain that if σ is a cyclic
permutation on P, then the coefficients of the characteristic
polynomial of Aσ are all odd integers and hence nonzero. |
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| ISSN: | 0161-1712 1687-0425 |