Puri, Yash; Ward, Thomas A dynamical property unique to the Lucas sequence. (English) Zbl 1018.37009 Fibonacci Q. 39, No. 5, 398-402 (2001). For a compact metric space \(X\) and a homeomorphism \(f:X\to X\) denote \(\text{ Per} _n (f) = \# \{x\in X \mid f^n x =x\}\). A sequence \((U_n)\) of nonnegative integers is said to be exactly realizable if there is a dynamical system \(f:X\to X\) with \(U_n = \text{ Per} _n (f)\) for all \(n\geq 1\). The main result says that the sequence \((U_n)\) defined by \(U_{n+2} = U_{n+1} +U_n\), \(n\geq 1\), \(U_1=a\), \(U_2=b\), \(a,b\geq 0\), is exactly realizable if and only if \(b=3a\). This enables to obtain several (known) congruences for the Lucas sequence. Reviewer: Ľubomír Snoha (Banská Bystrica) Cited in 5 Documents MSC: 37B10 Symbolic dynamics 11B39 Fibonacci and Lucas numbers and polynomials and generalizations Keywords:periodic point; Lucas numbers; congruences PDF BibTeX XML Cite \textit{Y. Puri} and \textit{T. Ward}, Fibonacci Q. 39, No. 5, 398--402 (2001; Zbl 1018.37009) Full Text: arXiv OpenURL Online Encyclopedia of Integer Sequences: Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3. Characteristic function of nonprimes: 0 if n is prime, else 1. Number of orbits of length n under the map whose periodic points are counted by A000364. Number of orbits of length n under the map whose periodic points are counted by A000984. Number of orbits of length n under the map whose periodic points are counted by A001641. Number of orbits of length n under the map whose periodic points are counted by A001642. Number of orbits of length n under the map whose periodic points are counted by A001643. Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945. Number of orbits of length n under the map whose periodic points are counted by A005809. Number of orbits of length n under a map whose periodic points seem to be counted by A006953. Number of orbits of length n under a map whose periodic points are counted by A027306. Number of orbits of length n under a map whose periodic points are counted by A056045. Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022). Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023). Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024). Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025). Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026). Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027). Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029). Number of orbits of length n under the map whose periodic points are counted by A000670. Number of orbits of length n under the map whose periodic points are counted by A047863. Number of orbits of length n in map whose periodic points are A000051.