A dynamical property unique to the Lucas sequence (Q2765399)

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.