Abstract
Du, Huang and Li showed in 2003 that the class of Dold–Fermat sequences coincides with the class of Newton sequences, which are defined in terms of so-called generating sequences. The sequences of Lefschetz numbers of iterates form an important subclass of Dold–Fermat (thus also Newton) sequences. In this paper we characterize generating sequences of Lefschetz numbers of iterates.
Highlights
Dold–Fermat sequences constitute a class of integer sequences satisfying some congruences [cf. (2.3)] that play important role both in number theory and dynamical systems
Let us mention that the class is called Fermat sequences by some authors [5,6], while some others name it Dold sequences
The sequences of fixed point indices of iterates are Dold–Fermat sequences, which shows the importance of this class of sequences in periodic point theory, where the existence of such congruences often leads to valuable information about behavior of orbits and dynamics of a map near fixed or periodic points
Summary
Dold–Fermat sequences constitute a class of integer sequences satisfying some congruences [cf. (2.3)] that play important role both in number theory and dynamical systems. The sequences of fixed point indices of iterates are Dold–Fermat sequences, which shows the importance of this class of sequences in periodic point theory, where the existence of such congruences often leads to valuable information about behavior of orbits and dynamics of a map near fixed or periodic points. 3 we find the formula for the generating sequence of the sum of two Newton sequences (Theorem 3.2) which allows us to interpret the operation of assigning a generating sequence to a given Newton sequence as a group homomorphism In this part of the paper we give a new, alternative proof of the fact that the classes of Dold–Fermat sequences and Newton sequences coincide (Theorems 4.3 and 4.5). We introduce three main classes of integer sequences considered in the article
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