Fixed point indices of iterated smooth maps in arbitrary dimension

Abstract

Let f be a smooth self-map of R m , when m is an arbitrary natural number. We give a complete description of possible sequences of indices of iterations of f at an isolated fixed point, answering in affirmative the Chow, Mallet-Paret and Yorke conjecture posed in [S.N. Chow, J. Mallet-Parret, J.A. Yorke, A periodic point index which is a bifurcation invariant, in: Geometric Dynamics, Rio de Janeiro,

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

“…In this section we recall the description of possible forms of indices for a fixed point in an open subset of R m given in [20,22]. We only consider the case of odd iterations, and thus the forms of indices are simpler.…”

“…We will use methods similar to those used in Section 9 in [20]. We will define a map on each of the spaces listed in Definition 3.17 as a discretization of some smooth flow.…”

“…It is possible to extend this maps to a C 1 self-map of the whole R m+1 + with 0 as the only periodic point, in such a way that the fixed point indices of all iterations for the extension are the same as for the restriction to W ∪ V 0 ∪ V (cf. [20] for details).…”

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

“…Define the set G = α : ∈ P α ≤ 2 α ∈ N By (11) we notice that B ⊂ G. On the other hand, it was proved in [12] that there exists a DD (1) sequence of the form Finally, we can change the condition for in Theorem 5.2, demanding that the following inequality is satisfied:…”

A sufficient condition for the realizability of the least number of periodic points of a smooth map

“…The map g may be chosen smooth if and only if the sequence of Lefschetz numbers (L(f k )) k|n is smoothly realizable in R m (see [9]). Since the last condition is easier to verify, it is natural to ask, what does it imply if the fundamental group is not trivial?…”