# Moscow Mathematical Journal

Volume 23, Issue 4, October–December 2023 pp. 533–544.

Gradient-Like Diffeomorphisms and Periodic Vector Fields

**Authors**:
V.Z. Grines (1) and L.M. Lerman (2)

**Author institution:**(1) National Research University, “Higher School of Economics” (Nizhny Novgorod branch)

(2) National Research University, “Higher School of Economics” (Nizhny Novgorod branch)

**Summary: **

A class of gradient-like nonautonomous vector fields (NVFs) on a smooth closed manifold $M$ is studied and the following problems are solved: 1) can a gradient-like NVF be constructed by means of the nonautonomous suspension over a diffeomorphism of this manifold, and if so, under what conditions on the diffeomorphism? 2) let a diffeomorphism $f$ be gradient-like (see the definition in the text) and diffeotopic to the identity map $\mathrm{id}_M$, when the NVF obtained by means of the nonautonomous suspension over $f$ be gradient-like? Necessary and sufficient conditions to this have been found in the paper. All these questions arise, when studying NVFs on $M$ admitting the uniform classification and a description via combinatorial type invariants.

2020 Math. Subj. Class. 34C40, 37B35, 37C60

**Keywords:**Nonautonomous vector field, uniform equivalence, exponential dichotomy, gradient-like, nonautonomous suspension.

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