Abstract
ABSTRACTIn this article we introduce a one-parameter family of skew product (Gt)t ∈ [−ε, ε] maps exhibiting a heterodimensional cycle such that the number of isolated periodic orbits inside it has not super-exponential growth. The dynamics in the central direction of the maps Gt is described by a one-parameter family of system of iterated functions.
Highlights
Let M be a compact manifold of dimension greater or equal to 2 and, for r ∈ N, Cr(M, M ) be the space of Cr mappings of M into itself, endowed by the Cr topology
In [K1], Kaloshin proved that the set of A-M diffeomorphisms having only hyperbolic periodic orbits is dense in Diffr(M )
Let us recall that the index of an hyperbolic periodic point is the dimension of the unstable manifold, and that the homoclinic class of a hyperbolic saddle P of a diffeomorphism f, denoted by
Summary
Let M be a compact manifold of dimension greater or equal to 2 and, for r ∈ N, Cr(M, M ) be the space of Cr mappings of M into itself, endowed by the Cr topology. Let us recall that the index of an hyperbolic periodic point is the dimension of the unstable manifold, and that the homoclinic class of a hyperbolic saddle P of a diffeomorphism f , denoted by. In this paper, following the model of [DER], we present systems Gt with a heterodimensional cycle at t = 0 and we prove that the growth of the number of periodic orbits for this systems is at most exponential (see Theorem 1.1 for the precise statement). Let f be a diffeomorphism, defined on a closed manifold of dimension equal to n ∈ N, with two hyperbolic periodic points P and Q with indices p and n − p + 1, respectively.
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