Planar homoclinic points

Homoclinic Points

Homoclinic Points for Area-preserving Surface Diffeomorphisms

We show a Cr connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic Cr , r = 1, 2, . . ., ∞, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a Cr, r = 1, 2, . . ., ∞ generic time periodic Hami...

HOMOCLINIC POINTS OF ALGEBRAIC Zd-ACTIONS

Let α be an action of Z by continuous automorphisms of a compact abelian group X. A point x in X is called homoclinic for α if αx → 0X as ‖n‖ → ∞. We study the set ∆α(X) of homoclinic points for α, which is a subgroup of X. If α is expansive then ∆α(X) is at most countable. Our main results are that if α is expansive, then (1) ∆α(x) is nontrivial if and only if α has positive entropy and (2) ∆α...

Homoclinic Points of Algebraic Z–Actions

Let α be an action of Z by continuous automorphisms of a compact abelian group X. A point x in X is called homoclinic for α if αx→ 0X as ‖n‖ → ∞. We study the set ∆α(X) of homoclinic points for α, which is a subgroup of X. If α is expansive then ∆α(X) is at most countable. Our main results are that if α is expansive, then (1) ∆α(x) is nontrivial if and only if α has positive entropy and (2) ∆α(...

Homoclinic points of non-expansive automorphisms

We study homoclinic points of non-expansive automorphisms of compact abelian groups. Connections between the existence of non-trivial homoclinic points, expansiveness, entropy and adjoint automorphisms (in the sense of Einsiedler and Schmidt) are explored. Some implications for countable abelian group actions by automorphisms of compact abelian groups are also considered and it is shown that if...