Dichotomies for linear difference equations and homoclinic orbits of noninvertible maps

n this article, we first showed that conditions given by Hale and Lin, Steinlein and Walther, and Sander, which ensured the presence of chaotic dynamics near a homoclinic orbit of a non-invertible map, were equivalent to the exponential dichotomy of the variational equation along the homoclinic orbit. Next, we studied the notion of generalized exponential dichotomy, which arose from Steinlein and Walther’s notion of hyperbolicity. Finally, we corrected a slight mistake in our article “Exponential dichotomy for noninvertible linear difference equations”, which appeared in volume 27 of the Journal of Difference Equations and Applications.