Bursts in average lifetime of transients for chaotic logistic map with a hole

Chaotic transients are studied for a logistic map at r=4, with an inserted narrow hole. We find that average lifetime τ of chaotic transients that are dependent on the hole position roughly follows the Frobenius-Perron semicircle pattern in most of the unit interval, but at the positions that correspond to the low period (1,2,3,ldots), unstable periodic orbits of the logistic map at r=4 there are bursts of τ. An asymptotic relation between the Frobenius-Perron and Kantz-Grassberger average lifetimes, at these positions, is obtained and explained in terms of missing preimages determined from a transient time map. The addition of noise leads to the destruction of bursts of average lifetime.