Cardinality of survival set for the chaotic Tent map with holes.

Abstract

The aim of this paper is to give an overview of the dynamics of one dimensionaldiscrete dynamical systems:  Tent map family T3, Doubling map E2and shift map σ are investigated.  Let I-intervals(Holes) lie in the interval [0,1) and let E2be a Doubling map.  The survivor set Ω(I) :={x∈[0,1) :Enx /∈ I, n≥0}.  Depending on location and size of the intervals we will characterize the survivor set Ω(I) infinite or finite.  Also we will show conjugacy of some maps that used in this paper.  By using conjugacy of functions we will show that the Survivor set is infinite or finite in another composition of maps.  The Cantor sets Λ that occur as non-survivor sets for c >2 from Tent map family Tc.[1]   Keywords:  dynamical systems, symbolic dynamics, interval maps, survivor sets, chaos, open systems, irregular sets.