Stability

We begin with the definition of stability and instability. Let ℜ be a topological space whose points we denote by þ, and let a be a certain point in ℜ. By a neighborhood here we will always mean a neighborhood of a in ℜ. Let þ₁ = Sþ be a topological mapping of a neighborhood U₁ onto a neighborhood B₁ whereby a = Sa is mapped onto itself. The inverse mapping p_-1 = S-1þ then carries B₁ onto U₁, and in general þ_n = Snþ (n = 0, ± 1, ± 2,…) is a topological mapping of a neighborhood U_n onto a neighborhood B_n, having a as a fixed-point. For each point þ = þ₀ in the intersection U₁∩B₁=M we construct the successive images þ_k+1 =Sþk (k = 0,1,…), as long as þ_k lies in U₁ and similarly þ_-k-1 =S-1þ_-k as long as þ_-k lies in B₁. If the process terminates with a largest k+ 1 = n, then þ₀,…, þ_n-1 all still lie in U₁, but þ_n no longer does; similarly for the negative indices. In this way, to each þ in M there is associated a sequence of image points þ_k (k =…, — 1,0,1,…), which is finite, infinite on one side, or infinite on both sides.