OK,
Let me sketch a proof that in a cycle with n terms and k odd terms, all elements are < 3^k. I will rely on the result that all elements are < 2^n. My proof uses the tiles but I think you could easily get it without too.
The result mean that you can always write α within the space of the parity sequence i.e. we’re always in the case below:
The fact that you read T^n(alpha) in ternary on the red line is an instance of Corollary 1.36 in my thesis, illustrated in Figure 1.16.
From there, just rotate the parity vector, this operation preserves k and you will read all the successive iterates of the cycle in ternary on the red vertical line, on k ternary digits, meaning that all elements are < 3^k.