For more general questions on the monoid of (complex) polynomials under composition, e.g. if your chains involve more than one nonlinear polynomial, many of the principle results go back to Ritt's work in the 1920's.

Strangely, I was led to reading these results yesterday by a question here - a reference to Clauwens, Commuting polynomials and $\lambda$-ring structures on ${\mathbb Z}[x]$, J. Pure Appl. Algebra 95 (1994) in Borger-de Smit's paper leads to a classification of the polynomials that commute (the Chebyshev polynomials make a star appearance here). Further references therein, such as Dorey-Whaples Prime and composite polynomials J. Algebra 28 (1974) give nice algebraic proofs of Ritt's results, saying for instance when one can have non-equivalent decompositions $f_1\circ f_2=g_1\circ g_2$.

For more general questions on the monoid of (complex) polynomials under composition, e.g. if your chains involve more than one nonlinear polynomial, many of the principle results go back to Ritt's work in the 1920's.

Strangely, I was led to reading these results yesterday by a question here - a reference to Clauwens, Commuting polynomials and $\lambda$-ring structures on ${\mathbb Z}[x]$, J. Pure Appl. Algebra 95 (1994) in Borger-de Smit's paper leads to a classification of the polynomials that commute (the Chebyshev polynomials make a star appearance here). Further references therein, such as Dorey-Whaples Prime and composite polynomials J. Algebra 28 (1974) give nice algebraic proofs of Ritt's results, saying for instance when one can have non-equivalent decompositions $f_1\circ f_2=g_1\circ g_2$.