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A classic result of Ritt shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see here for a modern presentation.

A classic result of Ritt shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebyshev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see here for a modern presentation.

A [classic result of Ritt][1] shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see [here][3] for a modern presentation. [0]:http://www.jstor.org/stable/2038036 [1]:http://www.ams.org/journals/tran/1923-025-03/S0002-9947-1923-1501252-3/S0002-9947-1923-1501252-3.pdf [3]:http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.8482

A classic result of Ritt shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see here for a modern presentation.

A [classic result of Ritt][1] shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. [0]:http://www.jstor.org/stable/2038036 [1]:http://www.ams.org/journals/tran/1923-025-03/S0002-9947-1923-1501252-3/S0002-9947-1923-1501252-3.pdf

A [classic result of Ritt][1] shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see [here][3] for a modern presentation. [0]:http://www.jstor.org/stable/2038036 [1]:http://www.ams.org/journals/tran/1923-025-03/S0002-9947-1923-1501252-3/S0002-9947-1923-1501252-3.pdf [3]:http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.8482