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replaced incorrect statements by correct ones
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Margaret Friedland
Margaret Friedland
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Re question 1:I am replacing my previous incorrect answer by this one. I just learned about a complete list of all pairs $f,g$ such thatrecent preprint by Hexi Ye,

http://arxiv.org/pdf/1211.4303.pdf

Among other things, he proves, for general $f^m=g^n=h$ would in particular require making a complete list of$f$ with degree $h$ which can be embedded into a continuous iteration semigroup$d \geq 3$, that $\{h_t: t>0, h_1=h\}$$\mu_f=\mu_g$ implies that ($f$ and $g$ are then iterational roots, aka fractional iterates, of $h$ of suitable orders). This problem is very hard indeed, and special cases are still being investigated share an iterate (e.g. in numerous papers on "analytic iteration"the converse is well known). Some material can be found in the monograph MR1067720The symbol (92f:39002) Kuczma, Marek; Choczewski, Bogdan; Ger, Roman Iterative functional equations. Encyclopedia of Mathematics and its Applications, 32. Cambridge University Press, Cambridge, 1990. xx+552 pp. ISBN: 0-521-35561-3

Re question 3: if$\mu_f$ denotes the unique $f$ and $g$ are iterational roots-invariant measure of maximal entropy for $h$ which embeds into a semigroup as above, then they would commute$f$ (by the "semigroup law"and similarly for $g$). But there may be other cases of commutingHe also analyzes generic maps with a common iterate. I am not aware of any particular results for rationaldegree $2$. The proof involves some holomorphic maps from $t \in \mathbb{C}$ to $f_t \in \rm{Rat}_d$, but for formal power series in the complex plane sufficient conditions for commuting roots were established in MR1331878 Reich, Ludwig Familiesset of commuting formal power series, semicanonical forms and iterative roots. Polish-Austrian Seminar on Functional Equations and Iteration Theoryrational functions of degree $d$ (Cieszynnot semigroups, 1994which you point out to be impossible). Ann. Math. Sil. No. 8 (1994)As far as I can tell at the first glance, 189–201 (to be found here: http://www.sbc.org.pl/Content/34237/1994_15.pdf)he does not seem to address the commutativity question.

Re question 1: a complete list of all pairs $f,g$ such that $f^m=g^n=h$ would in particular require making a complete list of $h$ which can be embedded into a continuous iteration semigroup $\{h_t: t>0, h_1=h\}$ ($f$ and $g$ are then iterational roots, aka fractional iterates, of $h$ of suitable orders). This problem is very hard indeed, and special cases are still being investigated (e.g. in numerous papers on "analytic iteration"). Some material can be found in the monograph MR1067720 (92f:39002) Kuczma, Marek; Choczewski, Bogdan; Ger, Roman Iterative functional equations. Encyclopedia of Mathematics and its Applications, 32. Cambridge University Press, Cambridge, 1990. xx+552 pp. ISBN: 0-521-35561-3

Re question 3: if $f$ and $g$ are iterational roots of $h$ which embeds into a semigroup as above, then they would commute (by the "semigroup law"). But there may be other cases of commuting maps with a common iterate. I am not aware of any particular results for rational maps, but for formal power series in the complex plane sufficient conditions for commuting roots were established in MR1331878 Reich, Ludwig Families of commuting formal power series, semicanonical forms and iterative roots. Polish-Austrian Seminar on Functional Equations and Iteration Theory (Cieszyn, 1994). Ann. Math. Sil. No. 8 (1994), 189–201 (to be found here: http://www.sbc.org.pl/Content/34237/1994_15.pdf)

I am replacing my previous incorrect answer by this one. I just learned about a recent preprint by Hexi Ye,

http://arxiv.org/pdf/1211.4303.pdf

Among other things, he proves, for general $f$ with degree $d \geq 3$, that $\mu_f=\mu_g$ implies that $f$ and $g$ share an iterate (the converse is well known). The symbol $\mu_f$ denotes the unique $f$-invariant measure of maximal entropy for $f$ (and similarly for $g$). He also analyzes generic maps of degree $2$. The proof involves some holomorphic maps from $t \in \mathbb{C}$ to $f_t \in \rm{Rat}_d$, the set of rational functions of degree $d$ (not semigroups, which you point out to be impossible). As far as I can tell at the first glance, he does not seem to address the commutativity question.

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Margaret Friedland
Margaret Friedland
  • 3.7k
  • 2
  • 50
  • 41

Re question 1: a complete list of all pairs $f,g$ such that $f^m=g^n=h$ would in particular require making a complete list of $h$ which can be embedded into a continuous iteration semigroup $\{h_t: t>0, h_1=h\}$ ($f$ and $g$ are then iterational roots, aka fractional iterates, of $h$ of suitable orders). This problem is very hard indeed, and special cases are still being investigated (e.g. in numerous papers on "analytic iteration"). Some material can be found in the monograph MR1067720 (92f:39002) Kuczma, Marek; Choczewski, Bogdan; Ger, Roman Iterative functional equations. Encyclopedia of Mathematics and its Applications, 32. Cambridge University Press, Cambridge, 1990. xx+552 pp. ISBN: 0-521-35561-3

Re question 3: if $f$ and $g$ are iterational roots of $h$ which embeds into a semigroup as above, then they would commute (by the "semigroup law"). But there may be other cases of commuting maps with a common iterate. I am not aware of any particular results for rational maps, but for formal power series in the complex plane sufficient conditions for commuting roots were established in MR1331878 Reich, Ludwig Families of commuting formal power series, semicanonical forms and iterative roots. Polish-Austrian Seminar on Functional Equations and Iteration Theory (Cieszyn, 1994). Ann. Math. Sil. No. 8 (1994), 189–201 (to be found here: http://www.sbc.org.pl/Content/34237/1994_15.pdf)