Members of the Tetration Forum have been interested in extending the domain and range of tetration to the real numbers. The problem is that the results of Schroeder's functional equation are exponential functions whose domain is the complex numbers. But the results of Abel's functional equation's allows for polynomial functions such that the domain and range are real numbers.
Now the ugly part. According to the Classification of Fixed Points while setting the fixed point to zero, Abel's functional equation only works when \( f'(0)=1 \). Can you get a paper published with this mistake? Well, people are doing it. To my knowledge, the only published paper handling Schroeder's functional equation is by Aldrovandi and Freitas. My research from the Nineties produces the same results that Aldrovandi and Freitas obtain.
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps,
J. Math. Phys. 39, 5324 (199
Now the ugly part. According to the Classification of Fixed Points while setting the fixed point to zero, Abel's functional equation only works when \( f'(0)=1 \). Can you get a paper published with this mistake? Well, people are doing it. To my knowledge, the only published paper handling Schroeder's functional equation is by Aldrovandi and Freitas. My research from the Nineties produces the same results that Aldrovandi and Freitas obtain.
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps,
J. Math. Phys. 39, 5324 (199
Daniel