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Yes, there exists a real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the "sickle" $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2].

For the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential), see [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.

Yes, there exists a real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the "sickle" $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line segment connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2].

For the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential), see [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.

Yes, there exists real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the sickel $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2].

For the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential) [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.

Yes, there exists a real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the "sickle" $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2].

For the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential), see [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.

Yes, there exists real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the sickel $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2] or for the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential) [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.

Yes, there exists real-analytic injective $E$.

Its inverse $E^{-1}$ is constructed in the paper of Kneser [1].

Uniqueness: $E^{-1}$ is the only holomorphic and injective Abel function (up to an additive constant) on the sickel $G$ given in the following way: Let $z_0$ and its conjugate $z_0^\ast$ be the both fixpoints of $e^z$ closest to the real axis, let $\ell$ be the straight line connecting $z_0$ and $z_0^\ast$ and let $\ell_1$ be the image of $\ell$ under $e^z$. Now let $G$ be the region bounded by $\ell$ and $\ell_1$ inclusive and $z_0$, $z_0^\ast$ exclusive. See also [2].

For the general theory of Abel functions between two complex fixed points (which is a relatively new development in holomorphic dynamics which is unaware of its application to the exponential) [3].

[1] Kneser, H. (1949). Reelle analytische Lösungen der Gleichung $\phi(\phi(x))=e^x$ und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187, 56–67.

[2] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.

[3] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363.