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Dynamic rays of bounded-type transcendental self-maps of the punctured plane

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The first author was partially supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168 and by the Spanish grants MTM2011-26995-C02-02 and MTM2014-52209-C2-2-P. The second author was supported by The Open University, by a Formula Santander Scholarship and by the Spanish grant MTM2011-26995-C02-02

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  • We study the escaping set of functions in the class $\mathcal{B}^*$, that is, transcendental self-maps of $\mathbb{C}^*$ for which the set of singular values is contained in a compact annulus of $\mathbb{C}^*$ that separates zero from infinity. For functions in the class $\mathcal{B}^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb{C}^*$ (and hence, in the class $\mathcal{B}^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence $e∈\{0,∞\}^{\mathbb{N}_0}$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to the set $\{0,∞\}$ according to $e$ under iteration by $f$.

    Mathematics Subject Classification: Primary: 37F20; Secondary: 30D05.


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  • Figure 1.  Period 8 cycle of rays landing on a repelling period 4 orbit in the unit circle for the function $f_{\alpha\beta}(z)=ze^{i\alpha}e^{\beta(z-1/z)/2}$ from the Arnol'd standard family, with $\alpha=0.19725$ and $\beta=0.48348$. Such points lie in the set $I_e(f_{\alpha\beta})$ with $e=\overline{\infty 0 \infty\infty 0 \infty 0 0}$ (see (2)).

    Figure 2.  Logarithmic coordinates for a function $f\in\mathcal{B}^*$.

    Figure 3.  Phase space of the function $f(z)=\exp(0.3(z+1/z))$ which has a disjoint-type logarithmic transform (see Example 3.12). In orange, the basin of attraction of the fixed point $z_0\simeq 2.2373$. Left, $z\in [-16,16]+i[-16,16]$; right, $z\in [-0.3,0.3]+i[-0.3,0.3]$.

    Figure 4.  Logarithmic tracts of functions of finite order with $\rho_\infty(f)=3$ and $\rho_0(f)=2$ (left) and infinite order (right). The color of every point $z\in\mathbb{C}^*$ has been chosen according to the modulus (luminosity) and argument (hue) of $f(z)$.

    Figure 5.  Fundamental domains of a function $f$ in the class $\mathcal{B}^*$.

    Figure 6.  In the left, we have the phase space of the function $f(z)=z\exp(z^2+\exp(-1/z^2))$ from Example 7.3. In the right, the graph of the restriction of this function to the positive real line

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