Hausdorff dimension of elliptic functions with critical values approaching infinity
AbstractWe consider the escaping parameters in the family \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$, i.e. these parameters for which the orbits of critical values of \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$ approach infinity, where \$\\textbackslash\wp\_\\textbackslash\Lambda\$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a non-atomic, \$\\textbackslash\sigma\$-finite, ergodic, conservative and invariant measure \$\\textbackslash\mu\$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the \$\\textbackslash\wp\_\\textbackslash\Lambda\$-function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$, and compare it with the Hausdorff dimension of escaping set in dynamical space, proving a similarity between parameter plane and dynamical space.
|Journal series||arXiv:1105.1021, (0 pkt)|
|Keywords in English||Mathematics - Complex Variables, Mathematics - Dynamical Systems, Primary 37F35. Secondary 37F10, 30D05|
|Publication indicators||= 0|
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.