On orbit complexity of dynamical systems: intermediate value property and level set related to a Furstenberg problem

Abstract For symbolicdynamics with some mild conditions, we solve the lowering topological entropy problem for subsystems and determine the Hausdorff dimension of the level set with given complexity, where the complexity is represented by Hausdorff dimension of orbit closure. These results can be applied to some dynamical systems such as $\beta $-transformations, conformal expanding repellers, etc. For two integers $p, q\geq 2$, let $F_{p, q}^{s}$ be the set of points in $[0, 1)$ whose sum of complexities under the $\times p$ map and $\times q$ map is equal to $s$, our second main result gives an explicit formula for the Hausdorff dimension of $F_{p, q}^{s}$.