# On the wandering property in Dirichlet spaces

## TL;DR

The paper demonstrates that in weighted Dirichlet spaces, any finite Blaschke product can be made to satisfy the wandering subspace property through an equivalent norm, extending previous results in the field.

## Contribution

It introduces a method to construct equivalent norms in Dirichlet spaces where finite Blaschke products exhibit the wandering property, generalizing earlier findings.

## Key findings

- Existence of equivalent norms for wandering property in Dirichlet spaces
- Extension of previous results by Carswell, Duren, and Stessin
- Special case with standard norm for certain parameters

## Abstract

We show that in a scale of weighted Dirichlet spaces $D_{\alpha}$, including the Bergman space, given any finite Blaschke product $B$ there exists an equivalent norm in $D_{\alpha}$ such that $B$ satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell, Duren and Stessin. As a particular instance, when $B(z)=z^k$ and $|\alpha| \leq \frac{\log (2)}{\log(k+1)}$, the chosen norm is the usual one in $D_\alpha$.