# Radial projections along chains

## TL;DR

This paper establishes strong Marstrand-type projection properties for certain fractals in Heisenberg groups, showing that their radial projections have constant dimension regardless of the base point, using entropy methods.

## Contribution

It proves that for specific fractals in Heisenberg groups, the dimension of radial projections is invariant across all centers, extending Marstrand's projection theorem to this setting.

## Key findings

- Radial projections of fractals have constant dimension across all centers.
- The results apply to limit sets of Schottky groups and self-similar IFS in Heisenberg groups.
- The proof uses local entropy averages and Hochman-Shmerkin techniques.

## Abstract

We state strong Marstrand properties for two related families of fractals in Heisenberg groups $\mathcal{H}^d$: limit sets of Schottky groups in good position, and attractors of self-similar IFS enjoying the open set condition in the quotient $\mathcal{H}^d/Z$. For such a fractal $X$, we show that the dimension of $\pi_x X$ does not depend on $x \in \mathcal{H}^d$, where $\pi_x$ denotes the radial projection along chains passing through $x$. This follows from a local entropy averages argument due to Hochman and Shmerkin.