# The $\epsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

## TL;DR

This paper estimates the Kolmogorov epsilon-entropy of certain infinite-dimensional ellipsoids in Hilbert spaces and applies these results to bound the fractal dimension of attractors in nonlinear parabolic equations.

## Contribution

It provides new bounds on epsilon-entropy for infinite-dimensional ellipsoids and links these bounds to fractal dimensions of attractors in nonlinear PDEs.

## Key findings

- Established epsilon-entropy estimates for infinite-dimensional ellipsoids.
- Derived bounds on fractal dimensions of attractors in nonlinear parabolic equations.
- Connected entropy estimates to attractor complexity via Zelik's result.

## Abstract

We prove an estimation of the Kolmogorov $\epsilon$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.