ERC Starting Grant · 2025
Dimension Theory of Stationary Fractal Measures
We consider the dimension of stationary fractal measures. Our examples include self-affine, self-similar, and Furstenberg measures, which are among the most fundamental and studied fractal objects. The dimension of such a measure has a natural upper bound defined in terms of entropy, Lyapunov exponents, and the dimension of the ambient space. Equality to the upper bound is expected in the absence of obvious algebraic obstructions. In very simple situations, this equality is easy to demonstrate and was known to hold long ago. In more complicated cases, it has been shown to hold almost surely under a natural randomization of the parameters. On the other hand, proving the expected equality…
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