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Abstract We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $$\mathbb {C}\times \mathbb {R}$$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $$\mathbb {C}$$ and $$\mathbb {R}$$ . 

The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? It is known that many different parameter settings $$\theta$$ can determine the same function $$f$$. Moreover, the degree of this redundancy is inhomogeneous: for some networks, the only symmetries are permutation of neurons in a layer and positive scaling of parameters at a neuron, while other networks admit additional hidden symmetries. In this work, we prove that, for any network architecture where no layer is narrower than the input, there exist parameter settings with no hidden symmetries. We also describe a number of mechanisms through which hidden symmetries can arise, and empirically approximate the functional dimension of different network architectures at initialization. These experiments indicate that the probability that a network has no hidden symmetries decreases towards 0 as depth increases, while increasing towards 1 as width and input dimension increase. 

Abstract Let $$f$$ be a degree $$d$$ bicritical rational map with critical point set $$\mathcal{C}_f$$ and critical value set $$\mathcal{V}_f$$. Using the group $$\textrm{Deck}(f^k)$$ of deck transformations of $f^k$, we show that if $$g$$ is a bicritical rational map that shares an iterate with $$f$$, then $$\mathcal{C}_f = \mathcal{C}_g$$ and $$\mathcal{V}_f = \mathcal{V}_g$$. Using this, we show that if two bicritical rational maps of even degree $$d$$ share an iterate, then they share a second iterate, and both maps belong to the symmetry locus of degree $$d$$ bicritical rational maps.