Higher-dimensional Contou-Carr\`ere symbol and continuous automorphisms

We prove that the higher-dimensional Contou-Carr\`ere symbol is invariant under continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new explicit formula for the higher-dimensional Contou-Carr\`ere symbol. Unlike previously known formulas, this formula is given over an arbitrary ring, not necessarily a $\mathbb Q$-algebra, and does not involve algebraic $K$-theory.