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Jordan tori and polynomial endomorphisms in 2

Manfred DenkerStefan Heinemann — 1998

Fundamenta Mathematicae

For a class of quadratic polynomial endomorphisms f : 2 2 close to the standard torus map ( x , y ) ( x 2 , y 2 ) , we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).

Markov partitions for fibre expanding systems

Manfred DenkerHajo Holzmann — 2008

Colloquium Mathematicae

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

Dirichlet forms on quotients of shift spaces

Manfred DenkerAtsushi ImaiSusanne Koch — 2007

Colloquium Mathematicae

We define thin equivalence relations ∼ on shift spaces and derive Dirichlet forms on the quotient space Σ = / in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.

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