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Jordan tori and polynomial endomorphisms in 2

Manfred Denker, Stefan 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).

Jumps of entropy for C r interval maps

David Burguet (2015)

Fundamenta Mathematicae

We study the jumps of topological entropy for C r interval or circle maps. We prove in particular that the topological entropy is continuous at any f C r ( [ 0 , 1 ] ) with h t o p ( f ) > ( l o g | | f ' | | ) / r . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to C r interval maps.

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