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Period doubling, entropy, and renormalization

100%

Hu J., Tresser C.

Fundamenta Mathematicae

1998

tom 155

nr 3

237-249

We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point. Similar techniques then allow us to show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable with the diameters of all periodic intervals going to zero as the period goes to infinity.

Extending the Dehn quandle to shears and foliations on the torus

81%

Chamanara R., Hu J., Zablow J.

Fundamenta Mathematicae

2014

tom 225

nr 1

1-22

The Dehn quandle, Q, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, 𝕋², we generalize this to show the existence of a quandle Q̂ extending Q and whose elements are measured geodesic foliations. The quandle action in Q̂ is given by applying a shear along such a foliation to another foliation. We extend some results which related Dehn quandle homology to the monodromy of Lefschetz fibrations. We apply certain quandle 2-cycles to yield factorizations of elements of SL₂(ℝ) fixing specified vectors (circles, foliations) and give examples. Using these, we show the quandle homology of Q̂ is nontrivial in all dimensions.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Hu J.

1 Chamanara R.

1 Tresser C.

1 Zablow J.

1 2014

1 1998

Wyniki wyszukiwania

Period doubling, entropy, and renormalization

Extending the Dehn quandle to shears and foliations on the torus