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For almost every tent map, the turning point is typical

Henk Bruin — 1998

Fundamenta Mathematicae

Let T a be the tent map with slope a. Let c be its turning point, and μ a the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, ʃ g d μ a = l i m n 1 n i = 0 n - 1 g ( T a i ( c ) ) . As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

Subcontinua of inverse limit spaces of unimodal maps

Karen BrucksHenk Bruin — 1999

Fundamenta Mathematicae

We discuss the inverse limit spaces of unimodal interval maps as topological spaces. Based on the combinatorial properties of the unimodal maps, properties of the subcontinua of the inverse limit spaces are studied. Among other results, we give combinatorial conditions for an inverse limit space to have only arc+ray subcontinua as proper (non-trivial) subcontinua. Also, maps are constructed whose inverse limit spaces have the inverse limit spaces of a prescribed set of periodic unimodal maps as...

Equilibrium states for interval maps: the potential - t log | D f |

Henk BruinMike Todd — 2009

Annales scientifiques de l'École Normale Supérieure

Let f : I I be a C 2 multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential φ t : x - t log | D f ( x ) | for t close to 1 , and also that the pressure function t P ( φ t ) is analytic on an appropriate interval near t = 1 .

Minimal nonhomogeneous continua

Henk BruinSergiǐ KolyadaL'ubomír Snoha — 2003

Colloquium Mathematicae

We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.

Existence of quadratic Hubbard trees

Henk BruinAlexandra KafflDierk Schleicher — 2009

Fundamenta Mathematicae

A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a postcritically finite (quadratic) polynomial. It is easy to read off the kneading sequences from a quadratic Hubbard tree; the result in this paper handles the converse direction. Not every sequence on two symbols is realized as the kneading sequence of a real or complex quadratic polynomial. Milnor and Thurston classified all real-admissible sequences, and we give a classification of all complex-admissible...

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