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K. Gelfert

Publications and source records attributed to K. Gelfert.

6 recordsLinked to original sources
arXiv2023

Loosely Bernoulli zero exponent measures for elliptic matrix cocycles

For an open and dense subset of elliptic ${\rm SL}(2,\mathbb R)$ matrix cocycles, we construct a family of loosely Bernoulli ergodic measures with zero top Lyapunov exponent. This provides a counterpart to a classical result by Furstenberg. The construction gives also an $\bar f$-connected set of measures with these properties whose entropies vary continuously from zero to almost the maximal possible value. We also obtain an analogous result for an open class of nonhyperbolic step skew products with $\mathbb S^1$ diffeomorphism fiber maps. Our approach combines substitution schemes between finite letter alphabets and differentiable dynamics.

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arXiv2020

Mingled hyperbolicities: ergodic properties and bifurcation phenomena (an approach using concavity)

We consider skew-products with concave interval fiber maps over a certain subshift obtained as the projection of orbits staying in a given region. It generates a new type of (essentially) coded shift. The fiber maps have expanding and contracting regions which dynamically interact. The dynamics also exhibits pairs of horseshoes of different type of hyperbolicity which, in some cases, are cyclically related. The space of ergodic measures on the base is an entropy-dense Poulsen simplex. Those measures lift canonically to ergodic measures for the skew-product. We explain when and how the spaces of (fiber) contracting and expanding ergodic measures glue along the nonhyperbolic ones. A key step is the approximation (in the weak$\ast$ topology and in entropy) of nonhyperbolic measures by ergodic ones, obtained only by means of concavity. Concavity is not merely a technical artificial hypothesis, but it prevents the presence of additional independent subsystems. The description of homoclinic relations is also a key instrument. These skew-products are embedded in non-decreasing entropy one-parameter family of diffeomorphisms stretching from a heterodimensional cycle to a collision of homoclinic classes. Associated bifurcation phenomena involve a jump of the space of ergodic measures and, in some cases, of entropy.

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arXiv2016

Nonhyperbolic step skew-products: Ergodic approximation

We study transitive step skew-product maps modeled over a complete shift of $k$, $k\ge2$, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak$\ast$ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.

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arXiv2013

Abundant rich phase transitions in step skew products

We study phase transitions for the topological pressure of geometric potentials of transitive sets. The sets considered are partially hyperbolic having a step skew product dynamics over a horseshoe with one-dimensional fibers corresponding to the central direction. The sets are genuinely non-hyperbolic containing intermingled horseshoes of different hyperbolic behavior (contracting and expanding center). We prove that for every $k\ge 1$ there is a diffeomorphism $F$ with a transitive set $Λ$ as above such that the pressure map $P(t)=P(t\, φ)$ of the potential $φ= -\log \,\lVert dF|_{E^c}\rVert$ ($E^c$ the central direction) defined on $Λ$ has $k$ rich phase transitions. This means that there are parameters $t_\ell$, $\ell=1,...,k$, where $P(t)$ is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of $t_\ell\,φ$ with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum of $F$ on $Λ$.

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arXiv2011

Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions

We study a partially hyperbolic and topologically transitive local diffeomorphism $F$ that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set $Λ$ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of $F|_Λ$ contains a gap and hence gives rise to a first order phase transition. A major part of the proofs relies on the analysis of an associated iterated function system that is genuinely non-contracting.

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