arXiv subjects

L. J. Díaz

Publications and source records attributed to L. J. Díaz.

5 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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arXiv2021

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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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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arXiv2010

Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.

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