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

DOI10.3934/DCDS.2022100arXiv2004.06742MaRDI QIDQ6338695

Michał Rams, Katrin Gelfert, Lorenzo J. Díaz

Publication date: 14 April 2020

Abstract: 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

a s t

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.

Mathematics Subject Classification ID

Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Symbolic dynamics (37B10) Partially hyperbolic systems and dominated splittings (37D30) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25) Dynamical systems involving maps of the interval (37E05) Homoclinic and heteroclinic orbits for dynamical systems (37C29) Integration and disintegration of measures (28A50)

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