Thermodynamic formalism for transient dynamics on the real line

We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic Z -extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the full dimension spectrum with respect to α-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.

Multifractal analysis in non-uniformly hyperbolic interval maps

In this paper, we establish a framework for the construction of Moran set driven by dynamics. Under this framework, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic measure in a class of non-uniformly hyperbolic interval maps with finitely many branches.

Invariant measures for interval maps without Lyapunov exponents

We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

Generic homeomorphisms have full metric mean dimension

We prove that for $C^0$ -generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$ -generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$ -dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$ -generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

Small C1-smooth perturbations of skew products and the partial integrability property

In this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.