Related papers: Generic expanding maps without absolutely continuo…

We show that a dissipative, ergodic measure preserving transformation of a sigma-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these.

In this short note, we establish a quantitative description of the genericity of transversality of $C^1$-submanifolds in $\mathbb{R}^n$: Let $\Sigma \subset \mathbb{R}^n$ be a $d$-dimensional $C^1$-embedded submanifold where $n \geq d+1$.…

We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In…

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses…

For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the…

We study permanence results for almost quasi-isometries, the maps arising from the Gromov construction of finitely generated random groups that contain expanders (and hence that are not C*-exact). We show that the image of a sequence of…

We prove existence and uniqueness of absolutely continuous invariant measures for generalizations of Viana maps admitting a higher order critical point introduced in arXiv:2312.00906. As a consequence of our approach, we obtain…

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota-Yorke inequality…

We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings on $C$ with the topology of pointwise convergence. We introduce the notion of a somewhat…

A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

We prove the entropic continuity of Lyapunov exponent for C^r maps of the interval or of the circle with large entropy for r>1, without making any assumptions on the set of critical points. A consequence is the upper semi-continuity of…

We construct a $C^1$ symplectic twist map $g$ of the annulus that has an essential invariant curve $\Gamma$ such that $\Gamma$ is not differentiable and $g$ restricted to $\Gamma$ is minimal.

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from…

In this paper, we prove that for real analytic expanding circle maps, all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to constant. We use the group structure of the…

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…

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are…

We show that no C^2 circle diffeomorphism of irrational rotation number has invariant 1-distributions other than (scalar multiples of) the invariant measure. We also show that this is false in the C^1 context by giving both minimal and…

We exhibit a local residual set of surface $C^1$ diffeomorphisms that are continuum-wise expansive but are not expansive. We also exhibit an open and dense set of surface $C^1$ diffeomorphisms where expansiveness implies being Anosov.

For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which…

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an…