Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

INVARIANT MEASURES FOR RANDOM MAPS VIA INTERPOLATION

Let T = {τ1, τ2, …, τK; p1, p2, …, pK} be a position dependent random map on [0, 1], where {τ1, τ2, …, τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, …, pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

Optimal transport and dynamics of expanding circle maps acting on measures

In this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

Dynamical properties of the negative beta-transformation

We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

MAXIMUM ENTROPY METHOD FOR POSITION DEPENDENT RANDOM MAPS

Let {τ1, τ2,…,τK} be a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2,…,pK} be a collection of position dependent probabilities on [0, 1]. We consider position dependent random maps T = {τ1,τ2,…,τK;p1,p2,…,pK} such that T preserves an absolutely continuous invariant measure with density f*. A maximum entropy method for approximating f* is developed. We present a proof of convergence of the maximum entropy method for random maps.

BRANCHING STRUCTURE FOR THE TRANSIENT (1, R)-RANDOM WALK IN RANDOM ENVIRONMENT AND ITS APPLICATIONS

An intrinsic multi-type branching structure within the transient (1, R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the Markov chain of environments seen from the particle and reprove the LLN with a drift explicitly in terms of the environment.