A family of cubic rational maps and matings of cubic polynomials.
We construct a large family of ergodic non-Markovian processes with infinite memory having the same p-dimensional marginal laws of an arbitrary ergodic Markov chain or projection of Markov chains. Some of their spectral and mixing properties are given. We show that the Chapman-Kolmogorov equation for the ergodic transition matrix is generically satisfied by infinite memory processes.
A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing...
We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.
It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has...
In this paper, we investigate Egoroff’s theorem with respect to monotone set function, and show that a necessary and sufficient condition that Egoroff’s theorem remain valid for monotone set function is that the monotone set function fulfill condition (E). Therefore Egoroff’s theorem for non-additive measure is formulated in full generality.
Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.
A class of functional equations with nonlinear iterates is discussed on the unit circle . By lifting maps on and maps on the torus to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.
We prove a generalisation of the entropy formula for certain algebraic -actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.