Jerome Buzzi

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy.In this way, we reduce the study ofCr interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.

Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

We show that a class of robustly transitive diffeomorphisms originally described by Mane are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mane example and one obtained through a Hopf bifurcation.

Weakly expanding skew-products of quadratic maps

We consider quadratic skew-products over angle-doubling of the circle and prove that they admit positive Lyapunov exponents almost everywhere and an absolutely continuous invariant probability measure. This extends corresponding results of M. Viana and J. F. Alves for skew-products over the linear strongly expanding map of the circle.

Zeta functions and transfer operators for multi-dimensional piecewise affine and expanding maps

Let X\subset\mathbb{R}^2 be a finite union of bounded polytopes and let T:X\to X be piecewise affine and eventually expanding. Then the Perron–Frobenius operator \mathcal{L} of T is quasicompact as an operator on the space of functions of bounded variation on \mathbb{R}^2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T . In higher dimensions the result remains true under an additional generically satisfied transversality assumption.

Absolutely continuous invariant probability measures for arbitrary expanding piecewise

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

\mathbb{R}

We consider the discontinuous dynamical systems on [0, 1]d defined by expanding affine maps considered modulo ℤd. We study their invariant probability measures which maximize entropy. We show that they form a non-empty, finite-dimensional simplex and reduce the question of their multiplicity to a topological problem. We also give a description of these measures. These results are obtained by using a generalization of F. Hofbauers Markov Diagram previously developed by the author for the study of non-piecewise monotonic, smooth interval maps. This paper is intended as a simple but non-trivial application of this technique in higher dimension.

-analytic mappings of the plane

Abstract Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy-conjugacy of the natural extensions of smooth entropy-expanding maps, e.g., C ∞ smooth interval maps with non-zero topological entropy.

Intrinsic ergodicity of affine maps in [0, 1] d

We introduce subshifts of quasi-finite type as a generalization of the well-known subshifts of finite type. This generalization is much less rigid and therefore contains the symbolic dynamics of many non-uniform systems, e.g., piecewise monotonic maps of the interval with positive entropy. Yet many properties remain: existence of finitely many ergodic invariant probabilities of maximum entropy; lots of periodic points; meromorphic extension of the Artin-Mazur zeta function.

Almost isomorphism for countable state Markov shifts

After discussing usual approaches to measuring blur, we show theoretically that there is essentially a unique way to quantify blur by a single number and we confirm the usefulness of that measure by some experiment on a natural image.