Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

We establish an equidistribution result for the pull-back of a (1,1)-closed positive current in ℂ² by a proper polynomial map of small topological degree. We also study convergence at infinity on good compactifications of ℂ². We make use of a lemma that enables us to control the blow-up of some integrals in the neighborhood of a big logarithmic singularity of a plurisubharmonic function. Finally, we discuss the importance of the properness hypothesis, and we give some results in the case where this...

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space ${\mathbb{P}}^k$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number ${\kappa }_f>0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup\left(\varphi \right)-inf\left(\varphi \right)<{\kappa }_f$, then ϕ admits a unique equilibrium state ${\mu }_{\varphi }$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,{\mu }_{\varphi })$ is K-mixing, whence ergodic. Proving...

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic...

The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...

If the ergodic transformations S, T generate a free ${\mathbb{Z}}^2$ action on a finite non-atomic measure space (X,S,µ) then for any $c_1,c_2\in ℝ$ there exists a measurable function f on X for which ${\left(N+1\right)}^{-1}{\sum }_{j=0}^{N}f\left(S^{j}x\right)\to c_1$ and ${(N+1)}^{-1}{\sum }_{j=0}^{N}f\left(T^{j}x\right)\to c_2\mu$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

We study the convergence of the ergodic averages $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ T^{u_k}$ where ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ is a bounded sequence and ${\left(u_k\right)}_{k\in \mathbb{N}}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ for some $\delta \&lt;1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.

Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified....