Smale endomorphisms over graph-directed Markov systems

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

The Unexpected Fractal Signatures in Fibonacci Chains

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L / S = ϕ . The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L / S ratio.

Boltzmann and the Statistical Multifractals

We extend the Boltzmann’s ideas that describe the evolution to the equilibrium of many body systems to the multifractal decomposition of the unitary interval 𝕀, in terms of sets Jα conformed by points with the same pointwise dimension, and obtain the D(α) singularity spectrum.

Dimension of ergodic measures and currents on

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^{2}$ of degree $d\geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current $S$ containing a measure of entropy $h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension ${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.

Dimension, recurrence via entropy and Lyapunov exponents for map with singularities

Let $f:M\rightarrow M$ be a $C^{1}$ self-map of a smooth Riemannian manifold $M$ and $\unicode[STIX]{x1D707}$ be an $f$-invariant ergodic Borel probability measure with a compact support $\unicode[STIX]{x1D6EC}$. We prove that if $f$ is Hölder mild on the intersection of the singularity set and $\unicode[STIX]{x1D6EC}$, then the pointwise dimension of $\unicode[STIX]{x1D707}$ can be controlled by the Lyapunov exponents of $\unicode[STIX]{x1D707}$ with respect to $f$ and the entropy of $f$. Moreover, we establish the distinction of the Hausdorff dimension of the critical points sets of maps between the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity and Hölder mildness conditions. Consequently, this shows that the Hölder mildness condition is much weaker than the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity condition. As applications of our result, if we study the recurrence rate of $f$ instead of the pointwise dimension of $\unicode[STIX]{x1D707}$, then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.

Weak Gibbs measures: speed of convergence to entropy, topological and geometrical aspects

In this paper we obtain exponential large-deviation bounds in the Shannon–McMillan–Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of finite type. We also prove almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shifts. Finally, we establish a topological characterization of large-deviation bounds for Gibbs measures and deduce some of their topological and geometrical aspects: the local entropy is zero and the topological pressure of positive measure sets is total. Some applications include large-deviation estimates for Lyapunov exponents, pointwise dimension and slow return times.