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Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu$ defined on the completion of the Borel $\sigma$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of…
Systems which can spontaneously reveal periodic evolution are dubbed time crystals. This is in analogy with space crystals that display periodic behavior in configuration space. While space crystals are modelled with the help of space…
In this paper we consider closed orientable surfaces $S$ of positive genus and $C^r$-diffeomorphisms $f:S\rightarrow S$ isotopic to the identity ($r\geq 1)$. The main objective is to study periodic open topological disks which are…
We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover…
We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures…
By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which…
We study the renormalizable quantum gravity formulated as a perturbed theory from conformal field theory (CFT) on the basis of conformal gravity in four dimensions. The conformal mode in the metric field is managed non-perturbatively…
We construct a $C^\infty$ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose…
We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|\lambda\gamma| = 1$. For a proper three-parameter unfolding that splits the tangency, varies…
An important question in dynamical systems is the classification problem, i.e., the ability to distinguish between two isomorphic systems. In this work, we study the topological factors between a family of multidimensional substitutive…
A diffeomorphism f has a heterodimensional cycle if it displays two (transitive) hyperbolic sets K and K' with different indices such that the unstable set of K intersects the stable one of K' and vice versa. We prove that it is possible to…
We provide a new angle and obtain new results on a class of metrics on length-normalized curves in $d$ dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the…
We develop a renormalization theory of non-perturbative dissipative H\'enon-like maps with combinatorics of bounded type. The main novelty of our approach is the incorporation of Pesin theoretic ideas to the renormalization method, which…
In asymptotically anti-de Sitter gravity, diffeomorphisms that change the conformal boundary data can be promoted to genuine physical degrees of freedom. I show that in 2+1 dimensions, the dynamics of these degrees of freedom is described…
In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small…
The necessity of renormalization arises from the infinite integrals which are caused by the discrepancy between the orders of differential and integral operators in the four dimensional QFTs. Therefore in view of the fact that finiteness…
We consider a parametrised perturbation of a $\mathscr C^r$ diffeomorphism on a closed smooth Riemannian manifold with $r\geq 1$, modeled by nonautonomous dynamical systems. A point without time averages for a (nonautonomous) dynamical…
We discuss the fate of diffeomorphism symmetry in discrete gravity. Diffeomorphism symmetry is typically broken by the discretization. This has repercussions for the observable content and the canonical formulation of the theory. It might…
We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for…
Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering.…