Search

The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They naturally form a dynamical system, and "random Brody curves" in the title refers to invariant probability measures on it. We study their geometric and dynamical properties. Given an invariant probability measure $\mu$ on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a "potential function" over $\mu$. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this "Ruelle inequality for Brody curves". The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms., Comment: 60 pages, 2 figures

Let $X$ be a full-shift on the alphabet $[0, 1]^a$ and let $(Y, S)$ be an arbitrary dynamical system. We prove that any equivariant continuous map from $X$ to $Y$ has conditional metric mean dimension not less than $a-\mathrm{mdim}(Y, S)$. This solves a problem posed in a paper of Shi--Tsukamoto (2023)., Comment: 11 pages

We develop a variational principle for mean dimension with potential of $\mathbb{R}^d$-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term. A basic strategy of the proof is the same as the case of $\mathbb{Z}$-actions. However measure theoretic details are more involved because $\mathbb{R}^d$ is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for $\mathbb{R}^d$-actions., Comment: 56 pages, 3 figures

We explain that Hadamard's global inverse function theorem very simply follows from the Hopf--Rinow theorem in Riemannian geometry., Comment: 4 pages. Comments are very welcome!

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps between dynamical systems and assume that the mean dimension of the domain is larger than the mean dimension of the target. We exhibit several situations for which the maps necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss about minimal dynamical systems non-embeddable in the shift on the Hilbert cube., Comment: 23 pages

Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2) The coincidence of mean Hausdorff dimension and metric mean dimension for self-similar systems. (3) The calculation of mean Hausdorff dimension of infinite dimensional carpets., Comment: 47 pages, 1 figure

The Hurewicz theorem is a fundamental result in classical dimension theory concerning continuous maps which lower topological dimension. We study whether or not its analogue holds for mean dimension of dynamical systems. Our first main result shows that an analogue of the Hurewicz theorem does not hold for mean dimension in general. Our second main result shows that it holds true if a base system has zero mean dimension., Comment: 22 pages, 4 figures

Motivated by fractal geometry of self-affine carpets and sponges, Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng--Huang. The equivalence of the two definitions seems highly nontrivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford--McMullen carpet in purely topological terms., Comment: 31 pages

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of $p$-periodic points admits a natural free action of $\mathbb{Z}/p\mathbb{Z}$ for each prime number $p$. We are interested in the growth of its index and coindex as $p\to \infty$. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by [TTY20]., Comment: 11 pages. arXiv admin note: text overlap with arXiv:2102.12197

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean dimension. We also explain the generalization to $\mathbb{R}^D$-actions and an illustrating example., Comment: 20 pages

Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical system, we study the $\mathbb{Z}_p$-index of the set of $p$-periodic points. We find that its growth is at most linear in $p$. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years., Comment: 23 pages

Furstenberg (1967) calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^2$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and mean Hausdorff dimension of $\mathbb{Z}^2$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg's theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^2$-subshifts in terms of Kolmogorov-Sinai entropy.

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension., Comment: 67 pages, 1 figure

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory., Comment: 46 pages, 3 figures. arXiv admin note: text overlap with arXiv:1901.05623

We prove that an $\mathbb{R}$-action on a compact metric space embeds equivariantly in the space of one-Lipschitz functions $\mathbb{R}\to[0,1]$ if its fixed point set can be topologically embedded in the unit interval. This is a refinement of the classical Bebutov--Kakutani theorem (1968)., Comment: To be published in "Journal of Dynamics and Differential Equations"

Let $K$ be a finite dimensional compact metric space and $K^\mathbb{Z}$ the full shift on the alphabet $K$. We prove that its mean dimension is given by $\dim K$ or $\dim K-1$ depending on the "type" of $K$. We propose a problem which seems interesting from the view point of infinite dimensional topology., Comment: 9 pages

Ma\~n\'e (1979) proved that if a compact metric space admits an expansive homeomorphism then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts "averaged dimension" of a dynamical system. We prove that if $T:\mathbb{Z}^k\times X\to X$ is expansive and if $R:\mathbb{Z}^{k-1}\times X\to X$ commutes with $T$ then $R$ has finite mean dimension. When $k=1$, this statement reduces to Ma\~{n}\'{e}'s theorem. We also study several related issues, especially the connection with entropy theory., Comment: 27 pages

We study the problem of embedding arbitrary $\mathbb{Z}^k$-actions into the shift action on the infinite dimensional cube $\left([0,1]^D\right)^{\mathbb{Z}^k}$. We prove that if a $\mathbb{Z}^k$-action satisfies the marker property (in particular if it is a minimal system without periodic points) and if its mean dimension is smaller than $D/2$ then we can embed it in the shift on $\left([0,1]^D\right)^{\mathbb{Z}^k}$. The value $D/2$ here is optimal. The proof goes through signal analysis. We develop the theory of encoding $\mathbb{Z}^k$-actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of the Euclidean space into a band-limited function which is constructed from another tiling., Comment: 59 pages, 6 figures

The purpose of this paper is to point out a new connection between information theory and dynamical systems. In the information theory side, we consider rate distortion theory, which studies lossy data compression of stochastic processes under distortion constraints. In the dynamical systems side, we consider mean dimension theory, which studies how many parameters per second we need to describe a dynamical system. The main results are new variational principles connecting rate distortion function to metric mean dimension., Comment: 38 pages, 1 figure

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $\left([0,1]^N\right)^{\mathbb{Z}}$. This problem is intimately related to the theory of mean dimension, which counts the averaged number of parameters of dynamical systems. Lindenstrauss proved that minimal systems of mean dimension less than $N/36$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$, and he proposed the problem of finding the optimal value of the mean dimension for the embedding. We solve this problem by proving that minimal systems of mean dimension less than $N/2$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$. The value $N/2$ is optimal. The proof uses Fourier and complex analysis., Comment: 38 pages

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $\mathbb{Z}^k$-action on a compact metric space $X$, we study the following three problems closely related to mean dimension. (1) When is $X$ isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy $h_{\mathrm{top}}(X)$ is infinite. How much topological entropy can be detected if one considers $X$ only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed $X$ into the $\mathbb{Z}^k$-shift on the infinite dimensional cube $([0,1]^D)^{\mathbb{Z}^k}$? These were investigated for $\mathbb{Z}$-actions in [Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{89} (1999) 227-262], but the generalization to $\mathbb{Z}^k$ remained an open problem. When $X$ has the marker property, in particular when $X$ has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3). A key ingredient is a new method to continuously partition every orbit into good pieces., Comment: 44 pages

Mean dimension measures the size of an infinite dimensional dynamical system. Brody curves are one-Lipschitz entire holomorphic curves in the projective space, and they form a topological dynamical system. Gromov started the problem of estimating its mean dimension in the paper of 1999. We solve this problem. Namely we prove the exact mean dimension formula of the dynamical system of Brody curves. Our formula expresses the mean dimension by the energy density of Brody curves. The proof is based on a novel application of the metric mean dimension theory of Lindenstrauss and Weiss., Comment: 27 pages

This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study the mean dimension of this huge dynamical system. Mean dimension is a topological invariant of dynamical systems introduced by Gromov. We prove the exact formula of the mean dimension by developing a new technique based on the metric mean dimension theory of Lindenstrauss--Weiss., Comment: 40 pages

We study an infinite dimensional ASD moduli space over the cylinder. Our main result is the formula of its local mean dimension. A key ingredient of the argument is the notion of non-degenerate ASD connections. We develop its deformation theory and show that there exist sufficiently many non-degenerate ASD connections by using the method of gluing infinitely many instantons., Comment: 33 pages

We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,...,l}^{\mathbb{Z}},\mathrm{shift})$ for some $l\in\mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})$. The result is sharp. If $(X,T)$ is an extension of an aperiodic zero-dimensional system then it embeds equivariantly in $(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$.

We study the growth of the numbers of critical points in one-dimensional lattice systems by using (real) algebraic geometry and the theory of homoclinic tangency., Comment: 19 pages

We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder., Comment: 5 pages

We introduce several definitions of energy density of Brody curves and show that they give the same value in an appropriate situation., Comment: 9 pages

We study the mean dimensions of the spaces of Brody curves. In particular we give the formula of the mean dimension of the space of Brody curves in the Riemann sphere., Comment: 26 pages

Taubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold having no non-flat instanton., Comment: 38 pages

We study a moduli space of ASD connections over $S^3\times \mathbb{R}$. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the mean dimension by using a "Runge-approximation" for ASD connections, and we prove its lower bound by constructing an infinite dimensional deformation theory of periodic ASD connections., Comment: 55 pages

We show estimates of the "macroscopic dimension" of the $\ell^p$-ball with respect to the $\ell^q$-norm., Comment: 10 pages

The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves., Comment: 18 pages

We study the mean dimension of the moduli space of Brody curves. We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension., Comment: 24 pages

We study entire holomorphic curves in the algebraic torus, and show that they can be characterized by the ``growth rate'' of their derivatives., Comment: 12 pages

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study the relation between theta functions and our packing problem. Applying the Nevanlinna theory, we investigate the packing densities of entire holomorphic curves in the complement of hyperplanes., Comment: 33 pages

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons have infinite energy in general. Moreover we show that they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson's ``alternating method''., Comment: Some explanations are added