arXiv subjects

Philipp Kunde

Publications and source records attributed to Philipp Kunde.

At least 19 recordsLinked to original sources

arXiv2026

Visiting time statistics

Many mixing dynamical systems $(X,T,\mu)$ are known to satisfy the hitting time statistics result \[ \lim_{r \to 0} \mu \{\, x : \tau_{B(y,r)} (x) > t/\mu(B(y,r))\,\} = e^{-t}, \] for $\mu$-almost every $y$, where $\tau_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$. Taking a different point of view, we fix $x$ and consider $\tau_{B(y,r)} (x)$ as a function of $y$. We call this the visiting time of $y$ from $x$, i.e. the time it takes for $y$ to get a visit from $x$ within a neighbourhood of radius $r$. We prove that \[ \lim_{r \to 0} \mu \{\, y : \tau_{B(y,r)} (x) > t/\mu(B(y,r)) \,\} = e^{-t}, \] for $\mu$-almost every $x$. As a byproduct we obtain a new method of proof for hitting time statistics.

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arXiv2025

Almost sure orbits closeness

We consider the minimal distance between orbits of measure preserving dynamical systems. In the spirit of dynamical shrinking target problems we identify distance rates for which almost sure asymptotic closeness properties can be ensured. More precisely, we consider the set $E_n$ of pairs of points whose orbits up to time $n$ have minimal distance to each other less than the threshold $r_n$. We obtain bounds on the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and $\liminf_{n} E_n$ are sets of measure 0 or 1. Results for the measure 0 case are obtained in broad generality while the measure one case requires assumptions of exponential mixing for at least one of the systems. We also consider the analogous question of the minimal distance of points within a single orbit of one dimensional exponentially mixing dynamical systems.

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arXiv2025

Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism

We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also $C^{\infty}$ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.'' In our argument we show that for a uniquely and totally ergodic automorphism $U$ and a particular family of automorphisms $\mathcal{S}$, if $T\times U$ is isomorphic to $T^{-1}\times U$ with $T\in\mathcal{S}$ then $T$ is isomorphic to ${T^{-1}}$. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lema\'nczyk of two weakly mixing automorphisms $T$ and $S$ and an irrational rotation $R$ such that $T\times R$ is isomorphic to $S\times R$, but $T$ and $S$ are not isomorphic.

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arXiv2024

Weak mixing behavior for the projectivized derivative extension

In both smooth and analytic categories, we construct examples of diffeomorphisms of topological entropy zero with intricate ergodic properties. On any smooth compact connected manifold of dimension 2 admitting a nontrivial circle action, we construct a smooth diffeomorphism whose differential is weakly mixing with respect to a smooth measure in the projectivization of the tangent bundle. In case of the 2-torus, we also obtain the analytic counterpart of such a diffeomorphism. The constructions are based on a quantitative version of the ``Approximation by Conjugation'' method, which involves explicitly defined conjugation maps, partial partitions, and the adaptation of a specific analytic approximation technique.

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arXiv2023

A Koopman-Takens theorem: Linear least squares prediction of nonlinear time series

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens' well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem for invertible measure-preserving maps and the Koopman operator on square-integrable functions.

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arXiv2023

Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism

We consider the problem of classifying Kolmogorov automorphisms (or $K$-automorphisms for brevity) up to isomorphism. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and $K$-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. J. Feldman observed that unlike Bernoulli shifts, the family of $K$-automorphisms cannot be classified up to isomorphism by a complete numerical Borel invariant. This left open the possibility of classifying $K$-automorphisms with a more complex type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to $K$-automorphisms, considered as a subset of the Cartesian product of the set of $K$-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to $K$-automorphisms that are also $C^{\infty}$ diffeomorphisms. This shows in a concrete way that the problem of classifying $K$-automorphisms up to isomorphism is intractible.

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arXiv2023

Anti-classification results for weakly mixing diffeomorphisms

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of ``computability''. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the $2$-torus.

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arXiv2022

There is no complete numerical invariant for smooth conjugacy of circle diffeomorphisms

Classical results by Poincar\'e and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by H\"older homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory and a quantitative version of the Approximation by Conjugation method for circle diffeomorphisms.

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arXiv2021

Slow entropy of some combinatorial constructions

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank.

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arXiv2021

Slow entropy for some Anosov-Katok diffeomorphisms

The Anosov-Katok method is one of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties. To measure the complexity of systems with entropy zero, invariants like slow entropy have been introduced. In this article we develop several mechanisms facilitating computation of topological and measure-theoretic slow entropy of Anosov-Katok diffeomorphisms.

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arXiv2021

Non-classifiability of Ergodic Flows up to Time Change

A time change of a flow $\{T_t\}$, ${t\in\mathbb{R}}$, is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. If a flow $\{S_t\}$ is isomorphic to a flow obtained by a reparametrization of a flow $\{T_t\}$, then we say that $\{S_t\}$ and $\{T_t\}$ are isomorphic up to a time change. For ergodic flows $\{S_t\}$ and $\{T_t\}$, Kakutani showed that this happens if and only if the two flows have Kakutani equivalent transformations as cross-sections. We prove that the Kakutani equivalence relation on ergodic invertible measure-preserving transformations of a standard non-atomic probability space is not a Borel set. This shows in a precise way that classification of ergodic transformations up to Kakutani equivalence is impossible. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change. Moreover, we obtain anti-classification results under isomorphism for ergodic invertible transformations of a sigma-finite measure space. We also obtain anti-classification results under Kakutani equivalence for ergodic area-preserving smooth diffeomorphisms of the disk, annulus, and 2-torus, as well as real-analytic diffeomorphisms of the $2$-torus. Our work generalizes the anti-classification results under isomorphism for ergodic transformations obtained by Foreman, Rudolph, and Weiss.

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arXiv2021

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Suppose $(f,\mathcal{X},μ)$ is a measure preserving dynamical system and $ϕ\colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=ϕ\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $μ(X_1>u_n)\to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $μ(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specific examples are piecewise expanding interval maps including the Gauss map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

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arXiv2021

On shrinking targets and self-returning points

We consider the set $\mathcal{R}_\mathrm{io}$ of points returning infinitely many times to a sequence of shrinking targets around themselves. Under additional assumptions we improve Boshernitzan's pioneering result on the speed of recurrence. In the case of the doubling map as well as some linear maps on the $d$ dimensional torus, we even obtain a dichotomy condition for $\mathcal{R}_\mathrm{io}$ to have measure zero or one. Moreover, we study the set of points eventually always returning and prove an analogue of Boshernitzan's result in similar generality.

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arXiv2020

Slow entropy of higher rank abelian unipotent actions

We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/Γ$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $\operatorname{Lie}(G)$ induced by $\operatorname{Lie}(U)$. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $G$. This generalizes the rank one results from [A. Kanigowski, K. Vinhage, D. Wei, Commun. Math. Phys. 370 (2019), no. 2, 449-474.] to higher rank abelian actions.

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arXiv2020

Real-analytic realization of Uniform Circular Systems and some applications

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth ergodic diffeomorphisms of a compact manifold are unclassifiable up to measure-isomorphism. In this paper we show that the uniform circular systems used in the work of Foreman-Weiss admit real-analytic realizations on the torus. As a consequence we obtain the same anti-classification result for real-analytic ergodic diffeomorphisms on the torus. In another application we show the existence of an uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms on the torus.

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arXiv2019

Loosely Bernoulli Odometer-Based Systems Whose Corresponding Circular Systems Are Not Loosely Bernoulli

M. Foreman and B. Weiss obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal{F}$ mapping odometer-based systems, $\mathcal{OB}$, to circular systems, $\mathcal{CB}$. This functor transfers the classification problem from $\mathcal{OB}$ to $\mathcal{CB}$, and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal{F}$ preserves other dynamical properties. We show that $\mathcal{F}$ does not preserve the loosely Bernoulli property by providing positive and zero entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

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arXiv2019

Shrinking targets and eventually always hitting points for interval maps

We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of eventually always hitting points. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. We derive necessary and sufficient conditions on the shrinking rate of the targets for $\mathcal{E}_{\text{ah}}$ to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauss map and correspondingly for the maximal digits in continued fractions expansions. In the case of the doubling map we also compute the packing dimension of $\mathcal{E}_{\text{ah}}$ complementing already known results on the Hausdorff dimension of $\mathcal{E}_{\text{ah}}$.

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arXiv2018

A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli

Let $M$ be a smooth compact connected manifold of dimension $d\geq 2$, possibly with boundary, that admits a smooth effective $\mathbb{T}^2$-action $\mathcal{S}=\left\{S_{α,β}\right\}_{(α,β) \in \mathbb{T}^2}$ preserving a smooth volume $ν$, and let $\mathcal{B}$ be the $C^{\infty}$ closure of $\left\{h \circ S_{α,β} \circ h^{-1} \;:\;h \in \text{Diff}^{\infty}\left(M,ν\right), (α,β) \in \mathbb{T}^2\right\}$. We construct a $C^{\infty}$ diffeomorphism $T \in \mathcal{B}$ with topological entropy $0$ such that $T \times T$ is loosely Bernoulli. Moreover, we show that the set of such $T \in \mathcal{B}$ contains a dense $G_δ$ subset of $\mathcal{B}$. The proofs are based on a two-dimensional version of the approximation-by-conjugation method.

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