# CONUNDRUMS FOR CONTINUOUS LEBESGUE MEASURE-PRESERVING INTERVAL MAPS

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

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## To cite this version:

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy. CONUNDRUMS FOR CONTINUOUS LEBESGUE MEASURE-PRESERVING INTERVAL MAPS. 2024. hal-04575887

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# Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy

<figure><img src="image_1.png" /></figure>

## To cite this version:

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy. Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.. 2025. hal-04575887v2

HAL Id: hal-04575887

https://hal.science/hal-04575887v2

Preprint submitted on 13 Mar 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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CONUNDRUMS FOR CONTINUOUS LEBESGUE
MEASURE-PRESERVING INTERVAL MAPS

JOZEF BOBOK, JERNEJ ČINČ, PIOTR OPROCHA, AND SERGE TROUBETZKOY

ABSTRACT. We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or 1 − id all such maps have topological entropy at least log 2 / 2 and generically they have infinite topological entropy. In this article we show that the generic map has zero measure-theoretic entropy. This implies that there are dramatic differences in the topological versus metric behavior both for injectivity as well as for the structure of the level sets of generic maps.

1. INTRODUCTION

Genericity of zero metric entropy has a long history starting with a result of Rokhlin. Let ( X, µ ) denote a Lebesgue probability space and Aut( X, µ ) the group of measure preserving automorphisms of X . Consider the weak topology: a sequence { f$_{i}$ } ⊂ Aut( X, µ ) converges to f if for every measurable subset A ⊂ X we have µ ( f$_{i}$ ( A ) △ f ( A )) → 0 as i → ∞ . This makes Aut( X, µ ) into a Polish space. In 1959 Rokhlin showed that the subset of all transformations T ∈ Aut( X, µ ) that have zero entropy with respect to µ contains a dense G$_{δ}$ subset [18]. Later Rudolf extended this result to countable amenable actions [12] and L. Bowen further extended this to the non-amenable case [8].

In the late 60s Katok and Stepin developed a quantitative approximation tech-
nique and were able to apply it not only for automorphisms but also for home-
omorphisms of a compact manifold M of dimension at least 2 with respect to
the uniform topology, i.e., induced by the metric

$$
\rho(f, g) = \sup_{x \in M} d(f(x), g(x))
$$

where *d* is the standard metric on *M*. In particular they showed that the
generic Lebesgue measure-preserving homeomorphisms of *M* have zero metric
entropy [15, 16]. These results have modern proofs established by Guihéneuf
[14].

On the other hand Parry proved similar results in two non-invertible settings.
First he considered the one-sided full 2-shift. He considered the weak-topology

<footnote>Date: May 15, 2024.</footnote>
GENERIC CONTINUOUS LEBESGUE
MEASURE-PRESERVING INTERVAL MAPS ARE NOWHERE
MONOTONE BUT INVERTIBLE A.E.

JOZEF BOBOK, JERNEJ ČINČ, PIOTR OPROCHA, AND SERGE TROUBETZKOY

ABSTRACT. We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or 1 - id all such maps have topological entropy at least log 2/2 and generically they have infinite topological entropy. In this article we show that the generic map has zero measure-theoretic entropy. This implies that there are dramatic differences in the topological versus measure-theoretic behavior both for injectivity as well as for the structure of the level sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arcs.

# 1. INTRODUCTION

Genericity of zero measure-theoretic entropy has a long history starting with a result of Rokhlin. Let $(X, \mu)$ denote a Lebesgue probability space and $\text{Aut}(X, \mu)$ the group of measure preserving automorphisms of $X$. Consider the weak topology: a sequence $\{f_i\} \subset \text{Aut}(X, \mu)$ converges to $f$ if for every measurable subset $A \subset X$ we have $\mu(f_i(A) \Delta f(A)) \to 0$ as $i \to \infty$. This makes $\text{Aut}(X, \mu)$ into a Polish space. In 1959 Rokhlin showed that the subset of all transformations $T \in \text{Aut}(X, \mu)$ that have zero entropy with respect to $\mu$ contains a dense $G_\delta$ subset [24]. Later Rudolf extended this result to countable amenable actions [14] and L. Bowen further extended this to the non-amenable case [9].

In the late 60s Katok and Stepin developed a quantitative approximation tech-
nique and were able to apply it not only for automorphisms but also for home-
omorphisms of a compact manifold $M$ of dimension at least 2 with respect to
the uniform topology, i.e., induced by the metric

$$
\rho(f, g) := \sup_{x \in M} d(f(x), g(x))
$$

<footnote>Date: March 13, 2025.</footnote>
on the set of invariant ergodic probability measures, and showed that the set of zero entropy (ergodic, fully supported, and non-atomic) measures is a dense $G_\delta$ [17]. This result immediately translates to the doubling map on the circle. Then he considered the space $S$ of strictly increasing degree 2 maps of the circle which preserve the Lebesgue measure with the uniform topology. In this setting he showed that the set of ergodic zero entropy maps is a dense $G_\delta$.

It is clear that Parry's result also holds for full tent maps preserving the Lebesgue measure. We consider the generalization of this question when we drop the (quite strong) unimodal restriction. Let $I := [0, 1]$. Define $C_\lambda$ to be the set of continuous maps of $I$ which preserve the Lebesgue measure $\lambda$. We consider the uniform topology. The main result of this article is that there is a dense $G_\delta$ set of maps in $C_\lambda$ with zero metric entropy, more precisely

**Theorem 1.** The set of zero metric entropy maps is a dense $G_\delta$ set $G \subset C_\lambda$.

Theorem 1 holds more generally for $C_\mu$ where $\mu$ is any non-atomic probability measure with full support.

By combining and adapting various known results we obtain several corollaries of Theorem 1 which contrast the measure theoretic and topological behaviors of generic maps in $C_\lambda$. Before turning to them we adapt the results of Bruckner and Garg [9] to show that generic maps in $C_\lambda$ are nowhere differentiable, even stronger

**Theorem 2.** Generic maps in $C_\lambda$ are of nonmonotonic type, i.e., for each generic $f$ we have $\overline{D}f(x) = +\infty$ and $\underline{D}f(x) = -\infty$ for every $x$.

Clearly there are no invertible interval maps that are nowhere differentiable. A map $f \in C_\lambda$ is called *invertible a.e.* if there is an $f$-invariant set $X \subset I$ of full Lebesgue measure such that the restriction $f|_X$ is invertible. According to [21, Corollary 4.14.3] $h(f) = 0$ implies that $f$ is invertible a.e.. Combining this with Theorem 2 yields the following surprising corollary.

**Corollary 3.** Generic maps in $C_\lambda$ are of nonmonotonic type and invertible a.e., thus for each generic $f$ there exists a set $X_f \subset I$ of Lebesgue measure 1 such that $f$ is a bijection on $X_f$ and $\lambda(M) = \lambda(M \cap X_f) = \lambda(f^{-1}(M \cap X_f))$ for each Borel set $M \subset I$.

**Remark 4.** Corollary 3 is in a strong contrast to the topological behavior of typical maps in $C_\lambda$. Let

$$B_0 := \{x \in I : \#f^{-1}(x) > 1\}.$$
where $d$ is the standard metric on $M$. In particular they showed that the generic Lebesgue measure-preserving homeomorphisms of $M$ have zero measure-theoretic entropy [17, 18]. These results have modern proofs established by Guihéneuf [16].

Zero entropy systems are invertible almost surely, thus such genericity results as described above might not be surprising for invertible systems. It is less clear that similar results could hold in the non-invertible setting. There are only a few known results for non-invertible maps in this direction. Namely, in the setting of the one-sided full 2-shift, Parry considered the weak-topology on the set of invariant ergodic probability measures, and showed that the set of zero entropy (ergodic, fully supported, and non-atomic) measures is a dense $G_\delta$ [21]. This result immediately translates to the doubling map on the circle. Then he considered the space $S$ of strictly increasing degree 2 maps of the circle which preserve the Lebesgue measure with the uniform topology. In this setting he showed that the set of ergodic zero entropy maps is a dense $G_\delta$.

It is clear that Parry’s result also holds for full branch unimodal interval maps preserving the Lebesgue measure. The only other result in this direction is the generalization of Parry’s result to full branch $k$-modal interval maps preserving Lebesgue measure by Friedland and Weiss [15]. We consider the generalization of this question when we omit the (quite strong) piecewise monotone restriction. Let $I := [0, 1]$. Define $C_\lambda$ to be the set of continuous maps of $I$ which preserve the Lebesgue measure $\lambda$. We consider the uniform topology. The first result of this article is that there is a dense $G_\delta$ set of maps in $C_\lambda$ with zero measure-theoretic entropy, more precisely

**Theorem 1.** *The set of zero measure-theoretic entropy maps is a dense $G_\delta$ set $Q \subset C_\lambda$.*

Let us note that Bobok and Troubetzkoy [3] proved that the set of maps with measure-theoretic entropy equal to $c$ is dense, but only for each $c \in (0, \infty)$. The proof of Theorem 1 follows a different path than the proofs of Parry [21] or Friedland and Weiss [15]. By conjugating, we see that the theorem also holds more generally for $C_\mu$ where $\mu$ is any non-atomic probability measure with full support [7, Remark 1.1].

By adapting various known results and combining them with our results we obtain a contrast between measure-theoretic and topological behaviors of generic maps in $C_\lambda$. In order to adapt the results of Bruckner and Garg [10], we recall a strong notion of nowhere differentiability: a map $f$ is of *nonmonotonic type* if the upper derivative $\overline{D}f(x) = +\infty$ and the lower derivative $\underline{D}f(x) = -\infty$ for every $x$.
It follows from [6, Corollary 3.2] that except for the two exception maps id and 1 − id the set $B_0$ contains an interval, and thus we always have a set of positive measure on which $f$ is not one-to-one.<fnref n="1" />

**Remark 5.** The reverse implication, $f$ is invethmBrtible a.e. implies that $h(f) = 0$, is not true in general. It is known that any compact manifold of dimension $d > 1$ which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy [1] and the measure can be assumed to be the Lebesgue measure. It seems unknown if this reverse implication holds in $C_\lambda$.

Our next result further contrasts the topological and metric behavior of generic
maps. For every $c \in I$ the set $\{x : f(x) = c\}$ is called a level set of $f$.

**Theorem 6.** For each generic map $f$ in $C_\lambda$ there exists a set $X_f \subset I$ of Lebesgue measure 1 such that for each $c \in (0,1)$ the level set $\{x: f(x) = c\}$ contains a perfect set, while for each $c$ the set $X_f \cap \{x: f(x) = c\}$ contains at most one point.

For comparison we resume other generic properties of maps from $C_\lambda$; in our
context the item (6) in the following theorem is particularly interesting. For
the definitions of the notions we will mention in the following theorem we refer
the reader to the associated articles.

**Theorem.** There is a dense $G_\delta$ subset of $C_\lambda$ such that for each $f \in G$

(1) $f$ has infinite topological entropy [3],

(2) $f$ is weakly mixing with respect to $\lambda$ [3],

(3) $f$ is leo and thus satisfies the specification property [3],

(4) $f$ has the shadowing property [4],

(5) periodic points have Cantor set structure, see [4] for details.

(6) $f$ has $\lambda$-a.e. point a knot point [2],

(7) the graph of $f$ has Hausdorff dimension and lower box dimension 1 and upper box dimension 2 [3], [19].

(8) $f$ is $\delta$-crooked [11].

The leo and specification properties are actually satisfied not only generically
but on an open dense subset of $C_\lambda$ [7].

Two other related results were proven by Sigmund. In his first result he con-
sidered the full shift on $\sigma : \mathbb{R}^\mathbb{Z} \to \mathbb{R}^\mathbb{Z}$ and showed that the set of probability
measures with zero entropy form a dense $G_\delta$ in the weak topology. He also
consider the set of all subshifts with a fixed finite alphabet with respect to the

<footnote n="1">¹On the first page of the article [3] there is an incorrect statement that maps other than id and 1 − id have non-zero metric entropy. This need not be the case as exhibited by the results of this article.</footnote>
Let $f \in C(I)$. For every $c \in \mathbb{R}$ the set $\{x : f(x) = c\}$ is called a level set of $f$. The level sets of a function $f \in C(I)$ are said to be normal if there is a countable set $E_f \subset (\min f, \max f)$ such that the level set $\{x : f(x) = c\}$ of $f$ is

(i) a non-empty perfect set when $c \notin E_f \cup \{\min f, \max f\}$,

(ii) a single point when $c = \min f$ or $c = \max f$,

(iii) the union of a nonempty perfect set $P$ with an isolated point $x \notin P$ when $c \in E_f$.

**Theorem 2.** The following two assertions hold.

(1) Generic maps in $C_\lambda$ are of nonmonotonic type.

(2) The level sets of generic maps in $C_\lambda$ are normal.

Clearly there are no continuous invertible interval maps that are nowhere differentiable. A map $f \in C_\lambda$ is called *invertible a.e.* if there is an $f$-invariant set $X \subset I$ of full Lebesgue measure such that the restriction $f|_X$ is invertible. Also let

$$B_0 := \{x \in I : \#f^{-1}(x) > 1\}.$$

The following corollary contrasts the topological and measure-theoretic behavior of generic maps.

**Corollary 3.** There is a dense $G_\delta$ subset $Q$ of $C_\lambda$ such that:

(1) For each generic $f \in Q$ there exists a set $X_f \subset I$ of Lebesgue measure 1 such that $f$ is a bijection on $X_f$ and $\lambda(M) = \lambda(M \cap X_f) = \lambda(f^{-1}(M \cap X_f))$ for each Borel set $M \subset I$.

(2) For each generic map $f \in Q$ in $C_\lambda$ there exists a set $X_f \subset I$ of Lebesgue measure 1 such that for each $c \in (0,1)$ the level set $\{x : f(x) = c\}$ is a perfect set, while for each $c$ the set $X_f \cap \{x : f(x) = c\}$ contains at most one point.

(3) Except for the two exceptional maps id and 1 - id in $C_\lambda$ the set $B_0$ contains an interval, and thus we always have a set of positive Lebesgue measure on which $f$ is not one-to-one.<fnref n="1" />

*Proof.* (1) According to [27, Corollary 4.14.3] $h_\lambda(f) = 0$ implies that $f$ is invertible a.e.. Thus combining this with Theorems 1 and 2 yields statement (1). Statement (2) follows immediately by combining (1) with Theorem 2. Statement (3) follows from [6, Corollary 3.2] that except for the two exceptional maps id and 1 - id the set $B_0$ contains an interval, and thus we always have a set of positive measure on which $f$ is not one-to-one. □

<footnote n="1">¹On the first page of the article [3] there is an incorrect statement that maps other than id and 1 - id have non-zero measure-theoretic entropy. This need not be the case as exhibited by the results of this article.</footnote>
Vietoris topology and showed that the generic subshift has zero topological entropy [20].

A generalization of Parry's result on genericity of zero entropy measures was obtained by Carvalho and Condori [10]. They considered uniformly continuous maps $f$ defined over a Polish metric space and showed that the set of $f$-invariant measures with zero metric entropy is a $G_\delta$ set in the weak topology and that this set is dense and thus generic if the set of $f$-periodic measures is dense in the set of $f$-invariant measures.

## 2. PROOF OF THEOREM 1

Let $\mathcal{P} := \{P_1, \dots, P_k\}$ be a finite partition of $I$ and let

$$H(\mathcal{P}) := - \sum_{i=1}^{k} \lambda(P_i) \log \lambda(P_i).$$

For $f \in C_\lambda$ we denote

$$h_n(f, \mathcal{P}) := \frac{1}{n} H(\bigvee_{i=0}^{n-1} f^{-i} \mathcal{P}).$$

This function is decreasing in $n$ thus the limit $h(f, \mathcal{P}) := \lim_{n \to \infty} h_n(f, \mathcal{P})$ exists. Finally we define the entropy of $f$ by

$$h(f) := \sup\{h(f, \mathcal{P}) : \mathcal{P} \text{ is a finite partition of } I\}.$$

Let $\mathcal{P}_i$ be the partition of $I$ into dyadic intervals $[\frac{j}{2^i}, \frac{j+1}{2^i}]$ and $\mathcal{A}_i$ the subalgebra generated by $\mathcal{P}_i$. Then $\mathcal{A}_1 \subseteq \mathcal{A}_2 \subseteq \dots$ and $\bigvee_{n=1}^\infty \mathcal{A}_n = \mathcal{B}$ hence by [21, Theorem 4.22] $h(f) = \lim_{i \to \infty} h(f, \mathcal{P}_i)$.

**Lemma 7.** Fix $f \in C_\lambda$ and choose an arbitrary interval $J = [a, b] \subset I$. Then $\forall \varepsilon > 0 \exists \delta > 0$ such that $\lambda(f^{-1}(J) \Delta g^{-1}(J)) < \varepsilon$ for all $g \in B(f, \delta)$.

*Proof.* Fix $\varepsilon > 0$ and let $\delta < \frac{\varepsilon}{4}$. If we put $H_\delta = f^{-1}([a+\delta, b-\delta])$, since $f \in C_\lambda$, (1) $\lambda(f^{-1}(J)) = \lambda(J) > \lambda(H_\delta) > \lambda(J) - \frac{\varepsilon}{2}$.

For each $g \in B(f, \delta)$ we can write analogously

$$ (2) \qquad \lambda(g^{-1}(J)) = \lambda(J) > \lambda(H_\delta) > \lambda(J) - \frac{\varepsilon}{2}. $$

Clearly $H_\delta \subset (f^{-1}(J) \cap g^{-1}(J))$, so (1) and (2) imply

$$ \lambda(f^{-1}(J) \Delta g^{-1}(J)) < \varepsilon. \quad \square $$
Unlike the approach given in [21] and [15], it is not clear if one can constructively define the set $X_f$ in Corollary 3 for a generic map in $C_\lambda$.

**Remark 4.** The reverse implication, *f* is invertible a.e. implies that $h_\lambda(f) = 0$, is not true in general. It is known that any compact manifold of dimension $d > 1$ which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy [1] and the measure can be assumed to be the Lebesgue measure. It seems unknown if this reverse implication holds in $C_\lambda$ - compare with [23, Theorem 1.9.7] and [27, Corollary 4.18.1].

For comparison we resume other generic properties of maps from $C_\lambda$; in our context the item (6) in the following theorem is particularly interesting. For the definitions of the notions we will mention in the following theorem we refer the reader to the associated articles.

**Theorem.** There is a dense $G_\delta$ subset $G$ of $C_\lambda$ such that for each $f \in G$

(1) *f* has infinite topological entropy [3],

(2) *f* is weakly mixing with respect to $\lambda$ [3],

(3) *f* is leo and thus satisfies the specification property [3],

(4) *f* has the shadowing property [4],

(5) periodic points have Cantor set structure, see [4] for details.

(6) *f* has $\lambda$-a.e. point a knot point [2],

(7) the graph of *f* has Hausdorff dimension and lower box dimension 1 and upper box dimension 2 [3], [25].

(8) *f* is $\delta$-crooked [12].

The leo and specification properties are actually satisfied not only generically but on an open dense subset of $C_\lambda$ [7].

Two other related results were proven by Sigmund. In his first result he considered the full shift on $\sigma: \mathbb{R}^Z \to \mathbb{R}^Z$ and showed that the set of probability measures with zero entropy form a dense $G_\delta$ in the weak topology. He also consider the set of all subshifts with a fixed finite alphabet with respect to the Vietoris topology and showed that the generic subshift has zero topological entropy [26].

A generalization of Parry's result on genericity of zero entropy measures was obtained by Carvalho and Condori [11]. They considered uniformly continuous maps $f$ defined over a Polish metric space and showed that the set of $f$-invariant measures with zero measure-theoretic entropy is a $G_\delta$ set in the weak topology and that this set is dense and thus generic if the set of $f$-periodic measures is dense in the set of $f$-invariant measures.
**Lemma 8.** For $\ell \in \mathbb{N}$ let positive numbers $a_1, \dots, a_\ell$ satisfy $\sum_{i=1}^\ell a_i = \eta < 1$. Then

$$ -\sum_{j=1}^{\ell} a_j \log a_j \le \eta \log \ell - \eta \log \eta. $$

*Proof.* Let $b_i := a_i/\eta$, then $\sum_{i=1}^\ell b_i = 1$ and thus $-\sum_{i=1}^\ell b_i \log b_i \le \log \ell$ ([21, Cor 4.2.1]) which is equivalent to the announced inequality. □

**Proposition 9.** Let $\mathcal{P} = \{P_1, \dots, P_k\}$ be a partition of $I$ consisting of intervals and $n \in \mathbb{N}$. The map

$$ f \to h_n(f, \mathcal{P}), \quad f \in C_\lambda $$

is continuous.

*Proof.* Fix $\varepsilon > 0$ and $f \in C_\lambda$. Put $\bigvee_{i=0}^{n-1} f^{-i}(\mathcal{P}) = \{A_1, \dots, A_m\}$, where $\lambda(A_j) > 0$ for each $j \in \{1, \dots, m\}$ and $\sum_{j=1}^m \lambda(A_j) = 1$. It means that there are exactly $m \le k^n$ nonempty intersections

$$ A_j = P_{i_0(j)} \cap f^{-1}(P_{i_1(j)}) \cap \dots \cap f^{-(n-1)}(P_{i_{n-1}(j)}), \quad j \in \{1, \dots, m\}, $$

and all other $(k^n - m)$ intersections are empty (mod 0). For $g$ close to $f$ we consider $\bigvee_{i=0}^{n-1} g^{-i}(\mathcal{P})$, and label its positive measure elements by $B_{j,g}$ ($1 \le j \le \ell_g \le k^m$). We choose sufficiently small $\delta > 0$ so that $\ell_g \ge m$ and applying Lemma 7 to the maps $f^0, f^1, \dots, f^{n-1} \in C_\lambda$ we obtain that each set

$$ B_{j,g} = P_{i_0(j)} \cap g^{-1}(P_{i_1(j)}) \cap \dots \cap g^{-(n-1)}(P_{i_{n-1}(j)}), \quad j \in \{1, \dots, m\}, $$

corresponds to the $A_j$ sufficiently well so that,

$$ -\sum_{j=1}^{m} \lambda(A_j) \log \lambda(A_j) + \sum_{j=1}^{m} \lambda(B_{j,g}) \log \lambda(B_{j,g}) < \frac{n\varepsilon}{2}, $$

and $\sum_{j=m+1}^{\ell_g} \lambda(B_{j,g}) = 1 - \sum_{j=1}^m \lambda(B_{j,g}) = \eta$ for which

$$ \eta \log(k^n - m) - \eta \log \eta < \frac{n\varepsilon}{2} $$

for each $g \in B(f, \delta)$. Since $\ell_g \le k^n$, from Lemma 8 we obtain

$$ (3) \quad 0 < - \sum_{j=m+1}^{\ell_g} \lambda(B_{j,g}) \log \lambda(B_{j,g}) \le \eta \log(k^n - m) - \eta \log \eta < \frac{n\varepsilon}{2}. $$

From (3) we obtain

$$ |h_n(f, \mathcal{P}) - h_n(g, \mathcal{P})| = \left| \frac{1}{n} H\left(\bigvee_{i=0}^{n-1} f^{-i}(\mathcal{P})\right) - \frac{1}{n} H\left(\bigvee_{i=0}^{n-1} g^{-i}(\mathcal{P})\right) \right| < \varepsilon. \quad \square $$
Let us give an outline of the article. In Section 2 we prove Theorem 1. In Section 3 we give a construction of Lebesgue measure-preserving maps that have measure-theoretic entropy 0. In Section 4 we prove Theorem 2, which is the second main result of our article. In Section 5 we provide an application of our main results. Part (8) from the previous theorem implies that inverse limit of any generic map from $C_\lambda$ is a curious topological object called the pseudo-arc. We apply Theorem 1 and Corollary 3 to give additional information on some measure-theoretic aspects of the construction of planar homeomorphisms with pseudo-arc attractors from [12].

## 2. PROOF OF THEOREM 1

Let $\mathcal{P} := \{P_1, \dots, P_k\}$ be a finite partition of $I$ and let

$$H(\mathcal{P}) := - \sum_{i=1}^{k} \lambda(P_i) \log \lambda(P_i).$$

For $f \in C_\lambda$ we denote

$$h_n(f, \mathcal{P}) := \frac{1}{n} H(\bigvee_{i=0}^{n-1} f^{-i} \mathcal{P}).$$

This function is decreasing in $n$ thus the limit $h(f, \mathcal{P}) := \lim_{n \to \infty} h_n(f, \mathcal{P})$ exists. Finally we define the entropy of $f$ by

$$h_{\lambda}(f) := \sup\{h(f, \mathcal{P}) : \mathcal{P} \text{ is a finite partition of } I\}.$$

Let $\mathcal{P}_i$ be the partition of $I$ into dyadic intervals $[\frac{j}{2^i}, \frac{j+1}{2^i}]$ and $\mathcal{A}_i$ the sub-algebra generated by $\mathcal{P}_i$. Then $\mathcal{A}_1 \subseteq \mathcal{A}_2 \subseteq \dots$ and $\bigvee_{n=1}^{\infty} \mathcal{A}_n \cong \mathcal{B}$ hence by [27, Theorem 4.22] $h_\lambda(f) = \lim_{i \to \infty} h(f, \mathcal{P}_i)$.

**Lemma 5.** Fix $f \in C_\lambda$ and choose an arbitrary interval $J = [a, b] \subset I$. Then $\forall \varepsilon > 0 \exists \delta > 0$ such that $\lambda(f^{-1}(J) \Delta g^{-1}(J)) < \varepsilon$ for all $g \in B(f, \delta)$.

*Proof.* Fix $\varepsilon > 0$ and let $\delta < \frac{\varepsilon}{4}$. If we put $H_\delta = f^{-1}([a+\delta, b-\delta])$, since $f \in C_\lambda$,

$$(1) \qquad \lambda(f^{-1}(J)) = \lambda(J) > \lambda(H_\delta) > \lambda(J) - \frac{\varepsilon}{2}.$$

For each $g \in B(f, \delta)$ we can write analogously

$$(2) \qquad \lambda(g^{-1}(J)) = \lambda(J) > \lambda(H_\delta) > \lambda(J) - \frac{\varepsilon}{2}.$$

Clearly $H_\delta \subset (f^{-1}(J) \cap g^{-1}(J))$, so (1) and (2) imply

$$\lambda(f^{-1}(J) \Delta g^{-1}(J)) < \varepsilon. \qquad \square$$
Proof of Theorem 1. Fix $\beta > 0$ and let

$$G_{\beta} := \bigcap_{k \ge 1} \bigcup_{i \ge k} \bigcup_{n \ge k} \{f \in C_{\lambda} : h_n(f, \mathcal{P}_i) < \beta\}.$$

We claim that

(i) $G_{\beta}$ a dense $G_{\delta}$ set and

(ii) $G_{\beta}$ coincides with the set of maps with entropy less than $\beta$.

Once the claim is established the theorem follows since $G := \bigcap_{n \ge 1} G_{1/n}$ is a dense $G_{\delta}$ set and it coincides with the set of zero entropy maps.

We turn to the proof of the claim. Let us first prove (ii). If $h(f) < \beta$ then $h_n(f, \mathcal{P}_i) < \beta$ for each $i$ for all sufficiently large $n$, so $f \in G_{\beta}$. Conversely to see that the entropy of the maps in $G_{\beta}$ is at most $\beta$ we fix $f \in G$. By the definition of $G_{\beta}$, for each $f \in G_{\beta}$ there are sequences $i_k \ge k$ and $n_k \ge k$ such that $h_{n_k}(f, \mathcal{P}_{i_k}) < \beta$ for each $k \ge 1$. Since $h_n(f, \mathcal{P})$ is decreasing in $n$ we conclude $h(f, \mathcal{P}_{i_k}) < \beta$.

Since $\mathcal{P}_{i+1}$ refines $\mathcal{P}_i$ we have $h(f, \mathcal{P}_i)$ is monotonic in $i$ and we conclude $h(f, \mathcal{P}_i) < \beta$ for all $i$. Thus $h(f) = \lim_i h(f, \mathcal{P}_i) < \beta$ which finishes the proof of (ii).

To prove (i) we remark that the set $\{f \in C_{\lambda} : h_n(f, \mathcal{P}_i) < \beta\}$ is open by Proposition 9, thus $G_{\beta}$ is a $G_{\delta}$ set.

On the other hand the density of $G_{\beta}$ follows from [3] Proposition 24. This finishes the proof of the Theorem. $\square$

### 3. CONSTRUCTIONS OF ZERO ENTROPY MAPS

**Example 10.** Suppose $f \in C_{\lambda}$ be such that $\lambda(\text{Per}(f)) = 1$. If we use the ergodic decomposition theorem [21, p. 153] then

$$(4) \qquad h(f) = \int_{E(I, f)} h_{\mu}(f) \, d\nu(\mu)$$

where $E(I, f)$ is the set of all ergodic measures invariant for $f$ and $\nu$ is a Borel measure on $E(I, f)$ satisfying

$$\lambda = \int_{E(I, f)} \mu \, d\nu(\mu).$$

Since $\lambda(\text{Per}(f)) = 1$ necessarily $\nu$ a.e. measure ergodic $\mu$ is a CO-measure (i.e., a measure on a periodic orbit). In [4, Theorem 2] we constructed a dense set of leo maps for which the periodic points have full measure, and thus they all have zero entropy. This yields an alternative proof of the density of $G$ from the proof of Theorem 1.
**Lemma 6.** For $\ell \in \mathbb{N}$ let positive numbers $a_1, \dots, a_\ell$ satisfy $\sum_{i=1}^\ell a_i = \eta < 1$. Then

$$-\sum_{j=1}^{\ell} a_j \log a_j \le \eta \log \ell - \eta \log \eta.$$

*Proof.* Let $b_i := a_i/\eta$, then $\sum_{i=1}^\ell b_i = 1$ and thus $-\sum_{i=1}^\ell b_i \log b_i \le \log \ell$ ([27, Cor 4.2.1]) which is equivalent to the announced inequality. $\square$

**Proposition 7.** Let $\mathcal{P} = \{P_1, \dots, P_k\}$ be a partition of $I$ consisting of intervals and $n \in \mathbb{N}$. The map

$$f \to h_n(f, \mathcal{P}), \quad f \in C_\lambda$$

is continuous.

*Proof.* Fix $\varepsilon > 0$ and $f \in C_\lambda$. Put $\bigvee_{i=0}^{n-1} f^{-i}(\mathcal{P}) = \{A_1, \dots, A_m\}$, where $\lambda(A_j) > 0$ for each $j \in \{1, \dots, m\}$ and $\sum_{j=1}^m \lambda(A_j) = 1$. Each set $A_j$ can be written as

$$A_j = P_{i_0(j)} \cap f^{-1}(P_{i_1(j)}) \cap \dots \cap f^{-(n-1)}(P_{i_{n-1}(j)}), \quad j \in \{1, \dots, m\},$$

and all other $(k^n - m)$ possible intersections have measure 0.

For $g$ close to $f$ we consider $\bigvee_{i=0}^{n-1} g^{-i}(\mathcal{P})$ and let $\ell_g$ be the number of positive measure elements of this partition. Enumerate the sets $B_{j,g}$ in $\bigvee_{i=0}^{n-1} g^{-i}(\mathcal{P})$ in such a way that

$$B_{j,g} = P_{i_0(j)} \cap g^{-1}(P_{i_1(j)}) \cap \dots \cap g^{-(n-1)}(P_{i_{n-1}(j)}), \quad j \in \{1, \dots, m\}.$$

It is clear that $B_{j,f} = A_j$. Furthermore these sets vary continuously with respect to $g$ sufficiently close to $f$ in the following sense. Applying Lemma 5 to the maps $f^0, f^1, \dots, f^{n-1} \in C_\lambda$ we can choose sufficiently small $\delta > 0$ so that each $B_{j,g}$ has positive measure provided that $A_j$ has positive measure for $1 \le j \le m$. Hence $\ell_g \ge m$, we may assume that $\lambda(B_{j,g}) > 0$ for $1 \le j \le \ell_g$ and furthermore (as a consequence of Lemma 5; see also Lemma 6).

$$-\sum_{j=1}^{m} \lambda(A_j) \log \lambda(A_j) + \sum_{j=1}^{m} \lambda(B_{j,g}) \log \lambda(B_{j,g}) < \frac{n\varepsilon}{2},$$

and $\sum_{j=m+1}^{\ell_g} \lambda(B_{j,g}) = 1 - \sum_{j=1}^{m} \lambda(B_{j,g}) = \eta$ for which

$$\eta \log(k^n - m) - \eta \log \eta < \frac{n\varepsilon}{2}$$

for each $g \in B(f, \delta)$. Since $\ell_g \le k^n$, from Lemma 6 we obtain

$$(3) \quad 0 < - \sum_{j=m+1}^{\ell_g} \lambda(B_{j,g}) \log \lambda(B_{j,g}) \le \eta \log(k^n - m) - \eta \log \eta < \frac{n\varepsilon}{2}.$$
**Example 11.** Parry sketched a construction of maps $f \in C_\lambda$ which are piece-wise monotone with 2-full branches which are invertible $\lambda$-a.e. [17]; a more detailed version of this construction (with $k$ full branches) was given by Fried-land and Weiss [13].

## 4. PROOFS OF THEOREM 2 AND THEOREM 6

Let $C(I)$ denote the set of continuous maps $f: I \to I$. A function $f \in C(I)$ is said to be non-decreasing at a point $x \in I$ if there exists a $\delta > 0$ such that $f(t) \le f(x)$ when $t \in I \cap (x - \delta, x)$ and $f(t) \ge f(x)$ when $t \in I \cap (x, x - \delta)$. The function $f$ is called nonincreasing at $x$ if $-f$ is nondecreasing at $x$, and $f$ is called monotone at $x$ if it is either nondecreasing or nonincreasing at $x$. If $f \in C(I)$ and $\gamma \in \mathbb{R}$ then we define the map $f_{-\gamma}$ by:

$$f_{-\gamma}(x) := f(x) - \gamma x \text{ for } x \in I.$$

A map $f$ is *monotonic type at $x$* if there exists a real number $\gamma$ such that the function $f_{-\gamma}$ is monotone at $x$. If $f$ is not of monotonic type at any point of $I$, it will be said to be of *nonmonotonic type*. It is easy to see that $f$ is of nonmonotonic type if and only if for each $x, \overline{D}f(x) = +\infty$ and $\underline{D}f(x) = -\infty$, where $\overline{D}f, \underline{D}f$ are the upper and lower derivatives.

The proof of Theorem 2 requires two auxiliary lemmas. The first lemma is a minor modification of Lemma 11 from [5].

**Lemma 12.** Let $f$ be a piecewise affine continuous interval with nonzero slopes and such that its derivative does not exist at a finite set $E$. Then $f \in C_\lambda$ if and only if

$$(5) \qquad \forall y \in [0, 1) \setminus F(E): \sum_{x \in F^{-1}(y)} \frac{1}{|F'(x)|} = 1.$$

A *critical value* of a map $f$ is a local maximum or a local minimum. A *determining value* of a piecewise affine map $f$ is either its critical value or value at a point in which the derivative of $f$ does not exist.

**Lemma 13.** The set of piecewise affine maps with rational determining values is dense in $C_\lambda$.

*Proof.* Consider a dense collection of piecewise affine maps $\{f_m\} \subset C_\lambda$ (Figure 1 left), each $f_m$ has finitely many determining points. Fix $m$ and choose $n = n(m)$ so large that each of the intervals $[k/n, k + 1/n]$ contains at most one determining value. For each irrational determining value $c_i$ of $f_m$ we consider the map $g_{n,i} \circ f_m$, where $g_{n,i}$ is the map of Figure 1 right. The resulting map has the determining value $c_i$ replaced by $k/n$ and $(k+1)/n$. We progressively remove all irrational determining values of $f_m$ to obtain a map
From (3) we obtain

$$|h_n(f, \mathcal{P}) - h_n(g, \mathcal{P})| = \left| \frac{1}{n} H(\bigvee_{i=0}^{n-1} f^{-i}(\mathcal{P})) - \frac{1}{n} H(\bigvee_{i=0}^{n-1} g^{-i}(\mathcal{P})) \right| < \varepsilon. \quad \square$$

*Proof of Theorem 1.* Fix $\beta > 0$ and let

$$Q_{\beta} := \bigcap_{k \ge 1} \bigcup_{i \ge k} \bigcup_{n \ge k} \{f \in C_{\lambda} : h_n(f, \mathcal{P}_i) < \beta\}.$$

We claim that

(i) $Q_\beta$ a dense $G_\delta$ set and

(ii) $Q_\beta$ coincides with the set of maps with entropy less than $\beta$.

Once the claim is established the theorem follows since $Q := \bigcap_{n \ge 1} Q_{1/n}$ is a dense $G_\delta$ set and it coincides with the set of zero entropy maps.

We turn to the proof of the claim. Let us first prove (ii). If $h_\lambda(f) < \beta$ then $h_n(f, \mathcal{P}_i) < \beta$ for each $i$ for all sufficiently large $n$, so $f \in Q_\beta$. Conversely to see that the entropy of the maps in $Q_\beta$ is at most $\beta$ we fix $f \in Q_\beta$. By the definition of $Q_\beta$, for each $f \in Q_\beta$ there are sequences $i_k \ge k$ and $n_k \ge k$ such that $h_{n_k}(f, \mathcal{P}_{i_k}) < \beta$ for each $k \ge 1$. Since $h_n(f, \mathcal{P})$ is decreasing in $n$ we conclude $h(f, \mathcal{P}_{i_k}) < \beta$.

Since $\mathcal{P}_{i+1}$ refines $\mathcal{P}_i$ we have $h(f, \mathcal{P}_i)$ is monotonic in $i$ and we conclude $h(f, \mathcal{P}_i) < \beta$ for all $i \ge i_1$. Thus $h_\lambda(f) = \lim_i h(f, \mathcal{P}_i) < \beta$ which finishes the proof of (ii).

To prove (i) we remark that the set $\{f \in C_\lambda : h_n(f, \mathcal{P}_i) < \beta\}$ is open by Proposition 7, thus $Q_\beta$ is a $G_\delta$ set.

The set $Q_\beta$ coincides with the set of maps with entropy less than $\beta$, thus its density follows from [3] Proposition 24. This finishes the proof of the Theorem. $\square$

### 3. CONSTRUCTIONS OF ZERO ENTROPY MAPS

Suppose $f \in C_\lambda$ be such that $\lambda(\text{Per}(f)) = 1$. If we use the ergodic decomposition theorem [27, p. 153] then

$$(4) \qquad h_{\lambda}(f) = \int_{E(I,f)} h_{\mu}(f) \, d\nu(\mu)$$
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Removing irrational critical values.</figcaption></figure>

$h_m$ with only rational determining values. The set $\{h_m\}$ is dense in $C_\lambda$ since
$\rho(f_m, h_m) \to 0$. $\square$

The proof of Theorem 2 follows the strategy of the proof of [9, Theorem 2.2]. We use the terminology of the proof from [9]. In particular, for any $n \in \mathbb{N}$, the set $A_n$ denotes the set of functions $f \in C_\lambda$ for which there exist $\gamma \in [-n, n]$ and $x \in [0, 1]$ such that $f_{-\gamma}(t) \le f_{-\gamma}(x)$ when $t \in [0, 1] \cap (x - 1/n, x)$ and $f_{-\gamma}(t) \ge f_{-\gamma}(x)$ when $t \in [0, 1] \cap (x, x + 1/n)$ (here we changed the notation of [9] replacing their $\lambda$ by $\gamma$ to avoid a collision of terminologies).

*Proof of Theorem 2.* Except for one step, the strategy of the proof is the same as in [9]; this difference is due to the fact of the preserving of Lebesgue measure. The step which differs is showing that each $A_n$ is nowhere dense, this step is the most technical step in the proof in [9]. To show this we proceed as illustrated in Figure 2. We know that piecewise affine maps are dense in $C_\lambda$ [3]. So from now $n \in \mathbb{N}$ defining $A_n$ is fixed.

**I. A choice of f.** Using Lemma 13, given an open set $U \subset C_\lambda$ and $\varepsilon > 0$ we can choose a piecewise affine map $f \in U$ whose determining values are rational such that $B(f, \varepsilon) \subset U$. Let $q$ be the greatest common denominator of the determining values. Let $1/\tau$ be a positive integer multiple of $q$ such that $\tau < \varepsilon/3$. For each integer $1 \le i \le 1/\tau$ consider the interval $J_i = [(i-1)\tau, i\tau]$. These intervals satisfy

(i) $\lambda(J_i) = \tau$ for all $i$,

(ii) $J_i^\circ \cap J_j^\circ = \emptyset$ for all $i \neq j$,

(iii) each $J_i$ has no determining value in its interior,

(iv) for each $i$, all points inside a $J_i$ have the same number $n(i)$ of preimages,

(v) $f^{-1}(J_i^\circ)$ has $n(i)$ connected components $[u_i^k, v_i^k]$, $1 \le k \le n(i)$,
where $E(I, f)$ is the set of all ergodic measures invariant for $f$ and $\nu$ is a Borel measure on $E(I, f)$ satisfying

$$ \lambda = \int_{E(I,f)} \mu \, d\nu(\mu). $$

Since $\lambda(\text{Per}(f)) = 1$ necessarily $\nu$ a.e. measure ergodic $\mu$ is a CO-measure (i.e., a measure on a periodic orbit). In [4, Theorem 2] we constructed a dense set of leo maps for which the periodic points have full measure, and thus they all have zero entropy. This yields an alternative proof of the density of $Q$ from the proof of Theorem 1.

**Remark 8.** Parry sketched a construction of maps $f \in C_\lambda$ which are piecewise monotone with 2 full branches which are invertible $\lambda$-a.e. [21]; a more detailed version of this construction (with $k$ full branches) was given by Friedland and Weiss [15].

## 4. PROOF OF THEOREM 2

Let $C(I)$ denote the set of continuous maps $f: I \to I$. A function $f \in C(I)$ is said to be *non-decreasing at a point* $x \in I$ if there exists a $\delta > 0$ such that $f(t) \le f(x)$ when $t \in I \cap (x - \delta, x)$ and $f(t) \ge f(x)$ when $t \in I \cap (x, x - \delta)$. The function $f$ is called *nonincreasing at* $x$ if $-f$ is nondecreasing at $x$, and $f$ is called *monotone at* $x$ if it is either nondecreasing or nonincreasing at $x$. If $f \in C(I)$ and $\gamma \in \mathbb{R}$ then we define the map $f_{-\gamma}$ by:

$$ (5) \qquad f_{-\gamma}(x) := f(x) - \gamma x \quad \text{for } x \in I. $$

A map $f$ is *monotonic type at* $x$ if there exists a real number $\gamma$ such that the function $f_{-\gamma}$ is monotone at $x$. If $f$ is not of monotonic type at any point of $I$, it will be said to be of *nonmonotonic type*. It is easy to see that $f$ is of nonmonotonic type if and only if for each $x$, $\overline{D}f(x) = +\infty$ and $\underline{D}f(x) = -\infty$, where $\overline{D}f$, $\underline{D}f$ are the upper and lower derivatives.

The proof of Theorem 2 requires two auxiliary lemmas. The first lemma is a minor modification of Lemma 11 from [5].

**Lemma 9.** Let $f$ be a piecewise affine continuous interval map with nonzero slopes and such that its derivative exists everywhere but on a finite set $E$. Then $f \in C_\lambda$ if and only if

$$ (6) \qquad \forall y \in [0, 1) \setminus F(E): \sum_{x \in F^{-1}(y)} \frac{1}{|F'(x)|} = 1. $$

A *critical value* of a map $f$ is the value of $f$ at a local maximum or a local minimum. A *determining value* of a piecewise affine map $f$ is either one of its critical values or its value at a point in which the derivative of $f$ does not exist.
(vi) $f([u_i^k, v_i^k]) = J_i$ for each $k$,

(vii) every window perturbations of $f$ on $[u_i^k, v_i^k]$ remains in $B(f, \tau) \subset B(f, \varepsilon/3)$; furthermore this containment holds if we make a finite number of such window perturbations (see section 2.1 of [7] for the definition of window perturbations).

## II. A perturbation $\tilde{f}$ of $f$. We choose $\delta > 0$ so small that

$$ (6) \qquad \delta < \min_{i,k} \left( \frac{\tau}{10n}, \frac{v_i^k - u_i^k}{2} \right). $$

If $x$ a common point of two adjacent $[u_i^k, v_i^k]$ intervals then we call the intervals $[x - \delta, x]$ and $[x, x + \delta]$ $\delta$-marked intervals. Each $\delta$-marked interval is contained in a single interval $[u_i^k, v_i^k]$ and its image is contained in a single $J_i$ and $\delta$-marked intervals have pairwise disjoint interiors. We consider a (not necessarily regular) piecewise affine window perturbation $\tilde{f} \in B(f, \tau)$ of $f$ on all $[u_i^k, v_i^k]$'s such that (see Figure 2, left)

(viii) if $K \subset [u_i^k, v_i^k]$ and $\lambda(K) = \delta$ then $K$ contains either at least three critical points of $\tilde{f}$ or two critical points of $\tilde{f}$ and one of the endpoints $u_i^k, v_i^k$,

(ix) if $[u_i^1, v_i^1]$ is the leftmost interval and $f$ is increasing on $[u_i^1, v_i^1]$ then $\tilde{f}(u_i^1) = f(v_i^1)$, $\tilde{f}(v_i^1) = f(v_i^1)$, the map $\tilde{f}$ has two laps on the $\delta$-marked interval $[v_i^1 - \delta, v_i^1]$,

(x) if $[u_i^{n(i)}, v_i^{n(i)}]$ is the rightmost interval and $f$ is increasing on $[u_i^{n(i)}, v_i^{n(i)}]$ then $\tilde{f}(u_i^{n(i)}) = f(u_i^{n(i)})$, $\tilde{f}(v_i^{n(i)}) = f(u_i^{n(i)})$, the map $\tilde{f}$ has two laps on the $\delta$-marked interval $[u_i^{n(i)}, u_i^{n(i)} + \delta]$,

(xi) otherwise $\tilde{f}(u_i^k) = f(u_i^k)$ and $\tilde{f}(v_i^k) = f(v_i^k)$ for each $i, k$, the map $\tilde{f}$ has two laps on the $\delta$-marked intervals $[u_i^k, u_i^k + \delta]$ and $[v_i^k - \delta, v_i^k]$.

By construction, the map $\tilde{f} \in B(f, \tau) \subset B(f, \varepsilon/3)$ fulfills (5), so $\tilde{f} \in C_\lambda$.

## III. Constructing $F$ from $\tilde{f}$. Finally we construct the map $F$, adapting the map $\tilde{f}$, such that (see Figure 2, right)

(xii) the graphs of $\tilde{f}$ and $F$ coincide outside of the $\delta$-marked intervals,

(xiii) if $J = [d, e]$ is a $\delta$-marked interval, then $\tilde{f}(d) = \tilde{f}(e)$; the graph of $F|_J$ is a copy of the graph $\tilde{f}|_J$ reflected in the line $y = f(d)$, i.e., $F(x) = 2\tilde{f}(d) - \tilde{f}(x)$; equivalently if $J = [d, e]$ is the union of the two consecutive $\delta$-marked intervals, then $\tilde{f}(d) = \tilde{f}(e)$; the graph of $F|_J$ is a copy of the graph $\tilde{f}|_J$ reversed, i.e., $F(x) = \tilde{f}(d + e - x)$.

By construction, the map $F$ fulfills (5), so $F \in C_\lambda$. By (vii), $F \in B(\tilde{f}, 2\tau) \subset B(f, \varepsilon) \subset U$. Let us show that $F \notin A_n$. Notice that by (6) and (viii) the map
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Removing irrational critical values.</figcaption></figure>

**Lemma 10.** The set of piecewise affine maps with rational determining values is dense in $C_\lambda$.

*Proof.* Consider a dense collection of piecewise affine maps $\{f_m\} \subset C_\lambda$ (Figure 1 left), each $f_m$ has finitely many determining points. Fix $m$ and choose $n = n(m)$ so large that each of the intervals $[k/n, k+1/n]$ contains at most one determining value. For each irrational determining value $c_i$ of $f_m$ we consider the map $g_{n,i} \circ f_m$, where $g_{n,i}$ is the map of Figure 1 right. The resulting map has the determining value $c_i$ replaced by $k/n$ and $(k+1)/n$. We progressively remove all irrational determining values of $f_m$ to obtain a map $h_m$ with only rational determining values. The set $\{h_m\}$ is dense in $C_\lambda$ since $\rho(f_m, h_m) \to 0$. $\square$

The proof of Theorem 2 follows the strategy of the proof of [10, Theorem 2.2]. We use the terminology of the proof from [10]. In particular, for any $n \in \mathbb{N}$, the set $A_n$ denotes the set of functions $f \in C_\lambda$ for which there exist $\gamma \in [-n, n]$ and $x \in [0, 1]$ such that $f_{-\gamma}(t) \le f_{-\gamma}(x)$ when $t \in [0, 1] \cap (x - 1/n, x)$ and $f_{-\gamma}(t) \ge f_{-\gamma}(x)$ when $t \in [0, 1] \cap (x, x+1/n)$. Remember that $f_{-\gamma}$ was defined in (5), here we changed the notation of [10] replacing their $\lambda$ by $\gamma$ to avoid a collision of terminologies.

*Proof of Theorem 2.* Except for one step, the strategy of the proof is the same as in [10]; this difference is due to the fact of the preserving of Lebesgue measure. The step which differs is showing that each $A_n$ is nowhere dense, this step is the most technical step in the proof in [10]. To show this we proceed as illustrated in Figure 2. We know that piecewise affine maps are dense in $C_\lambda$ [3]. So from now $n \in \mathbb{N}$ defining $A_n$ is fixed.
<figure><img src="image_4.png" /><figcaption>FIGURE 2. Left: the map $\tilde{f}$ in step II. Right: the map $F$ in step III.</figcaption></figure>

$F$ has absolute value of the slope greater than $10n$ on each affine part of its graph.

If $x \in [0, 1]$ and $F$ is decreasing on some one-sided neighborhood $V$ of $x$ then $F_{-\gamma}$, $\gamma \in [-n, n]$, has its derivative less than $-9n$ on the same neighborhood $V$ of $x$, so either there is $t \in (x, x + \frac{1}{n})$ with $F_{-\gamma}(x) > F_{-\gamma}(t)$ or $t \in (x - \frac{1}{n}, x)$ for which $F_{-\gamma}(t) > F_{-\gamma}(x)$. This is the case when $x$ is an endpoint or a critical point of $F$ or such that $F$ is decreasing on some two-sided neighborhood of $x$.

It remains to consider the case when $x \in (0, 1)$ and $F$ is increasing on the two-sided neighborhood of $x$. Let $x \in [u_i^k, v_i^k]$. We distinguish several possibilities.

For convenience denote the endpoints of the interval $J_i$ by $a_i = (i - 1)\tau$ and $b_i = i\tau$.

The first case is when $F(x) \in J_i$ and $F(x) \le (a_i + b_i)/2$. In this case by (viii) there has to be a point $y \in [u_i^k, x)$ for which $x - y < 3\delta$ and $F(y) = b_i$. Then since

$$3\delta < \frac{3\tau}{10n} < \frac{3}{10n} < \frac{1}{n} \quad \text{and} \quad \frac{F(x) - F(y)}{x - y} < -\frac{\tau}{6\delta} < -\frac{5n}{3},$$

we obtain

$$\frac{F_{-\gamma}(x) - F_{-\gamma}(y)}{x - y} < -\frac{2n}{3} \text{ for } y \in (x - \frac{1}{n}, x).$$

The second case is $F(x) \in J_i$ and $F(x) > (a_i + b_i)/2$. Again by (viii) there has to be a point $y \in (x, v_i^k)$ for which $y - x < 3\delta$ and $F(y) = a_i$. Analogously as
I. A choice of f. Using Lemma 10, given an open set $U \subset C_\lambda$ and $\varepsilon > 0$ we can choose a piecewise affine map $f \in U$ whose determining values are rational such that $B(f, \varepsilon) \subset U$. Let $q$ be the greatest common denominator of the determining values. Let $1/\tau$ be a positive integer multiple of $q$ such that $\tau < \varepsilon/3$. For each integer $1 \le i \le 1/\tau$ consider the interval $J_i = [(i-1)\tau, i\tau]$. These intervals satisfy

(i) $\lambda(J_i) = \tau$ for all $i$,

(ii) $J_i^\circ \cap J_j^\circ = \emptyset$ for all $i \neq j$,

(iii) each $J_i$ has no determining value in its interior,

(iv) for each $i$, all points inside a $J_i$ have the same number $n(i)$ of preimages,

(v) $f^{-1}(J_i^\circ)$ has $n(i)$ connected components $[u_i^k, v_i^k]$, $1 \le k \le n(i)$,

(vi) $f([u_i^k, v_i^k]) = J_i$ for each $k$,

(vii) every window perturbations of $f$ on $[u_i^k, v_i^k]$ remains in $B(f, \tau) \subset B(f, \varepsilon/3)$; furthermore this containment holds if we make a finite number of such window perturbations (see section 2.1 of [7] for the definition of window perturbations).

II. A perturbation $\tilde{f}$ of $f$. We choose $\delta > 0$ so small that

$$ \delta < \min_{i,k} ((v_i^k - u_i^k)/2) $$

and satisfying a further assumption which will be made in the next step.

If $x$ a common point of two adjacent $[u_i^k, v_i^k]$ intervals then we call the intervals $[x-\delta, x]$ and $[x, x+\delta]$ $\delta$-marked intervals. Each $\delta$-marked interval is contained in a single interval $[u_i^k, v_i^k]$ and its image is contained in a single $J_i$ and $\delta$-marked intervals have pairwise disjoint interiors. We consider a (not necessarily regular) piecewise affine window perturbation $\tilde{f} \in B(f, \tau)$ of $f$ on all $[u_i^k, v_i^k]$'s such that (see Figure 2, left)

(viii) if an interval $K \subset [u_i^k, v_i^k]$ and $\lambda(K) = \delta$ then $K$ contains either at least three critical points of $\tilde{f}$ or two critical points of $\tilde{f}$ and one of the endpoints $u_i^k, v_i^k$,

(ix) if $[u_i^1, v_i^1]$ is the leftmost interval and $f$ is increasing on $[u_i^1, v_i^1]$ then $\tilde{f}(u_i^1) = f(v_i^1)$, $\tilde{f}(v_i^1) = f(v_i^1)$, the map $\tilde{f}$ has two laps on the $\delta$-marked interval $[v_i^1 - \delta, v_i^1]$,

(x) if $[u_i^{n(i)}, v_i^{n(i)}]$ is the rightmost interval and $f$ is increasing on $[u_i^{n(i)}, v_i^{n(i)}]$ then $\tilde{f}(u_i^{n(i)}) = f(u_i^{n(i)})$, $\tilde{f}(v_i^{n(i)}) = f(v_i^{n(i)})$, the map $\tilde{f}$ has two laps on the $\delta$-marked interval $[u_i^{n(i)}, u_i^{n(i)} + \delta]$,

(xi) otherwise $\tilde{f}(u_i^k) = f(u_i^k)$ and $\tilde{f}(v_i^k) = f(v_i^k)$ for each $i, k$, the map $\tilde{f}$ has two laps on the $\delta$-marked intervals $[u_i^k, u_i^k + \delta]$ and $[v_i^k - \delta, v_i^k]$.

By construction, the map $\tilde{f} \in B(f, \tau) \subset B(f, \varepsilon/3)$ fulfills (6), so $\tilde{f} \in C_\lambda$.
before

$$ \frac{F(x) - F(y)}{x - y} < -\frac{5n}{3}, $$

hence

$$ \frac{F_{-\gamma}(x) - F_{-\gamma}(y)}{x - y} < -\frac{2n}{3} \text{ for } y \in (x, x + \frac{1}{n}). $$

The last case is when $F(x) \in J_\ell$ with $\ell \neq i$. Then

- if $\ell = i+1$, one can proceed analogously as above taking $y$ the closest critical point to the right of $x$ where the local minimum is attained,

- if $\ell = i-1$, one can proceed analogously as above taking $y$ the closest critical point to the left of $x$ where the local maximum is attained.

This finishes the proof. □

Let $f \in C(I)$. For every $c \in \mathbb{R}$ the set $\{x : f(x) = c\}$ is called a level set of $f$. The level sets of a function $f \in C(I)$ are said to be normal if there is a countable set $E_f$ in $(\min f, \max f)$ such that the level set $\{x : f(x) = c\}$ of $f$ is

(i) a non-empty perfect set when $c \notin E_f \cup \{\min f, \max f\}$,

(ii) a single point when $c = \min f$ or $c = \max f$,

(iii) the union of a nonempty perfect set $P$ with an isolated point $x \notin P$ when $c \in E_f$.

By replacing Theorem 2.2. from [9] with Theorem 2 we can prove the following theorem using exactly the same arguments as in Theorem 3.3 in [9]. Note that if $f \in C_\lambda$ then $\min f = 0$ and $\max f = 1$.

**Proposition 14.** *The level sets of a generic map in $C_\lambda$ are normal.*

*Proof of Theorem 6.* Theorem 6 follows immediately from Theorem 1, Corollary 3 and Proposition 14. □

## 5. ACKNOWLEDGMENTS

We would like to thank Mark Pollicott for pointing out Parry's article [17] to us. The authors gratefully acknowledge the support of Slovenian research agency ARIS grant J1-4632 which enabled us to meet in Maribor. Our collaboration was also partially supported by the project AIMet4AI No. CZ.02.1.01/0.0/0.0/17-049/0008414.
<figure><img src="image_4.png" /><figcaption>FIGURE 2. Left: the map $\tilde{f}$ in step II. Right: the map $F$ in step III.</figcaption></figure>

### III. Constructing $F$ from $\tilde{f}$.
Finally we construct the map $F$, adapting the map $\tilde{f}$, such that (see Figure 2, right)

(xii) the graphs of $\tilde{f}$ and $F$ coincide outside of the $\delta$-marked intervals,

(xiii) if $J = [d, e]$ is a $\delta$-marked interval, then $\tilde{f}(d) = \tilde{f}(e)$; the graph of $F|_J$ is a copy of the graph $\tilde{f}|_J$ reflected in the line $y = f(d)$, i.e., $F(x) = 2\tilde{f}(d) - \tilde{f}(x)$; equivalently if $J = [d, e]$ is the union of the two consecutive $\delta$-marked intervals, then $\tilde{f}(d) = \tilde{f}(e)$; the graph of $F|_J$ is a copy of the graph $\tilde{f}|_J$ reversed, i.e., $F(x) = \tilde{f}(d+e-x)$.

By construction, the map $F$ fulfills (6), so $F \in C_\lambda$. By (vii), $F \in B(\tilde{f}, 2\tau) \subset B(f, \varepsilon) \subset U$. Let us show that $F \notin A_n$. Notice that by (7) and (viii) the map $F$ has absolute value of the slope greater than $10n$ on each affine part of its graph.

If $x \in [0, 1]$ and $F$ is decreasing on some one-sided neighborhood $V$ of $x$ then $F_{-\gamma}, \gamma \in [-n, n]$, has its derivative less than $-9n$ on the same neighborhood $V$ of $x$, so either there is $t \in (x, x + 1/n)$ with $F_{-\gamma}(x) > F_{-\gamma}(t)$ or $t \in (x - 1/n, x)$ for which $F_{-\gamma}(t) > F_{-\gamma}(x)$. This is the case when $x$ is an endpoint or a critical point of $F$ or such that $F$ is decreasing on some two-sided neighborhood of $x$.

It remains to consider the case when $x \in (0, 1)$ and $F$ is increasing on the two-sided neighborhood of $x$. Let $x \in [u_i^k, v_i^k]$. We distinguish several possibilities.

For convenience denote the endpoints of the interval $J_i$ by $a_i = (i - 1)\tau$ and $b_i = i\tau$.
<figure><img src="image_5.png" /></figure>

This research is part of J. Činč's project that has received funding from the European Union's Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. HE-MSCA-PF-PFSAIL-101063512.

## REFERENCES

[1] F. Béguin, S. Crovisier, F. Le Roux, *Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique*, Annales Scientifiques de l'École Normale Supérieure, **40**, (2007) 251-308.

[2] J. Bobok, *On non-differentiable measure-preserving functions*, Real Analysis Exchange, **16** (1991) 119-129.

[3] J. Bobok, S. Troubetzkoy, *Typical properties of interval maps preserving the Lebesgue measure* Nonlinearity **33** (2020), 6461-6501.

[4] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Periodic points and shadowing property of Lebesgue measure-preserving interval maps*, Nonlinearity **35** (2022) 2534-2557.

[5] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, s-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps. Ergodic Theory Dynam. Systems **43** (2023), no. 1, 78-98.

[6] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Interval maps with dense periodicity*, arXiv:2402.05638, February 2024.

[7] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey*, to appear in Topology Appl.

[8] L. Bowen, *Zero entropy is generic Entropy* **18** (2016) 20 pp.

[9] A.M. Bruckner, K.M. Garg, *The level structure of a residual set of continuous functions*, Trans. Amer. Math. Soc. **232**(1977), 307-321.

[10] S.L. Carvalho, A. Condori, *On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces*, **296**(2023) 980-995.

[11] J. Činč, P. Oprocha, *Parametrized family of pseudo-arc attractors: physical measures and prime end rotations*, Proc. Lond. Math. Soc. **(3)**125 (2022) 318-357.

[12] M. Foreman, B. Weiss, *An anti-classification theorem for ergodic measure preserving transformations*, J. Eur. Math. Soc. **6** (2004) 277-292.

[13] S. Friedland, B. Weiss, *Generalized interval exchanges and the 2-3 conjecture*, Central European J. Math. **3** 2005 412-429.

[14] P.-A. Guihéneuf, *Propriétés dynamiques génériques des homéomorphismes conservatifs*. Ensaios Mat., **22** Sociedade Brasileira de Matemática, Rio de Janeiro, 2012. 115 pp.

[15] A. Katok, A. Stepin, *Approximations of ergodic dynamic systems by periodic transformations*, Dokl. Akad. Nauk SSSR **171** (1966) 1268-1271.

[16] A. Katok, A. Stepin, *Metric properties of measure preserving homeomorphisms*, Russ. Math. Surveys **25** (1970), 191-220.

[17] W. Parry, *In general a degree two map is an automorphism*, Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135) Ed. P. Walters. American Mathematical Society, Providence, RI, 1992, pp. 335-338

[18] V.A. Rokhlin, *Entropy of metric automorphism*, Dokl. Akad. Nauk SSSR, **124** (1959) 980-983.

[19] J. Schmeling, R. Winkler, *Typical dimension of the graph of certain functions*, Monatsh. Math. **119** (1995), 303-320.

[20] K. Sigmund, *On the prevalence of zero entropy*, Israel J. Math. **10** (1971) 281-288.

[21] P. Walters, *An introduction to ergodic theory*, Graduate Texts in Mathematics **79**, Springer Verlag, 1982, ix+250 pp.
We additionally assume that

$$ \delta < \frac{\tau}{10n}. $$

The first case is when $F(x) \in J_i$ and $F(x) \le (a_i + b_i)/2$. In this case by (viii) there has to be a point $y \in [u_i^k, x)$ for which $x - y < 3\delta$ and $F(y) = b_i$. Then since

$$ 3\delta < \frac{3\tau}{10n} < \frac{3}{10n} < \frac{1}{n} \quad \text{and} \quad \frac{F(x) - F(y)}{x - y} < -\frac{\tau}{6\delta} < -\frac{5n}{3}, $$

we obtain

$$ \frac{F_{-\gamma}(x) - F_{-\gamma}(y)}{x - y} < -\frac{2n}{3} \quad \text{for } y \in (x - \frac{1}{n}, x). $$

The second case is $F(x) \in J_i$ and $F(x) > (a_i + b_i)/2$. Again by (viii) there has to be a point $y \in (x, v_i^k)$ for which $y - x < 3\delta$ and $F(y) = a_i$. Analogously as before

$$ \frac{F(x) - F(y)}{x - y} < -\frac{5n}{3}, $$

hence

$$ \frac{F_{-\gamma}(x) - F_{-\gamma}(y)}{x - y} < -\frac{2n}{3} \quad \text{for } y \in (x, x + \frac{1}{n}). $$

The last case is when $F(x) \in J_\ell$ with $\ell \ne i$. Then

- if $\ell = i+1$, one can proceed analogously as above taking $y$ the closest critical point to the right of $x$ where the local minimum is attained,

- if $\ell = i-1$, one can proceed analogously as above taking $y$ the closest critical point to the left of $x$ where the local maximum is attained.

This finishes the proof of (i). The assertion (ii) on level sets follow using the same arguments as in Theorem 3.3 in [10], but replacing their Theorem 2.2 by (i). Note that if $f \in C_\lambda$ then $\min f = 0$ and $\max f = 1$. $\square$

### 5. CONSEQUENCES FOR A PARAMETRIZED FAMILY OF PLANAR ATTRACTORS

In this section, we will apply Theorem 1 and Corollary 3 providing a better understanding of some planar attractors from [12]. We find these results surprising, because it highlights the dichotomy between topological and measure-theoretical perception of these attractors. Before we proceed we need some additional terminology.

Let $X$ be a compact metric space and let $F: X \to X$ be continuous. Define

$$ \hat{X}_F := \lim_{\leftarrow\rightarrow} (X, F) = \{\hat{x} := (x_0, x_1, x_2, \dots) \in \Pi_{i=1}^{\infty} X : x_i = F(x_{i+1}), \forall i \ge 0\}, $$
(J. Bobok) DEPARTMENT OF MATHEMATICS OF FCE, CZECH TECHNICAL UNIVERSITY IN PRAGUE, THÁKUROVA 7, 166 29 PRAGUE 6, CZECH REPUBLIC

Email address: jozef.bobok@cvut.cz

(J. Činč) UNIVERSITY OF MARIBOR, KOROŠKA 160, 2000 MARIBOR, SLOVENIA – & –
– AND – ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP),
TRIESTE, ITALY

Email address: jernej.cinc@um.si

(P. Oprocha) AGH UNIVERSITY OF KRAKOW, FACULTY OF APPLIED MATHEMATICS, AL. MICKIEWICZA 30, 30-059 KRAKÓW, POLAND. – & – CENTRE OF EXCELLENCE IT4INNOVATIONS - INSTITUTE FOR RESEARCH AND APPLICATIONS OF FUZZY MODELING, UNIVERSITY OF OSTRAVA, 30. DUBNA 22, 701 03 OSTRAVA 1, CZECH REPUBLIC

Email address: oprocha@agh.edu.pl

(S. Troubetzkoy) AIX MARSEILLE UNIV, CNRS, I2M, MARSEILLE, FRANCE

Email address: serge.troubetzkoy@univ-amu.fr
where $F$ is called the bonding map. We equip the inverse limit $\hat{X}_F$ with the subspace metric which is induced by product metric in $\Pi_{i=1}^{\infty} X_i$. We can define the shift homeomorphism $\hat{F} : \hat{X}_F \to \hat{X}_F$ defined for $(x_0, x_1, x_2, \dots) \in \hat{X}_F$ by

$$ \hat{F}((x_0, x_1, x_2, \dots)) = (F(x_0), F(x_1), F(x_2), \dots) = (F(x_0), x_0, x_1, \dots). $$

Denote by $\pi_n : \hat{X}_F \to X$ the coordinate projection maps. Recall also that $\mathcal{B}(X)$ denotes the $\sigma$-algebra of Borel sets in $X$.

For the details of what follows we refer the reader to [12]. Let $\mathcal{L}$ be the Lebesgue measure on a topological disk $\mathcal{D}$ and $\lambda$ the Lebesgue measure on $I = [0, 1]$. Suppose that $I \times \{0\} \subset \text{int}(\mathcal{D})$ and let $f : I \to I$ be a continuous map. The map $f$ can be extended to a map $F$ defined as a uniform limit of homeomorphisms of $\mathcal{D}$. Let us denote $\hat{\mathcal{D}} := \lim_{\leftarrow\leftarrow\rightarrow} (\mathcal{D}, F)$; by Brown's theorem $\hat{\mathcal{D}}$ is again a topological disk. Let $\mathcal{B}(\hat{\mathcal{D}})$ be the smallest $\sigma$-algebra on $\hat{\mathcal{D}}$ such that all the projection maps $\pi_i$ are measurable. Then there exists a unique probability measure $\hat{\mathcal{L}}$ on $\mathcal{B}(\hat{\mathcal{D}})$ such that $\hat{\mathcal{L}}(\pi_n^{-1}(A)) = \mathcal{L}(A)$ for all $A \in \mathcal{B}(\mathcal{D})$ and each $n \in \mathbb{N}_0$ where (the measure $\hat{\lambda}$ is defined in an analogous way).

Let us recall a result from [22]. Let $\mu$ be an $F$-invariant Borel probability measure on $\mathcal{D}$. The set $B_\mu$ which consists of all points $x \in \mathcal{D}$ such that

$$ \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} g(F^i(x)) = \int g d\mu $$

holds for all continuous maps $g : \mathcal{D} \to \mathbb{R}$ is called the *basin of $\mu$ for $F$*. It is well known that $B_\mu$ is a Borel set (see for example [13, Prop. 4.7]). We call the measure $\mu$ *physical for $F$* if $\mathcal{L}(B_\mu) > 0$. An invariant measure $\hat{\nu}$ for the natural extension $\hat{F} : \hat{\mathcal{D}} \to \hat{\mathcal{D}}$ is called an *inverse limit physical measure* if $\hat{\nu}$ has a basin $\hat{B}_{\hat{\nu}}$ so that $\mathcal{L}(\pi_0(\hat{B}_{\hat{\nu}})) > 0$.

By results from [19] if $\mathcal{L}$ is a physical measure for $F$ then the induced measure $\hat{\mathcal{L}}$ on $\hat{\mathcal{D}}$ is an inverse limit physical measure for the natural extension $\hat{F}$. In particular, there is a basin $\hat{B}_{\hat{\mathcal{L}}} := \pi_0^{-1}(B)$ of $\hat{\mathcal{L}}$ for $\hat{F}$ with $\mathcal{L}(B) > 0$, In particular, this statement holds for $\lambda$, $I$ and $f$.

The pseudo-arc is a very curious object arising from Continuum Theory (see the survey of Lewis [20] and the introduction of [8] for the overview of results involving the pseudo-arc). Its complicated structure is reflected by the fact that it is *hereditarily indecomposable*, i.e., there are no proper subcontinua $A, B \subset H$ such that $A \cup B = H$ for every subcontinuum $H$ of the pseudo-arc $P$. In this sense its structure is in complete opposition to intuitive perception of an arc. Let us recall two recent results relating the maps in $C_\lambda$ with pseudo-arc and planar attractors.
**Theorem.** [12, Theorem 1.1] *The inverse limit with any $C_\lambda(I)$-generic map as the bonding map is the pseudo-arc.*

**Theorem.** [12, Theorem 1.6] *There exists a dense $G_\delta$ subset $G$ of $C_\lambda(I)$ and a parametrized family of homeomorphisms $\{\Phi_f\}_{f \in G} \subset \mathcal{H}(D, \mathcal{D})$ varying continuously with $f$, having $\Phi_f$-invariant pseudo-arc attractors $\Lambda_f \subset D$ for every $f \in G$ so that*

(1) $\Phi_f|_{\Lambda_f}$ is topologically conjugate to $\hat{f}: \hat{I}_f \to \hat{I}_f$.

(2) The attractors $\{\Lambda_f\}_{f \in G}$ vary continuously in the Hausdorff topology.

(3) For each $f \in G$ the measure $\mu_f := \hat{\lambda}$ induced by $(\lambda, f)$ is a weakly mixing, $\Phi_f$-invariant, inverse limit physical measure supported on the attractor $\Lambda_f$.<fnref n="2" />

(4) The measures $\mu_f$ vary continuously in the weak* topology.

We can now interpret our results in this setting. By Corollary 3 there is a set $X_f$ of full measure $\lambda$ such that $f|_{X_f}$ is invertible. Let $Z_f := \lim_{n \to \infty} (X_f, f) = \bigcap_{n=1}^\infty \pi_n^{-1}(X_f)$. Clearly $\hat{\lambda}(Z_f) = 1$ and $\pi_0|_{Z_f}$ is one-to-one onto $X_f$. Therefore, $\mu_f$ is isomorphic to $\lambda$ and from the ergodic theory point of view, dynamics of $(\Lambda_f, \Phi_f, \mu_f)$ and $(I, f, \lambda)$ are the same. Therefore, on one hand we observe very complicated topological structure of the attractor in the disc (pseudo-arc) and complicated topological dynamics on it (infinite topological entropy and topological mixing) but, on the other hand, from the physical perspective we see simple dynamics with zero measure-theoretic entropy (yet weakly mixing) governed by $\lambda$ which is inherited from the interval.

## 6. ACKNOWLEDGMENTS

We would like to thank Mark Pollicott for pointing out Parry’s article [21] to us. The authors gratefully acknowledge the support of Slovenian research agency ARIS grant J1-4632. Our collaboration was also partially supported by the project AIMet4AI No. CZ.02.1.01/0.0/0.0/17_049/0008414 and by the project from OP JAK, No. CZ.02.01.01/00/23_021/0008759, which is co-financed by the European Union.

<figure><img src="image_5.png" /></figure>

Co-funded by
the European Union

This research is part of J. Činč’s project that has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. HE-MSCA-PF-PFSAIL-101063512.

<footnote n="2">²Here we use the fact that $\mathcal{D}$ is homeomorphic to some $\hat{\mathcal{D}}_{F_f}$ where maps $F_f$ vary continuously with $f$.</footnote>
REFERENCES

[1] F. Béguin, S. Crovisier, F. Le Roux, *Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique*, Annales Scientifiques de l'École Normale Supérieure, **40**, (2007) 251–308.

[2] J. Bobok, *On non-differentiable measure-preserving functions*, Real Analysis Exchange, **16** (1991) 119–129.

[3] J. Bobok, S. Troubetzkoy, *Typical properties of interval maps preserving the Lebesgue measure*, Nonlinearity **33** (2020), 6461–6501.

[4] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Periodic points and shadowing property of Lebesgue measure-preserving interval maps*, Nonlinearity **35** (2022) 2534–2557.

[5] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, s-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps. Ergodic Theory Dynam. Systems **43** (2023), no. 1, 78–98.

[6] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Interval maps with dense periodicity*, arXiv:2402.05638, February 2024.

[7] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, *Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey*, Topology Appl. (2024), https://doi.org/10.1016/j.topol.2024.109101.

[8] J. Boroński, J. Činč, P. Oprocha, *Beyond 0 and $\infty$: The solution to the Barge entropy conjecture*, arXiv:2105.11133, May 2021.

[9] L. Bowen, *Zero entropy is generic Entropy* **18** (2016) 20 pp.

[10] A.M. Bruckner, K.M. Garg, *The level structure of a residual set of continuous functions*, Trans. Amer. Math. Soc. **232**(1977), 307–321.

[11] S.L. Carvalho, A. Condori, *On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces*, Mathematische Nachrichten **296**(2023) 980–995.

[12] J. Činč, P. Oprocha, *Parametrized family of pseudo-arc attractors: physical measures and prime end rotations*, Proc. Lond. Math. Soc. **(3)**125 (2022) 318–357.

[13] M. Denker, C. Grillenberger, K. Sigmund, Ergodic theory on compact spaces. Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976. iv+360 pp.

[14] M. Foreman, B. Weiss, *An anti-classification theorem for ergodic measure preserving transformations*, J. Eur. Math. Soc. **6** (2004) 277–292.

[15] S. Friedland, B. Weiss, *Generalized interval exchanges and the 2-3 conjecture*, Central European J. Math. **3** 2005 412–429.

[16] P.-A. Guihéneuf, *Propriétés dynamiques génériques des homéomorphismes conservatifs*. Ensaios Mat., **22** Sociedade Brasileira de Matemática, Rio de Janeiro, 2012. 115 pp.

[17] A. Katok, A. Stepin, *Approximations of ergodic dynamic systems by periodic transformations*, Dolk. Akad. Nauk SSSR **171** (1966) 1268–1271.

[18] A. Katok, A. Stepin, *Metric properties of measure preserving homeomorphisms*, Russ. Math. Surveys **25** (1970), 191–220.

[19] J. Kennedy, B. E. Raines, D. R. Stockman, *Basins of measures on inverse limit spaces for the induced homeomorphisms*, Ergod. Th. & Dynam. Sys. **30** (2010), 1119–1130.

[20] W. Lewis, *The pseudo-arc*. Continuum theory and dynamical systems (Arcata, CA, 1989), 103–123, Contemp. Math., **117**, Amer. Math. Soc., Providence, RI, 1991.

[21] W. Parry, *In general a degree two map is an automorphism*, Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135) Ed. P. Walters. American Mathematical Society, Providence, RI, 1992, pp. 335–338.

[22] K. R. Parthasarathy, *Probability Measures on Metric Spaces*. American Mathematical Society, Providence, RI, 2005. Reprint of the 1967 original.
[23] F. Przytycki, M. Urbański, *Conformal Fractals - Ergodic Theory Methods*, London Mathematical Society Lecture Note Series vol. 371, Cambridge University Press, 2010.

[24] V.A. Rokhlin, *Entropy of metric automorphism*, Dokl. Akad. Nauk SSSR, 124 (1959) 980–983.

[25] J. Schmeling, R. Winkler, *Typical dimension of the graph of certain functions*, Monatsh. Math. **119** (1995), 303–320.

[26] K. Sigmund, *On the prevalence of zero entropy*, Israel J. Math. 10 (1971) 281–288.

[27] P. Walters, *An introduction to ergodic theory*, Graduate Texts in Mathematics **79**, Springer Verlag, 1982, ix+250 pp.

(J. Bobok) DEPARTMENT OF MATHEMATICS OF FCE, CZECH TECHNICAL UNIVERSITY IN PRAGUE, THÁKUROVA 7, 166 29 PRAGUE 6, CZECH REPUBLIC

Email address: jozef.bobok@cvut.cz

(J. Činč) UNIVERSITY OF MARIBOR, KOROŠKA 160, 2000 MARIBOR, SLOVENIA – & – ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP), TRIESTE, ITALY

Email address: jernej.cinc@um.si

(P. Oprocha) AGH UNIVERSITY OF KRAKOW, FACULTY OF APPLIED MATHEMATICS, AL. MICKIEWICZA 30, 30-059 KRAKÓW, POLAND. – & – CENTRE OF EXCELLENCE IT4INNOVATIONS - INSTITUTE FOR RESEARCH AND APPLICATIONS OF FUZZY MODELING, UNIVERSITY OF OSTRAVA, 30. DUBNA 22, 701 03 OSTRAVA 1, CZECH REPUBLIC

Email address: piotr.oprocha@osu.cz

(S. Troubetzkoy) AIX MARSEILLE UNIV, CNRS, I2M, MARSEILLE, FRANCE

Email address: serge.troubetzkoy@univ-amu.fr